M a t he mati c a I Physics Proceedings of the 12th Regional Conference
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Ma t he mati c a I Phys ic s Proceedings of the 12th Regional Conference Islamabad, Pakistan
27 March - 1 April 2006
M. Jamil Aslam COMSATS Institute of Information Technology, Pakistan National Centre for Physics, Pakistan
Faheem Hussain COMSATS Institute of Information Technology, Pakistan National Centre for Physics, Pakistan
Asghar Qadir National University of Science & Technology, Pakistan
Riazuddin National Centre for Physics, Pakistan
Hamid Saleem Pakistan Institute of Nuclear Science & Technology, Pakistan
N E W JERSEY
*
LONDON
World Scientific K SINGAPORE
*
BElJlNG
*
SHANGHAI
*
HONG KONG
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TAIPEI
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CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd.
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MATHEMATICAL PHYSICS Proceedings of the 12th Regional Conference Copyright 02007 by World Scientific Publishing Co. Pte. Ltd
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ISBN-I 3 978-98 1-270-591-4 ISBN-10 98 1-270-591-0
Printed in Singapore by World Scientific Printers (S)Pte Ltd
INTERNATIONAL ADVISORY COMMITTEE
Farhad Ardalan, IPM, Tehran Hessameddin Arfaei, IPM, Tehran Alexander Belavin, ITP, Moscow Ugur Camci, Cannakale Ali Chamseddine, CAMS, Beirut John Ellis, CERN, Geneva George Jorjadze, RMI, Tbilisi Elias Kiritsis, Polytechnique, Paris Werner Nahm, IAS, Dublin Neda Sadooghi, IPM, Tehran Spenta Wadia, TIFR, Mumbai
LOCAL ORGANIZING COMMITTEE Kamaluddin Ahmed, CIIT, Islamabad M. Jamil Aslam, NCP, Islamabad Fazal-e- Aleem, CHEP, Lahore Faheem Hussain, CIIT/NCP, Islamabad Jameel-un-Nabi, GIKI, Topi Ghulam Murtaza, GCU, Lahore Asghar Qadir, CAMP-NUST, Rawalpindi Riazuddin, NCP, Islamabad Hamid Saleem, PINSTECH Muhammad Sharif, PU, Lahore Azad A. Siddiqui, CAMP-NUST, Rawalpindi Kashif Sabih, QAU, Islamabad
V
LIST OF SPEAKERS
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
B. Acharya* M. Ahmed* Z. Ahmed' M. Alishahiha F. Ardalan M. J . Aslam A. P. Balachandran U. Camci N. K. Dadhich 0. F. Dayi A. Dhar M. Duff I. H. Duru* Z. Ehsan* A. Fayyazudduin D. Ghoshal G. Hall M. Haseeb* P. A. Hoodbhoy T. Z. Hussain V. Hussain Jameel Un Nabi T. Jayaraman* A. Kadeer T. Kaladze A. R. Kashif M. Koca J. G. Korner A. Kumar D. A. Leites S. M. Mahajan
32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.
A. M. Mirza G. Murtaza* A. Naqvi J . V. Narlikar Maqsood-ul-Hassan Nasim* S. Panda J . Pasupathy' A. Qadir M. A. Rashid Riazuddin K. Saifullah H. Saleem M. Salimullah* S. Sazhenkov A. Sen* G. Shabbir H. A. Shah* M. Sharif S. S. Jabbari A. A. Siddiqui F . Tahir* I. Tarhan* G. Thompson* Hooft, G. 't N. Tsintsadze' G. Unal N. ijnal M. Varadarajan S. Wadia Y . Yan* A. F . Zakharov
*Their lectures could not be included in the proceedings.
vi
PREFACE These are the proceedings of the 12th Regional Conference on Mathematical Physics organized by the National Centre for Physics (NCP) in Islamabad, Pakistan, from 27 March to 1 April, 2006. This series of conferences was initiated by Iranian, Pakistani and Turkish physicists after the meeting at the International Centre for Theoretical Physics (ICTP) in 1986. Strangely, the first conference organized by this group of friends was called the 2nd Regional Conference which was held in Adana, Turkey in September 1987. The “region” referred to originally comprised Iran, Turkey and Pakistan. Over the years the region has expanded to encompass the broad West Asian region including Armenia, Georgia and other Central Asian former Soviet republics. In the east, the region was expanded to include India and Bangladesh. Over the years the regional conference has covered a broad range of topics in Mathematical Physics. In the 12th Regional Conference the topics treated were Particle Physics and String Theory; Relativity, Astrophysics and Cosmology; Plasma Physics; Formal Aspects. There were 23 plenary talks of 45 minutes each which took place in the mornings, which were for all participants. The afternoons were devoted to 38 specialized halfhour talks spread over two parallel sessions: one for Superstring Theory, High Energy Physics and Formal Aspects; the other for Relativity, Astrophysics, Cosmology and Plasma Physics. There were 163 participants from 15 countries, of whom 60 were speakers and 40 were from abroad. It was particularly heartening to see the enthusiastic participation of many young Pakistani students. There was also one evening lecture on “Abdus Salam at Imperial College” by Michael Duff, Abdus Salam Chair of Theoretical Physics at the Imperial College of Science, Technology and Medicine, in London, England. Apart from Michael Duff, participants included leading physicists from around the world, in particular Gerard ’t Hooft, the winner of the Nobel Prize for Physics in 1999. The organizers were particularly pleased
vii
viii
about the enthusiastic participation of a large contingent of colleagues from India. Our arrangement of the material is simple. We present all papers on Formal Aspects in Part I, Particle Physics and String Theory in Part 11, Plasma Physics in Part I11 and Relativity, Astrophysics and Cosmology in Part IV. Articles in each section are arranged in alphabetic order (by author’s name). Plenary talks are indicated by an asterisk. In our opinion the conference was a great success. To make this possible, financial and institutional support was necessary. The NCP provided the primary institutional support. We are most grateful to Prof. Qasim Jan, Vice-Chancellor of Quaid-i-Azam University, for also providing institutional support including use of the Geophysics and Physics Auditorium. The conference was co-sponsored by (as it is now known) the Abdus Salam ICTP, Trieste, Italy, the Higher Education Commission (HEC), Islamabad and the National University of Sciences and Technology (NUST), Rawalpindi. We are particularly grateful for the generous financial support provided by the Abdus Salam ICTP and the HEC. We are also indebted to the Pakistan Atomic Energy Commission (PAEC) for financial support. We would also like to thank the members of the International Advisory Committee and the Local Organising Committee for valuable advice. It is worth mentioning that two other Proceedings, of the 3rd and the 11th conferences that were held in 1988 and 2004 respectively, were also published by World Scientific in 1989 and 2005. Without the help of the staff of the NCP the conference would not have been a success. Our special thanks go to Ubaidullah Khalid, Gulzaman Khan, Kashif Sarfraz Khan, Mansoor Sheikh, Ashfaq Ahmad and the staff of the computer section. Special thanks go also to all the students of the NCP (far too many t o name here) who voluntarily did all the tough legwork to make the running of the conference a great success. We would also like to thank Alimjan Kadeer who came from afar t o help with organizing the conference.
M. Jamil Aslam Faheem Hussain Asghar Qadir Riazuddin Hamid Saleem Islamabad, Pakistan November 2006
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TABLE OF CONTENTS Formal Aspects
1
Noncommutative Geometry: Fuzzy Spaces, the Groenwald-Moyal Plane A.P. Balachandran* and B.A. Qureshi
3
S-Dualities in Noncommutative and Non-anticommutative Field Theories O.F. Dayi, L. T. Kelleyane, K. Ulker and B. Yapigkan
12
New Casimir Energy Calculations H. Ahmedov and I.H. Durn
21
Quaternionic and Octonionic Structures of the Exceptional Lie Algebras M. Koca
25
On Critical Dimensions of String Theories D. Leites and C. Sachse
31
Transition Amplitudes for Time-dependent Linear Harmonic Oscillator with Linear Time-dependent Terms Added to the Hamiltonian M. A. Rashid
39
Entropy Solutions to a Genuinely Nonlinear Ultraparabolic Kolmogorov-type Equation S. A. Sazhenkov
47
Fokker-Planck-Kolmogorov Equation for fBm: Derivation and Analytical Solutions
G. Unar
53
*Plenary speaker
xi
xii
Particle Physics and String Theory
61
Holography and de Sitter Space M. Ahhahiha*
63
Brane Cosmology with String Antisymmetric Field F. Ardalan
73
Bosonization of a Finite Number of Non-relativistic Fermions and Application A. DhaV
79
Recent Applications of the Weyl Anomaly M. J. D u f
90
Higher Dimensional Perspective on N = 2 Black Holes A. Fayyazuddin*
102
pstrings vs Strings D. Ghoshal
109
Detecting Two-photon Exchange Effects in Hard Scattering from Nucleon Targets P. Hood bhoy*
117
BPS M-brane Geometries T.Z. Husain
130
Ward Identities and Radiative Rare Leptonic B-decays M.J. Aslam, A.H.S. Gilani, M.S. Khan and Riazuddin
136
Calculation of NLO QCD Corrections to Polarization Effects in Top Quark Decays A . Kadeer
147
Selected Topics in Top Quark Physics J . G. Korner*
155
Moduli Stabilization using Open String Fluxes A. Kumar
169
Topological Strings and Special Holonomy Manifolds J . de Boer, A . Naqvi and A. Shomer
177
xiii
Vacua in N = 4 Gauged Supersymmetry S. Panda
190
Neutrino Physics Riazuddin*
199
What String Theory has Taught Us about the Quantum Structure of Space-time M. M. Sheikh- Jabbari*
210
Gauge Theory Description of the Fate of the Small Schwarzschild Blackhole S.R. Wadia*
220
Plasma Physics
235
Zonal Flow Generation by Magnetized Rossby Waves in Ionospheric E-layer T.D. Kaladze, D. J. Wu, O.A. Pokhotelov, R.Z. Sagdeev, L. StenfEo and P.K. Shukla
237
Quiescent and Catastrophic Events in Stellar Atmospheres S.M. Mahajan* and N.L. Shatashvili
252
Zero-dimensional MHD Modelling of Two Gas-puff Staged Pinch Plasma with Finite-P Effect A.M. Mirza, F. Deeba, K. Ahmed and M.Q. Haseeb
283
Does Quasi-neutrality Remain Valid in Pair-ion Plasmas? H. Saleem*
290
Relativity, Astrophysics and Cosmology
299
Invariance under Complex Transformations, and its Relevance to the Cosmological Constant Problem G. 't H o o p and S. Nobbenhuis
301
Ricci Collineations in Bianchi I1 Spacetime U. Camci
320
On the Gauss-Bonnet Gravity N. DadhicW
331
xiv
Geometry and Symmetry in General Relativity G. HalP
341
Gravitational Collapse in Quantum Gravity V. Husain*
354
Can 55C0 Give us the Desired Prompt Explosion of Massive Stars? Jamil-un-Nabi
362
Symmetry Classification and Invariant Characterization of Two-dimensional Geodesic Equations A . R . KashiA F.M. Mahomed and A . Qadar
369
Gravitational Collapse with Negative Energy Fields J. V. Narlika?
375
Quantum Non-locality, Black Holes and Quantum Gravity A . Qadi?
382
Homothetic and Conformal Motions K . Saifullah
393
Proper Projective Symrnetrics in Space-times G. Shabbir
400
Matter Symmetries of Non-static Plane Symmetric Spacetimes M. Sharif and T. Ismaeel
407
Spacetime Foliation A .A . Saddiqui
416
Spinning Particle: Electromagnetic and Gravitational Interactions N . Unal
424
Quantum Gravity and Hawking Radiation M. Varadarajan
430
Measuring Parameters of Supermassive Black Holes with Space Missions A .F. Zakharov'
436
Participants
445
Formal Aspects
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NONCOMMUTATIVE GEOMETRY: FUZZY SPACES, THE GROENWALD-MOYAL PLANE* A. P. BALACHANDRAN and B. A. QURESHI Department of Physics, Syracuse University Syracuse, NY 13210, USA balQphy.syr. edu In this talk, we review the basics concepts of fuzzy physics and quantum field theory on the Groenwald-Moyal Plane as examples of noncommutative spaces in physics. We introduce the basic ideas, and discuss some important results in these fields. In the end we outline some recent developments in the field. Keywords: Noncommutative Geometry; Quantum Algebra; Quantum Field Theory. SU-4252-832
1. Introduction Noncommutative Geometry is a branch of mathematics due to Connes, Gel’fand, Naimark, Rieffel and many others’. Physicists in a very short time vulgarized it and nowadays use this phrase whenever the spacetime algebra is noncommutative. There are two such particularly active fields in physics at present. (i) Fuzzy Physics, (ii) Quantum Field Theory (QFT) on the Groenwald-Moyal Plane. Item (i) is evolving into a tool to regulate QFT’s, and for numerical work. It is an alternative to lattice methods. Item (ii) is more a probe of Planck-scale Physics. This introductory talk will discuss both items (i) and (ii).
*Plenary talk given by A. P. B. at the 12th Regional Conference on Mathematical Physics, Islamabad (27 March-1 April, 2006).
3
4
2. History
The Groenwald-Moyal plane is an example where spacetime coordinates do not commute. The idea that spatial coordinates may not commute first occurs in a letter from Heisenberg to Peierls2. Heisenberg suggested that an uncertainty principle such as
can provide a short distance cut-off and regulate quantum field theories (qft’s). In this letter, he apparantly complains about his lack of mathematical skills to study this possibility. Peierls communicated this idea to Pauli, Pauli to Oppenheimer and finally Oppenheimer to Snyder. Snyder wrote the first paper on the subject3. This was followed by a paper of Yang4. In the mid-go’s, Doplicher, F’redenhagen and Roberts5 systematically constructed unitary quantum field theories on the Groenwald-Moyal Plane, even with time-space noncommutativity. Later string physics encountered these structures. 3. What is Noncommutative Geometry
According to Connes’, noncommutative geometry is a spectral triple,
where
A = a C*-algebra, possibly noncommutative, D = a Dirac operator, 7-1 = a Hilbert space on which they are represented. If A is a commutative C*-algebra, we can recover a Hausdorff topological space on which A are functions, using theorems of Gel’fand and Naimark. But that is not possible if A is not commutative. But it is still possible to formulate qft’s using the spectral triple. A class of examples of noncommutative geometry with A noncommutative is due to Connes and Landi‘. If some of the strict axioms are not enforced then the examples include SU(2),, fuzzy spaces, Groenwald-Moyal plane, and many more. The introduction of noncommutative geometry has produced a conceptual revolution. Manifolds are being replaced by their “duals”, algebras, and these duals are being ”quantized”, much as in quantum mechanics.
5
4. Fuzzy Physics In what follows, we sketch the contents of ”fuzzy physics”. Reference [7] contains a detailed survey.
4.1. W h at is f u zzy p h y s i ~ s ? ~ We explain the basic idea of fuzzy physics by a two-dimensional example: S; . Consider the two-sphere S2.We quantize it to regularize by introducing a short distance cut-off. For example in classical mechanics, the number of states in a phase space volume
AV
d3pd3q
=
is infinite. But we know since Planck and Bose that on quantization, it becomes
AV h3
- = finite.
This is the idea behind fuzzy regularization. In detail, this regularization works as follows on S 2 . We have
s2= [&R3
:
2 . 2 = T”.
(2)
Now consider angular momentum Li: 4 2
[ L i , L j ] = iEijkLk; L
= 1(1+ 1).
(3)
Set
Li
xi
=
=+
(4)
TJqmJ
+
+
-
where Bi E Matzl+l 3 space of (21 1) x (21 1) matrices. As 1 00, they become commutative. They give the fuzzy sphere S$ of radius r and dimensions 21 1.
+
4.2. W h y is this space fuzzy
As 2i-i ,?j (i # j ) do not commute, we can not sharply localize &. Roughly in a volume 47rr2 there are (21 1) states.
+
6 4.3.
Field theory on fuzzy sphere
A scalar field on fuzzy sphere is defined as a polynomial in 2i. i.e., A scalar field @ = A polynomial in 2i = A (21
+ 1)
-
dimensional matrix.
Differentiation is given by infinitesimal rotation: &@
=
[ L i ,@I.
A simple rotational invariant scalar field action is given by S ( @ )= p T r [ L i , Q I t [ L i ,@]
+
m
j-Tr(Qt@)
+ XTr(@+@)'.
(6)
Simulations have been performeds on the partition function 2 = d@ePs(@) of this model and the major findings include the following: Continum limit exists. 0
If clrn fim,
@ =
fim= spherical tensor,
then there are three phases:
(c
(a) Disordered : lclmI2) = 0. (b) Uniform ordered: ( \cool2) # 0 , ( Iclrnl') = 0 for 1 # 0. (c) Non-uniform ordered: ( IclmI2) # 0, ( 1 ~ ; ~=) 0 for 1 # 1. The last one is the analogue of Gupser-Sondhi phaseg.
4.3.1. Dirac operator
S$ has Dirac operator with no fermion doubling including instantons7. Also 5'; can nicely describe topological features. Hence it seems better suited for preserving symmetries than lattice approximations.
4.3.2. Supersymmetry Replacing S U ( 2 ) by OSp(2,1), the fuzzy sphere becomes the N = 1 supersymmetric fuzzy sphere and can be used for simulating supersymmetry. Simulations in this regard are already starting.
4.3.3. Strings'' If N D-branes are close together, the transverse coordinates N x N matrices with the action given by
s
=
x Tr [@z , @ j ] tpi, @j]
+ i fijk @i@j@k
@i
become
(7)
7
where f i j k are totally antisymmetric. The equations of motion
give solutions when f i j k are structure constants of a simple compact Lie group. Thus we can have Qi = cLi
f i j k = CEijk
, c = constant
If Li is irreducible, then we have 4
4
L . L = Z(Z+l),
(2Z+1) = N ,
and we have one fuzzy sphere. Or we can have a direct sum
Then we have many fuzzy spheres. Stability analysis of these solutions including numerical studies has been done by many groups. 5. The Groenwald-Moyal Plane 5.1. Quantum gravity and spacetime noncommutativity: heuristics
The following arguments were described by Doplicher, Redenhagen and Robert in their work in support of the necessity of noncommutative spacetime at Planck scale. 5.1.1. Space-space noncommutativity
In order to probe physics at the Planck scale L, the Compton wavelength of the probe must fulfill
ti ti - < L or M 2 - > Planckmass. Mc -
Lc -
(9)
Such a high mass in the small volume L3 will strongly effect gravity and can cause black holes to form. This suggests a fundamental length limiting spatial localization.
8
5.1.2. Time-space noncommutativity Similar arguments can be made about time localization. Observation of very short time scales requires very high energies. They can produce black holes and black hole horizons will then limit spatial resolution suggesting
At AlT'l 2 L2 ,
L = a fundamental length.
(10) The Groenwald-Moyal plane models the above spacetime uncertainties. 5.2. What is the Groenwald-Moyal plane?
The Groenwald-Moyal plane ds(Rd+') consists of functions a ,p, . . . on Rd+' with the *-product
a
+ .
p
= ae-+
+
a.eUUd,
P.
* xp
=
(11)
For spacetime coordinates, this implies,
lxp,
x y ~ *=
xp
* x,
-
x,
-iepv
(12)
The Groenwald-Moyal plane also emerges in string physics and quantum Hall effect. 5.3. How the G-M plane emerges from quantum Hall eflect and strings
5.3.1. Quantum Hall effect (the Landau problem)
+ Consider an electron in 1-2 plane and an external magnetic field B = (0, 0 , B ) perpendicular to the plane. Then the Lagrangian for the system is 1 2 L = -mx, 2
+ ex:A,
(13)
where
B ,bX a,b = 1,2, (14) 2 is the electromagnetic potential and xa are the coordinates of the electron. Now if eB -+ 00, then eB L -(X,X2 - i:2X1) (15) 2 This means that on quantization we will have i [?a , ? b ] = -Cab (16) eB which defines a Groenwald-Moyal plane.
A,
= --€
N
9
5.3.2. Strings Consider open strings ending on DpBranes. If there is a background two-form Neveu-Schwarz field given by the constants Bij = -Bji, then the action is given by Sc =
As a'
-
1
I 47ra
[gij aaxi d a x j - 27ra' Bij aaxi &xj eab (17)
+spinor termslda dt. 00,
+
e [Bij2j , 2.'"]= i&. or (20) i [ 2 j , ?.'"I = - ( B - ' ) j k (21) e which is just a Groenwald-Moyal plane. Figure [l]below indicates the different sources from where fuzzy physics and the Groenwald-Moyal plane emerge. The question mark is to indicate that the Groenwald-Moyal plane may not regularize qft's.
Fuzzy Physics
QFT Regularisation
l-kdl Effect
+>-
Groenwald-
Quantum Gravity Fig. 1. The Tangled Web: Emergence of Noncommutative spaces from different fields.
10
5.4. Prehistory (before 2004/2005)
Until 2004/2005 much work had been done on 0
0
QFT’s on Groenwald-Moyal plane and renormalization theory, uncovering the phenomenon of UV/IR mixing. Phenomenology, studying the effects on Lorentz violation ( from O,, in [z, , z,], = zO,,), C, C P and C P T
5.5. Modern e r a
In 2004/2005, Chaichian et. a1.l’ and Aschieri et. a1.12 popularized the Drinfeld twist13 which restores full diffeomorphism invariance (with a twist in the “coproduct”) despite the presence of constants Oh, in [Z, , Zv] = iOh,. This twist also twists statistics14. Much of this was known to Majid15, Oeckle16, Fiore17 and Watts?. So we have that the Drinfeld twist twists both (i) action of diffeomorphisms, and (ii) exchange statistics. This brings into question much of prehistory-analysis. Examples include: (i) Lorentz invariance need not be violated even if 0,” # 0. (ii) There need be no ultraviolet-infrared (UV-IR) mixing in absence of gauge fieldslg. There is also a striking, clean separation of matter from gauge fields due to the Drinfeld twist20, (in the sense that they have to be treated differently) reminiscent of separation of particles and waves in the classical theory. 6. Acknowledgments
The work was supported by DOE under grant number DE-FG0285ER40231.
References 1. Alain Comes, Noncommutative Geometry. (Academic Press, San Diego, CA, 1994); Joseph C. Varilly, Hector Figueroa and Jose M. Gracia-Bondia, EZ-
ements of Noncomutative Geometry. (Birkhauser, Boston,2000); G. Landi, Introduction to Noncommutative Spaces and their Geometries.(Springer Verlag, New York, 1997).
11
2. R. Jackiw, NucLPhys. Proc.Supp1. 108, 30 (2002), arXiv:hep-th/0110057; Letter of Heisenberg to Peierl (1 930), Wolfgang Pauli, Scientific Correspondence, Vol. 11, p.15, 3. H. Snyder, Phys. Rev. 7 1 , 38 (1947). 4. C. N. Yang, Phys. Rev. 7 2 (1947). 5. S . Doplicher, K. Fredenhagen and J. Roberts, Phys. Lett. B 331, 39 (1994); S. Doplicher, K. Fkedenhagen and J. Roberts, Comm. Math. Phys. 172, 187 (1995), arXiv:hep-th/0303037. 6. A. Connes and G. Landi, Commun. Math. Phys. 221, 141-159, 2001, arXiv:math.qa/O011194. 7. A. P. Balachandran, S. Kurkcuoglu and S. Vaidya, Lectures on Fuzzy and Fuzzy Susy Physics, arXiv:hep-th/O511114. 8. X. Martin, JHEP 0404 (2004) 077, arXiv:hep-th/0402230; F. G. Flores, D. O’Connor and X. Martin,PoS Lat 2005 (2005) 262. 9. S. S. Gubser and S. L. Sondhi, Nucl. Phys. B 605 (2001) 395-424, arXiv:hepth/0006119; J. Ambjorn and S. Catterall, Phys. Lett. B 549 (2002) 253259, arXiv:hep-lat/0209106; J. Medina, W. Bietenholz, F Hofheinz and D. O’Connor, PoS LAT 2005 263 (2005), arXiv:hep-lat/0509162. 10. R. J. Szabo, arXiv:hep-th/0512054. 11. M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, Phys. Lett. B 604, 98 (2004), arXiv:hep-th/0408069; M. Chaichian, P. Presnajder and A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005), arXiv:hep-th/0409096. 12. M. Dimitrijevic and J. Wess, arXiv:hep-th/0411224; P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, Class. Quant. Grav. 22 (2005) 3511-3532, arXiv:hep-th/0504183. 13. V. G. Drin’feld, Leningrad Math. J . 1 (1990) 1419-1457. 14. A. P. Balachandran, G. Mangano, A. Pinzul and S. Vaidya, arXive:hepth/0508002; B. A. Qureshi, arXiv:hep-th/0602040. 15. S. Majid, Foundations of quantum group theory, Cambridge University Press, 1995. 16. R. Oeckl, Nucl. Phys. B 581 (2000) 559, arXiv:hep-th/0003018. 17. G. Fiore and P. Schupp, published in Quantum Groups and Quantum Spaces, Banach Centre Publications vol. 40, Inst. of Mathematics, Polish Academy of Sciences, Warszawa (1997) 369-377; G. Fiore, J. Math. Phys. 39 (1998) 3437, arXiv:q-alg/9610005; G. Fiore and P. Schupp, Nucl. Phys. B 470 (1996) 211, arXiv: hep-th/9508047. 18. P. Watts, Phys. Lett. B 4 7 4 (2000) 295-302, arXiv:hep-th/9911026; P. Watts, arXiv: hep-th/0003234. 19. A. P . Balachandran, A. Pinzul and B. Qureshi, Phys. Lett. B 634 (2006) 434-436, arXiv: hep-t h/0508151. 20. A. P. Balachandran, A. Pinzul, B. A. Qureshi and S. Vaidya, work in preparation.
S-DUALITIES IN NONCOMMUTATIVE AND NON-ANTICOMMUTATIVE FIELD THEORIES * OMER F. DAYI Physics Department, Faculty of Science and Letters, Istanbul Technical University, TR-34469 Maslak-Istanbul, Turkey, and Feza Gursey Institute, P. 0.Box 6, TR-34684 Cengelkoy-Istanbul, Turkey LARA T. KELLEYANE Physics Department, Faculty of Science and Letters, Istanbul Technical University, TR-34469 Maslak-Istanbul, Turkey KAYHANULKER Feza Gursey Institute, P. 0.Box 6, TR-34684 Cengelkoy-Istanbul, Turkey BARIS YAPISKAN Physics Department, Faculty of Science and Letters, Istanbul Technical University TR-34469 Maslak-Istanbul, Turkey Parent actions to formulate (S-) duals of noncommutative and nonanticommutative N = 1/2 supersymmetric U(1) gauge theories are presented. The equivalence of partition functions of the N = 1/2 supersymmetric U(l) theory and its dual is demonstrated within the Hamiltonian approach. The results which we obtain are valid at the first order in the noncommutativity parameter 0,” or in the non-anticommutativity parameter Cao.
1. Introduction
(S-) Duality transformations map strong coupling domains to weak coupling domains of gauge theories. A method of studying duality invariance of pure U(1) gauge theory is the parent action formalism’. This approach allows the introduction of a dual formulation of the noncommutative U(1) gauge theory2, which was studied in terms of Hamiltonian methods3. Dual *Talk given by 0 . F. Dayi at the 12thRegional Conference on Mathematical Physics, Islamabad (27 March-1 April, 2006)
12
13
actions for supersymmetric U ( l ) gauge theories were derived utilizing a parent action when only bosonic coordinates are noncomrn~ting~. Actually, duality is helpful for inverting computations performed in weak coupling domains to strong coupling domains, when partition functions of the “original” and dual theories are equivalent, i.e. when there exists a duality symmetry. For noncommutative U (1) gauge theory without supersymmetry, this equivalence was established within the Hamiltonian formalism5. Moreover, for non-anticommutative N = 1/2 supersymmetric U(l) theory dual action has been obtained and the equivalence of their partition functions was shown6. In the next section we give the parent actions which produce the “original” and dual theory actions. In Section 3, we demonstrate the equivalence of partition functions of N = 1/2 supersymmetric U(l) theory and its dual. 2. Parent actions In terms of the dual gauge field AD one can introduce the parent action for an Abelian gauge theory (9”” = diag(-l,l,l, 1)):
S, =
J
1 d4x(--F
d92
”u
F””
+ -21~ ~ , , ~ ~ d ~ A l f , F ~ ~ ) .
One treats F as an independent variable without requiring any relation with the gauge field A. Performing the path integral over AD, which is equivalent to solving the equations of motion for AD in terms of F and replacing it in the action, leads to the Abelian gauge theory action
‘J
So = -d4~FPuF””, 4g2 with now F = d A . Performing the path integral over F , which is equivalent to solving the equations of motion for F in terms of AD and replacing it in the action, leads to the dual action
where FD = d A D . The duality transformation is 1 g+-. 9 Let us deal with the noncommutative space defined by the constants
6”” :
[x”,x”] = P ” .
14
Noncommuting variables should be treated as operators. However, one can retain them commuting under the usual product and introduce noncommutativity in terms of the star product
Now, the coordinates x@ satisfy the Moyal bracket xp
*
* xp
-
= epv.
(5)
Moreover, one can perform the Seiberg-Witten map to noncommutative gauge theory fields, to deal with the ordinary gauge transformations7. The related parent action is2
S,
=1 /d4x(FpvFpv
+ 28pvFvpFP"F,p - -Op"FvpFpoFuP) 1
4g2
2 d4x A
D
~
c
~
~
~
~
~
~
F
P
~
,
where F and AD are taken as independent field variables. Similar t o the previous case, one can solve equations of motion for AD or for F and plug them in (6) to obtain the actions which are dual to each other. By this method duality is generalized to noncommutative gauge theory as g+
1
-. 9
and epv
-+ @v
= g2Cpvp~epo.
yielding noncommutativity of space-time coordinates of the dual theory even if the original noncommutativity was between space-space coordinates. It is shown that3 noncommutativity of space-time coordinates resulting from duality does not require any change in the Hamiltonian formalism. Thus we derived Hamiltonian formulation of noncommutative D3-branes and studied some of its aspects. For the supersymmetric U(1) theory, a parent action was available in terms of restricted superfieldsg. We introduced a version in component fields4. Moreover, the Seiberg-Witten map between ordinary and noncommutative gauge fields is generalized to supersymmetric gauge theories. However, it is possible to give some different parent actions; thus some different dual actions for noncommuting supersymmetric U (1)theory were obtained. The formalism of superstring theory with pure spinors" in a graviphoton background" gives rise to a non-anticommutative s ~ p e r s p a c e ' ~ ~ ~ ~
15
which was introduced also in other contexts 14915. It can equivalently be introduced as a deformation of 4 dimensional N = 1 superspace by making the chiral fermionic coordinates 8,, (Y = 1 , 2 , non-anticommuting
{ea,e p } = cap, where C a p (0‘” = C@ep&’ 7) are constant deformation parameters. gh are intact. This breaks half of the supersymmetry. Moyal antibrackets (star products) are used. Thus, instead of operators, one deals with the usual superspace variables. In euclidean R4 chiral and antichiral fermions are not related by complex conjugation. Seiberg used the vector superfield of this deformed superspace to derive, after a change of variables like the Seiberg-Witten map, the N = supersymmetric Yang-Mills theory action12. Gauge transformations possess the usual form. Although we deal with euclidean R4,we use Minkowski space notation. We proposed the parent action as6
where
I. =
g2
/d4x(
-
i --CpuF,,(ix 4
i ~ 4 F p Y F p-y-X@x 2
1
- !$@$I 2
1
1
+ -D: 4 + ;D;
+ $4) ,
1 1 I1 = /d4x{ + p u X n F p v a ~ A ~~, X @ X D -XD@X - &@AD 2 2 1-
-
--AD@$ 2
+
+ ~i D D ( D-I D2)
+
1
.
Here Fpy are independent field variables. Solving the equations of motion with respect to the “dual” fields and plugging into the parent action (7), the non-anticommuting N = supersymmetric U (1) gauge theory action follows: -
Here we have FPv = a,A, - &A,.
- iX@X
1 + -D2
2
16
On the other hand, if one solves the equations of motion with respect to the “original” fields and substitutes them in the parent action (7), the dual theory action will follow:
where FD,, = a,AoV - dvADp. Observe that the original theory action and its dual possess the same form and the duality transformation is
9 4
1 -1
9
3. Equivalence of partition functions Partition functions for the parent actions are expected to produce partition functions of their daughter actions. Indeed, it was shown that partition functions for noncommutative U(1) theory and its dual are equivalent5. Here we will deal with the N = 1/2 supersymmetric theory (10) and its dual (11). In the parent action (7) there are some terms cubic in fields. Thus, it would be apposite to discuss its partition function in phase space, where integrations would be simplified due to Hamiltonian constraints. Let
be the canonical momenta corresponding to
17
Each of the canonical momenta resulting from the parent action (7) gives rise to a primary constraint, which we collectively denote as ( 0 " ) :
+? = poi x 0 , xy = l-Iy M 0 , X2& E fi2& @I
= PI Pg
$Dl
xg
E
11%
-
12
-$&@O&c'
@D
x M
x 0,
+ki = pij 2 -
XI&
-
+&;A,";
@2 3
8D2
0,
pi
P2 x 0 , $ijkFjk
-
+ $A"(T:&
0,
x D & E no&
M
- +JD&@O&"
&$&~O"U
0,
M
7
= fi1&- &A"C&
x, = II;
0 0, 0,
M
0, x 0,
PD x 0.
The canonical Hamiltonian associated with the parent action is
1- +ZADp$
-
i -DD(DI2 D2).
The extended Hamiltonian is obtained by adding the primary constraints 0" with the help of Lagrange multipliers ,Z to the canonical Hamiltonian: 'FIE = 'FIp
+ La@".
Consistency of the primary constraints with the equations of motion:
6" = {'FIE, 0 " ) M 0 gives rise to the secondary constraints 1 i {'FIp,P1}= --D1- 0 x~0 , 2g2 2 1 i A2 E {'FIp,P2}= - 7 0 2 ZDD M 0 , 29 i AD = {XP, P o } I= -(Dl - 0 2 ) M 0 , A1 E
+
2
1
.(
(Po = {'FIp,P$}= -EaJkdkFijX 0 2
, ig2
{'FIp, Poi} = Poi - g2fijkdjADk+ -coi(xx+ M0 . 2 In path integrals first and second class constraints are treated on different grounds. Thus, let us first identify the first class constraints: 401 is Cpyi E
4$)
18
obviously first class. Moreover, we observe that the linear combination
is also a first class constraint. There are no other first class constraints. However, the constraints q5b2 contain second class constraints which we should separate out. This is due to the fact that a vector can be completely described by giving its divergence and rotation (up to a boundary condition). We used divergence of so that there are still two linearly independent second class constraints following from the curl of pD2:
@A,,,
454
Kr@A2 = ICnicijkdj&,
M
0,
where n = 1,2. ICY are some constants whose explicit forms are not needed for the purposes of this work. Although all of them are second class, we would like to separate cpyi in a similar manner:
where Cnj are some constants. The reason of preferring this set of constraints will be clear when we perform the path integrals, though explicit forms of Cl play no role in our calculations. In phase space, the partition function can be written as
where Yi and Pyi embrace all of the fields and their momenta. Sz denotes all second class constraints: S, = (41, 4z , @ I , Qz, 404,Q D ,cpz , cp3, Ai , A2 , (PO, A D ,X I , 21,xz,Rz, X D , RD). We adopted the gauge fixing (auxiliary) conditions
for the first class constraints 4~~ and 4 ~N ~is a .normalization constant. M is the matrix of the generalized Poisson brackets of the second class constraints: M = { S z ,SZ!}.
19
When we integrate over the fields which do not carry the label "D"
2=
/
V A DV~ ~ VD P DV~D D V P D(det g2)(deta:)(detj3)2
~ ( D D ) S ( P Da) ( a . P D ) w . exp{ i
/
d3x
_-g42 FDi j FDa3
' '
AD)
[piA D i + P D D D - T291P D i P L - i c g P D i x D x D
- '2g 2 C g F D i j x D X D - i g 2 X D @ x D
+
will result. In the exponent we distinguish the first order lagrangian of the dual theory, where IIg and f i ~ bare integrated. On the other hand integration of the other fields leads to
z=J
V A D~ P ~ v x v xV D V P
(det g 2 ) (det @)(det p ) 2 S ( D ) B ( P ) S ( B . P ) 6 ( d .A)
In the exponent we recognize the first order lagrangian of the original theory after integrations over II?, f i 1 & , IIg and n 2 & are performed in its path integral. Let us adopt the normalization to write the partition function of nonanticommutative N = supersymmetric U(1) gauge theory as
20
Therefore, by applying the duality transformation (12)-( 13), partition function of its dual can be obtained as
ZNAD
J
VAi V P i DA V O X V D V P 6 ( D ) d ( P ) b ( dP)S(a. . A)
Here, we omitted the label “D”of the dual fields. Comparing the results one concludes that the partition functions of non-anticommutative N = supersymmetric U(1) gauge theory Z N A and its dual Z N A Dare equivalent:
3
Z N A= Z N A D . Therefore, under the strong-weak duality non-anticommutative N = supersymmetric U ( l ) gauge theory is invariant.
4
Acknowledgments:
O.F.D. thanks the organizers Professors Riazuddin and F. Hussain for invitation and support. References 1. T.H. Buscher, Phys. Lett. B 194,59 (1987); Phys. Lett. B 201,466 (1988). 2. O.J. Ganor, G. Rajesh and S. Sethi, Phys. Rev. D 62,125008 (2000). 3. O.F. Dayi and B. Yapiakan, JHEP 10,022 (2002). 4. O.F. Dayi, K. Ulker, and B. Yapqkan, JHEP 10,010 (2003) . 5. O F . Dayi and B. Yapigkan, JHEP 11,064 (2004) . 6. O.F. Dayi, L.T. Kelleyane and K. Ulker, JHEP 10,035 (2005). 7. N. Seiberg and E. Witten, JHEP 09,032 (1999). 8. J. Wess and J. Bagger, Supersymmetry and supergravity (Princeton University Press, 1992). 9. N. Seiberg and E. Witten, Nucl. Phys. B 426,19 (1994) . 10. N. Berkovits, JHEP 04,018 (2000) . 11. H. Ooguri and C. Vafa, Adv.Theor.Math.Phys. 7,53 (2003). 12. N. Seiberg, JHEP 06, 010 (2003). 13. J. de Boer, P.A. Grassi , P. van Nieuwenhuizen, Phys. Lett. B 574,98 (2003). 14. J. W. Moffat, Phys. Lett. B 506, 193 (2001) . 15. D. Klemm, S. Penati, L. Tamassia, Class.Quant.Grav. 20,2905 (2003) .
NEW CASIMIR ENERGY CALCULATIONS H. AHMEDOV Feza Gursey Institute, 81220, Istanbul, Turkey. E-mail:
[email protected]. tr
I. H. DURU Izmir Institute of Technology, 35430, Izmir, Turkey. E-mail:
[email protected] New Casimir energy results for massless scalar field in some 3 -dimensional cavities are presented. We attempted to discuss the correlation between the sign and the magnitude of the energy and the shape of the cavities.
1. Introduction
The sign of the Casimir energy is known to be dependent on the dimension, topology and geometry. In this note we present some new exact results for massless scalar fields in three dimensional cavities with the trivial topology. We then compare the known Casimir energy values for several three dimensional cavities. The conclusion we arrived a t is that the existence of corners lowers the vacuum energy. 2. New Casimir Energy Results in Some 3-dimensional
Cavities In this section we present Casimir Energies for a massless scalar field in some 3-dimensional cavities. These cavities are rather special regions, for all of them are fundamental domains for some crystallographic group generated by reflections with respect to the boundary walls. This property enables us to' obtain the wave functions satisfying the Dirichlet boundary conditions and then the correct energy spectrum. (i) A Pyramidal Cavity The region is defined by the planes
21
22
This is the fundamental domain of the group of order 48 generated by the reflection with respect to the above planes'. The Casimir energy for a massless scalar field in this cavity is ( in h = c = 1 units )
0.069 7 > 0.
EpYT
(ii) A Conical Cavity The conical cavity we consider is the one with height h = a and with a very special opening angle ,B = arcsin; '. The crystallographic group which admits this cavity as the fundamental domain is the Tetrahedral group. The Casimir energy due to the fluctuation of the massless scalar field is'.
Em,
0.080 => 0. a
(3)
(iii) Triangular Cylinders Three kinds of triangles are the fundamentals domains of some crystallographic groups in the plane. These are equilateral, right-angled isosceles and the right-angled triangle which is the half of the equilateral one3. Here we give the results for a cylindrical cavity of height b and with equilateral triangular cross-section of edges a. Three possibilities are distinguished: a) For b > a
EtTi N b) For a
0.053 a
--
+ -(0.029)' . a2 '
> b > 5 N 0.7a 0.013
L ( - h
EtTi N 2 c) For b <
+
(0.011) a b2
+--0.093 a
(0.048) b a2
5 0.039 (0.014) a EtTi N -b2 ' b +
The energy for height a is
0.022 EtTi N -. a 3. Some Known Casirn,; Energy Results for 3-dimens~nal
Cavities (i) Casimir energy for the spherical cavity of radius a is4
23
(ii) Cylinders with square cross-section of edges a and of height b5 have
0.013 a
E r e c t 2 --
+
(0.011) b for a2
,a
(9)
and
0.013 b
Erect N --
+
(0.011)a for b2
For the cube of edge a we have
EcUbN
0.002
--
which is very small.
4. Discussion To obtain a meaningful comparison of the results we consider cavities of equal volumes. Such an approach may help us to understand the shape dependence of the energy. The positive energies for the pyramidal, conical and triangular cylinder cavities can be expressed in terms of energy of spherical cavity of the same volume as:
The energy for the cube ( which is negative but very close t o zero ) of the same volume is
It seems that corners of the cavity reduce the energy. If we think in terms of the path integral picture we can say that the paths hitting the corners cannot bounce back, but disappear. Thus we can think that corners in the cavities reduce the phase space volume, and then reduce the vacuum energy.
Acknowledgements The authors H. Ahmedov and I. H. Duru thank the Turkish Academy of Sciences (TUBA) for its support.
24
References 1. H. Ahmedov and I.H. Duru, J. Math. Phys. 46,022303 (2005). 2. H. Ahmedov and I.H. Duru, J. Math. Phys. 46,022304 (2005). 3. H. Ahmedov and I.H. Duru, in preparation. See also H. Ahmedov and I.H. Duru, J. Math. Phys. 45,965 (2004). 4. T. H. Boyer, Phys. Rev. 174,1764 (1968); B. Davies, J . Math. Phys. 13,1324 (1972); R.Balian and B. Duplantier, Ann. Phys. 104,300 (1978); J Schwinger, L.L. De Raad and K . A. Milton, Ann. Phys. 115,1 (1978); ibid. 115,388 (1978). 5 . See for example V.M. Mostapanenko and N.N. Trunov , The Casimir Effect and i t s Applications. (Oxford Univ. Press, New York, 1997) and references therein..
QUATERNIONIC AND OCTONIONIC STRUCTURES OF THE EXCEPTIONAL LIE ALGEBRAS MEHMET KOCA Sultan Qaboos University College of Science, Physics Department P.O. Box 36, Al-Khoud, 123 Muscat Sultanate of Oman E-mail: [email protected] The Cayley-Dickson procedure has been used to construct the root systems of S 0 ( 8 ) , SO(9) and the exceptional Lie algebra F4 in terms of quaternions. The Aut(F4) has a simple presentation of the form ( 0 , O ) @ (O,O)* where 0 represents the binary octahedral group. Two sets of quaternionic root system of F4, symbolically written [F4,F4] = F4 e7F4, describe the octonionic root system of Es where the root system of the exceptional Lie algebra E7 are represented by the imaginary octonions. The automorphism group of the octonionic root system of E7 preserving the octonion algebra is the Chevalley group G2(2) where the maximal subgroups of G z ( 2 ) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E6, S U ( 2 ) x SO(12) and S U ( 8 ) of E7. Another pairing of the quaternionic roots of the form [F4,F4] = F4 aF4, constitutes the root system of E8 in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H4. The relevance of H4 to Es and the maximal subgroups of H4 have been discussed. Polyhedral structures obtained from the quaternionic root systems of H4 are described as the orbits of H3.
+
+
Keywords: Polyhedra,Quaternions and Octonions
1. Introduction
Integer elements of the quaternions and octonions play important roles in the description of the root systems of the Lie algebras of rank-4, S0(8),SO(9) and the exceptional Lie algebra F4 The imaginary integer elements of quaternions correspond to the root system of the Lie algebra SP(3)while those of octonions describe the root system of the exceptional Lie algebra E7 '. The Weyl groups of the rank-4 Lie algebras have presentations as the finite subgroups of O(4) M S U ( 2 ) x S U ( 2 ) where each S U ( 2 ) can be represented in terms of quaternionic polyhedral groups. We introduce the Cayley-Dixon doubling procedure to construct the root systems
'.
25
26
of those celebrated algebras. Denote by a, b, c,d the elements of a division algebra other than the octonions. The pairs satisfying the product rule (Cayley-Dickson doubling rule)
( a ,b)(c,d) = (UC - db,cb + ad)
(1)
define the elements of a higher rank division algebra. For a , b real numbers ( a ,b) is a complex number; if a, b are complex numbers the pair is a quaternion. Similarly a pair of quaternion defines an octonion under the rule of (1).To construct the root systems of higher rank Lie algebras we start with the roots and the fundamental weights of SU(2) : 0, f l ;fi and obtain first the roots of SO(4) and then SO(5) in the manner;
SO(4) : ( f ,0)
= f l ;(0, f l ) = fi,
1 1 1 f-)= -(*I f i). 2 2 2 Two sets of roots of SO(5) constitute the roots of SO(8) :
SO(5) : ( f , O ) = f l ;(0, f l ) = fi; (f-,
T
= Vo @
V+ CB V-,
(2)
(3)
i
where VO= { f l ,f e l , f e z , fes}, V+ = { f l f el f e2 f e3) (even number of (+) sign), V- = { f l f el f ez f e3) (odd number of (+) sign). Here imaginary quaternions satisfy the relations eiej = -6ij Eijkek, i , j , k = 1,2,3. The set of quaternions T represents also the binary tedrahedral group as well as the 4-dimensional polytope {3,4,3). The weights of three 8dimensional representations of SO(8) are given by the quaternions:
i
+
The roots of SO(9) are the quaternions T @ V1. We define T’ = V1@VZ @ V3. Then the set of quaternions T@T’represent the roots of the exceptional Lie algebra F4. Imaginary quaternions of the roots of F4 constitute the roots of the Lie algebra SP(3). We also note that the binary octahedral group of order 48 is given by 0 = T d T ’ . We define the O(4) transformations by the quaternions: [a,b] : q -+ q‘ = aqb , [c,d]* : q q“ = cqd where q is the quaternion conjugation. Using this notation we can write the AutF4 and the Weyl group of F4 as follows 3:
+
--f
AutF4
=
[O,O]@ [O,O]*,
W ( F 4 ) = [T,T]@ [T,T]* @ [T’,T’]@ [T’,T’]*. Many subgroups of the AutF4 have been studied in reference ’.
(5)
27 2. Octonionic Roots of Es
Following the Cayley-Dickson doubling procedure we construct the octonionic root system of Eg as follows:
( T ,0) = T , (0, T ) = e7T, (K , K ) = VI e 7 K , (E,V3) = V2
+
+ e7V3, (V3,V2) = V3 + e7v2
(6)
where ea = -1 and e7e1 = e4, e7e2 = e5,e7e3 = e 6 1 , 4 .The imaginary octonions of (6) constitute the roots of the Lie algebra E7. The octonion multiplication table is given by e a. ej .-- -6..a j + $a.j.k e k, with
i,j,k=1,2,...,7,
(7)
completely anti-symmetric and $123 = $246 = 4435 = $367 = $651 = $572 = $714 = 1The automophism group of the octonionic root system of E7 is a finite subgroup of the exceptional Lie group G2 called adjoint Chevalley group G2(2) of order 12096215. It has four maximal subgroups of orders 432,336,192 corresponding to the automorphism groups of the octonionic root systems of the Lie algebras E6,SU(8) and SO(12) respectively. The fourth maximal subgroup of order 192 preserves the structure ( H ,H ) = H e7H where H stand for the quaternions6. A quaternionic representation of the root system of Eg is also possible with the following pairing7: $ijk
+
E8
=
( T ,0) @ (0, T )@ (Vl,V2)@ (v2, v3) @ ( 6 K ,)
+
a
(8)
where (A, B) = A aB , a= 2 ’ 7 = 2 . The quaternionic Eg roots in (7) can also be written as E8 = I aI where I represents the quaternions of the binary icosahedral group. It is known that the quaternions of I represent the roots of non-crystallographic Coxeter group H4 . It has been shown that the symmetry group of the root system of H4 can be written as8
+
of order 14,400. The maximal subgroups of the Coxeter group W ( H 4 ) have been studied in [9] and found to be Aut(SU(5)) M W(SU(5)) : 2 2 , W ( H 3 ) X 2 2 , W ( s U ( 3 ) ) X W(sU(3)) : 2 4 , W ( H 2 ) X W ( H 2 ) : 2 s , W ( S O ( 8 ) ): Z3 of respective orders 240, 240, 144, 400, 576. Two of these maximal subgroups W ( H 3 ) x 2 2 and W(H2) x W ( H 2 ) : Z4 are the non-crystallographic subgroups of W ( H 4 ) . The others are the crystallographic groups. The non-crystallographic Coxeter group W ( H 4 ) is a
28
maximal subgroup of the crystallographic group W(E8) .The eigenvalues of the Coxeter element of W(E8)can be determined from the characteristic equation8
+ A' - x5 x4 - x3 + x2 + 1 = p(X)q(X) = 0, 21r p ( X ) = X4 + 7X3 + 7 X 2 + TX + 1 = 0 -+ exp(im-), m = 7,13,17,23, 30 21r q ( x ) = x4 + ax3 + (rx2 + (TX + 1 = o -+ exp(im-), 30 1x1 - MI = AS
-
m = 1,11,19,29(Coxeter exponents of The half of the Coxeter exponents of
E8
W(H4)).
are the Coxeter exponents of H4 .
3. Quaternions and Polyhedra
We have already noted that the root system of SO(8) represents the vertices of the 4-dimensional polytope {3,4,3} . Actually the weights of any one of these 8-dimensional representations of SO(8) constitute the vertices of an hyperoctahedra {3,3,4} known also as 16-cell. Then the 16 weights of the remaining two 8-dimensional representation of S 0 ( 8 ) , in other words, the weights of the spinor representation of SO(9) represent the vertices of a hypercube or the polytope {4,3,3}, dual of the polytope {3,3,4}1°. One can determine the vertices of the polyhedra in 3-dimension by projecting any representation of these Lie algabras S0(8), SO(9) and F4. The useful Weyl groups in these projections are the W(SP(3)) and WjSO(7)). We shall give an example of this projection when we determine the orbits of W ( H 3 ) in the polytopes {3,3,5} = I and {5,3,3}. The vertices of {3,3,5} and {5,3,3}, in other words, the elements of the binary icosahedral group 1 and its dual can be obtained as follows" 5
{3,3,5} =
5
5
C @p3T= j=1 C @T?, {5,3,3} = j,k=1 C @p3T'pk
(11)
j=1
+ +
el oe3) and T is given in (3) and the elements of T' are where p = ;(T given in (4). Two subgroups of W ( H 4 ) , namely, W ( H 3 ) and W(SU(5)) each can be embedded in W ( H 4 ) in 120 different ways. The polytope {3,3,5} is made of tedrahedra in 3-dimensions and the symmetry of the tedrahedron is isomorphic to the Weyl group W(SU(4)) x S4 of order 24, a subgroup of the group W(SU(5)). It is implicit that the group W(SU(4)) M 5'4 can be embedded 120 x 5 = 600 different ways. This proves that the polytope {3,3,5} is made of 600 tedrahedra and so it is called 600-cell. From the
a
h
d
e
c
Fig. 1. Polyhedra Projected from the Polytope (5,3,3} as orbits of W ( H 3 ) .
fact that the Coxeter group W ( H 3 ) is isomorphic to the icosahedral group with inversion and can be embedded in the Coxeter group W(H4 ) in 120 different ways it can be argued that the polytope {5,3,3} is made of 120 dodecahedra and hence we have the name 120-cell.It is difficult to visualize these polytopes in 4-dimensions. Therefore it is advisable to project them into 3-dimensions by determining the orbits of W ( H 3 ) in the polytopes {3,3,5} and {5,3,3}. One representation of the group W(Iy3)in W(W4) can be written as W W 3 ) = blF1 G3 I v l P I * ; P l P E 1.
(la)
The orbits of W ( H 3 ) in {3,3,5} are just the conjugacy classes of 6. They consist of four icosahedrons, two dodecahedrons and one icosidodecahedron. Including the poles f 1 they complete the vertices of {3,3,5} : 4 x 12 + 2 x 20 $- 30 2 = 120 . The orbits of W(H3) in the polytope {5,3,3} can be determined by acting W ( H 3 ) in (12) on the quaternions in (11)l2.One cuts the $sphere S3 with the hyperplane f(@ Eq) = d where c is a fixed quaternion. In our example it is the quaternion B which is left invariant under the transformations of (12). Therefore the scalar d represents the scalar product of 1 with the quaternions in a given orbit. Calcu~ation$show that we have 15 orbits with the values of d equal
+
+
30
*+,
to 0, 'f4 f&, 2 2 f "2,J z 4 1 3 ,k224 ,f224 .These orbits represent convex solids some of which are regular polyhedra, some are Archimedean solids and the rest are the convex solids with two edge lengths having pentagonal, triangular, square and non regular hexagonal faces. We illustrate some of them in the figures. The orbit with d = 0 is a convex solid with 60 vertices (with two edge lengths), 12 pentagonal , 20 triangular and 30 rectangular faces. The widths of the rectangles are the edges of the pentagons and the lengths of the rectangles are the sides of the triangular faces. The = 7'. It is depicted in Fig. length to the width ratio of the rectangle is l a . The orbit with d = 'f is a icosidodecahedron with 30 vertices, 32 4. faces( 12 pentagonal and 20 triangular) and 90 edges which is shown in Fig. l b . The convex solid of d = f L has 60 vertices with 12 pentagonal and 2Jz 20 semi-regular hexagonal faces and is shown in Fig. lc. This could be used as a model for CSO as it has two different edge lengths corresponding to two different bondings. The orbit with d = is similar to the one in Fig. l c but long and short edges are interchanged. The orbit with d = ZIZ& is a semi-regular polyhedron, which has also 60 vertices, 90 edges, 62 faces(l2 pentagonal, 20 triangular, 30 square). Fig. I d shows this Small Rhombicosidodecahedron. The three remaining orbits are the dodecahedra with different circumscribed radii, one of which is depicted in Fig. le. I would like to thank Dr. Ramazan Koc for illuminating discussions.
*&
References 1. M. Koca and N. Ozdes, J. M. Phys. A 22, 1469 (1989); M. Koca, R. Koq, M. Al-Barwani, J. M. Phys. 44, 03123 (2003). 2. F. Karsch and M. Koca, J. Phys. A 23, 4739 (1990); M. Koca and R. Koq J . Phys. A 27,2429 (1994). 3. M. Koca, R. Koq, M. Al-Barwani, J. M. Phys 47, 043507-1 (2006). 4. H. S. M. Coxeter, Duke Math. J . 13,561 (1946). 5. C. Chevalley, Tohoku. Math. J. 7,14 ( 1955); Am. J . Math. 77,778 ( 1955). 6. M. Koca, R. Koq, N. 0. Koca, hep-th /0509189. 7. M. Koca, J. Phys. A 22, 1949 (1989); J . Phys. A 22, 4125 (1989). 8. M. Koca, R. Koq, M. Al-Barwani, J. Phys. A 34, 11201 (2001). 9. M. Koca, R. Koq, M. Al-Barwani, S. Al-Farsi, Linear Algeb. Appl. 412, 441 (2006). 10. H. S. M. Coxeter, Regular Complex Polytopes, (Cambridge: Cambridge University Press, 1973). 11. P. du Val, Homographies, Quaternions and Rotations, (Oxford University Press,1964.) 12. M. Koca, R. Koq, M. Al-Ajmi, ' I Quaternionic Representation of the Coxeter Group W ( H 4 ) and the Polyhedra", Submitted for publication.
ON CRITICAL DIMENSIONS OF STRING THEORIES DIMITRY LEITES’ and CHRISTOPH SACHSEt Max-Planck-Institut fur Mathematik in den Naturwissenschaften (MPIMIS), Inselstr. 88, DE-04103 Leiptig, Germany; *E-mail: [email protected] t E-mail: [email protected] Exactly 10 of all simple Lie superalgebras of vector fields on 1IN-dimensional supercircles (superstrings) preserving a structure (either nothing, or a volume element, or a contact form) have nontrivial central extensions. The values of the central charges in (projective) spinor-oscillator representations of these stringy superalgebras associated with the adjoint module can be interpreted as critical dimensions of respective superstrings. Apart from the well-known values 26, 10, 2 and 0 corresponding to N = 0, 1, 2 and > 2, respectively, for the contact type stringy superalgebras (Neveu-Schwarz and Ramond types alike), there arc two more non-zero values of critical dimension: for the general and divergencefree algebras for N = 2. These dimensions, found here, are -1 and -2, respectively. We also mention related problems. Keywords: stringy Lie superalgebras, critical dimensions
1. Introduction We start by quoting from Ref. 1: “In various papers and books on string theories, a degenerate (in Dirac’s sense) Lagrangian is considered. The Fourier harmonics of the constraints of this Lagrangian can be endowed with a Lie algebra structure; this Lie algebra is isomorphic to the Lie algebra of the group of diffeomorphisms of the circle. The complexification of this Lie algebra is a versiona of the Witt algebra mitt: Either aer (@[x-l,x]) or its completion. The following types of mitt-modules (or their completions) are mainly studied: (1) FA,^ = @[x-’,x]x”(dx)’ (the answer to the question which of these modules are irreducible follows from Ref. 2, where the polynomial case is aEvery Lie algebra of vector fields on a manifold has several “incarnations”: The coefficients of the fields can be smooth functions, or polynomials, or power series, or Laurent polynomials, or (for completion) Laurent series, etc. depending on the problem.
31
32
considered) ; (2) highest (or lowest) weight modules (irreducible ones are completely described in Ref. 3) which are all realized as quotients of the spinor Spin(V) or oscillator Osc(V) representations constructed from modules V = FA+. Since the representations of type (2) are projective, they give rise to a nontrivial central extension of mitt, the Virasoro algebra bit, and actually are representations of oir, rather than of mitt. Realization of the elements of mitt by quadratic polynomials in creation and annihilation operators acting in the spaces of spinor and oscillator representations is interpreted in physics as quantization. The “critical dimension” CD that appears in physics papers is the (only) dimension of the Minkowski space (in which the string under study lives) for which quantization is free of anomalies4. There are several, perhaps inequivalent in super setting, ways to compute the critical dimensions. The first ones are related to a study of various action functionals, see, e.g., Ref. 4, 5.” Feigin interpreted the CD as the value of the central charge of bir in the spinor representation corresponding to the adjoint mitt-module. Feigin used this CD in the computation of the semi-infinite cohomology which he introd~ced~~~-~. For the list of super analogues of mitt, simple Lie superalgebras that appear in string theories, see Ref. 10. Among them, there are precisely ten distinguished ones, which have a nontrivial central extension. Superization of Polyakov’s action and Nambu’s action leads to interesting supersymmetric integrable systems (see, e.g., Ref. 11, 12, 13, 14, and references therein) , super versions of minimal surfaces and constant curvature surfaces, but we can not describe critical dimensions in these terms. Our result: We apply Feigin’s picture to consider two, hopefully new, examples. (The isomorphism, up to parity, Spin(V) Y Osc(V), as spaces is referred to as the “Fermi-Bose correspondence” in physics papers. The values of CDs in these spaces differ by a sign, and this gives us hope to be able to interpret negative CDs as reasonable dimensions.) 2. Stringy superalgebras and modules over them 2.1. Vectorial Lie superalgebras with polynomial or formal coefficients. Let F := C[[y]] be the supercommutative superalgebra of formal power series in y = (y1, . . . ,ynfm) = ( ~ 1 , .. . , z,
cm)
33
by the yi. Define the topology on 3 in which the ideals (y)", where T = 0, 1, 2, . . . , are neighborhoods of zero. We see that .F is complete with respect to this topology. Let oect(n1m) be the Lie superalgebra of formal vector fields, i.e., of derivations of C[[y]]continuous with respect to the above topology. In general questions we denote oect(n)m) and its infinite dimensional subalgebras by L. In L , define a descending filtration L = L-I 3 LO 3 C1 3 . . . , setting
L, = {D E oect(n1m) 1 D ( 3 )
c (y)'")
(1)
Denote by L = @Lr, where L, = L r / L r + l r the associated graded Lie superalgebra. Let us identify Lo with g[(nlm)setting Eij H yjdi.
Remark. For some simple Lie superalgebras, the spaces of the highest and lowest weight vectors is multidimensional, cf. Ref. 15; there are also various gradings listed in Ref. 16. 2.2. Tensor fields.
Let us consider a given representation p of LO = Lo/L1 in V as a representation of LO.Since Hom(C[d],C) F,we may set T(V) := H o m U ( . c o ) ( ~ (V). ~), Each vector field D = Didi acts in T(V) by means of the Lie derivative Lo:For any f E . F , v E V and Dij = (-l)P(Yi)P(Dj)diDj, we set
+ (-i)~(~)p(f)fC D~~~(E~~)(v).
L ~ ( ~=vD) ( ~ ) v
-
Having fixed coordinates, we define the divergence as the map
D=
d XfiG + x g j dw i
j
divD =
Set soect(n1m) = {D E oect(n1m) divD
c + x(-l)p(gj)-. 2
i
j
dgi 89,
= 0).
2.3. Modules over stringy superalgebras.
Simple stringy superalgebras and their nontrivial central extensions are listed in Ref. 10. These extensions only exist for contact algebras (and the corresponding CDs are known), and also for oectL(l12) = OerC[~-',h, Ez][[z]land
mectf(112) = {D E oect'(ll2)
I div(zXD)= 0).
Denote by I ( V ) the oect(1ln)-module that differs from T(V) by allowing Laurent polynomials as coefficients of its elements instead of polynomials.
34
Clearly, I ( V ) is a bect’(lln)-module. Set ?;L(V):= ‘T(V)xP.Some of such modules are described in Ref. 17 as an easy corollary of Ref. 18. None of these modules has a highest or lowest weight vector. The “simplest” such modules over the Lie superalgebras oect’( 1In) and soectf:(1In) are, clearly, the rank 1 modules over 3,the algebra of functions. They are constructed as follows: Let VoZP be the space of tensor fields corresponding to the “pth power” of the g I-module corresponding to the superdeterminant or Berezinian (infinitesimally, XvoZ = pstr(X)voZ). For p = 0, this is just the space of functions, 3. Over tL(N)c oect’(lIN), the contact superalgebra which preserves the Pfaff equation with the form
it is more natural to consider the modules defined in terms of
CYN:
Observe that VoZ’ = FA(^-^) and that the Lie superalgebras of series tL do not distinguish between and a-’: their transformation rules are identical. Over t’(ll2) (and similarly over tM(113)), there are also modules 3 ~ := , 3~ ~ ; , [;d J~ 1 ] ”where , [dell = d& mod C Y N . Let c# = C Y N - ~ xed0 be the “Mobius” version of the contact form Q N and let t M ( N )c oect’(1IN) be the Lie superalgebra that preserves the Pfaff equation a f ( X ) = 0 for X E oect’(1IN). We set (clearly, Kf E tL(N))
&
+
K f = (2 - E ) ( f ) a where E =
c@i& and Hf i
af
ax + (-l)P(f)Hf+ -E, ax = C %&is the hamiltonian field with j I N
Hamiltonian f that preserves daN and (clearly, Kf” E t M ( N ) )
Kf”= (2 - E)(f)’D+ D ( f ) E + (-l)P(f)HfM, where
35
3. The spinor and oscillator representations in super setting 3.1. The Lie superalgebra bei(2nlrn). Define the Heisenberg Lie superalgebra fjei(2nlm) as follows. Consider a (272 1(m)-dimensionalsuperspace W = V @@I, where p( 1)= 0, and let (., .) be an even nondegenerate skew-symmetric bilinear form on V . Let W be the superspace of fjei(2nlm)with the bracket
+
[w,w] = (w,w)n, for w,w
E
V ; [n, fjei] = 0.
3.1.1. The big and small Weyl superalgebras. Being primarily interested in irreducible representations of fjei, or, equivalently, of its enveloping algebra, we consider not the whole U(fjei(2nlm) (the “big” Weyl superalgebra) but its quotient (the “small” Weyl superalgebra)
Uh(fjei(2nlm)= U(fjei(2nlm))/(n- h) for h E C. The quotient Ufi(bei(2nlm),called the superalgebra of observables, is, by definition, an associative superalgebra isomorphic, for k = [ and m even, to the associative superalgebra of differential operators on the supermanifold PIkwith polynomial coefficients diff(n1k) = “Mat”(C[q,(1) whereas, for k = and m odd, it is isomorphic to (for the definition of the queer analog QMat of the general matrix superalgebra, see, e.g., Ref. 19, 20)
F]
[F]
+
(For example, take 7r = Crn &.) For n = 0, the algebras diff and qdiff with super structure ignored are known under the name of Clifford algebras.
..
3.2. Spinor (Clifford-Weil-wedge-. ) and oscillator representations. Let po(2nlm) be the Poisson Lie superalgebra realized on polynomials. As is easy to see, po(2nlm)oE 05p(m12n),the superspace of elements of degree 0 in the &graded Lie superalgebra or, which is the same, the superspace of quadratic elements in the representation by generating functions. The complete description of deformations of po(2nlm) was recently given in Ref. 21: There is only one (class of) deformations: quantization Q and
36
we denote Spin(V) the module given by the through mapb
We will denote this representation Spin(V) and set Osc(V) := Spin(II(V)), where V is the standard representation of osp(m12n) = 05p(v), and IT is the change of parity. For n = 0, it is called the spinor representation; for m = 0 the oscillator representation. Now, suppose that V is a g-module without any bilinear form. Then, consider the module W = V @ V' (in the infinite dimensional case, V* is the restricted dual whose elements are sums of finitely many terms, not series) endowed with the form (for any w1, w1 E V, w2, w2 E V') B((Wl,W2),
(w1, w2)) = WZ(W1) f (-1)p("')*(w2)w2(w1).
This form is symmetric for the plus sign and skew-symmetric otherwise. In the following tables we give the results of calculations of the highest weights - the labels ( c ,h;HI, . . . , H k ) with respect to the central elements z ; Kt and Kclql, . . . , Kckllhof the spinor representations Spin(.FA,,) of the contact superalgebras. For the oscillator representation 0 s c ( . F ~ , ~=) Spin(II(.FA,,)) the values of the highest weight are (-c, h;H I , . . . , I l k ) . Problem. For which projective representations of tL(1IN) and tM(1IN), where N = 3,4, the values of c are nonzero? (So far there is only a partial answer22.)
3.3. The values c and h of the highest weight of the tL(lIN)-and tM(1IN)-module Spin(Fx;p). The labels of the highest weight other than c, h are all 0 , except for tL(112) and tM(113): the highest weight of Spin(.FA,v;,) is ( c ,h; v). N 0
h for tL
C
12x2 - 12x
+2
1
3 3 - 12X)
2
2
(fi
+ 2 X ) ( p + 1) p+2x
2 p + 2x
+v
h for tM 2p
+ 3x - f
2p
+ 2x - ;
bThe subscript L makes an associative superalgebra a Lie one, replacing the dot product by the supercommutator.
37
Remark. For the contact superalgebras g, our choice of g-modules V = FA;+ from which we constructed Spin(V @ V " )is natural provided we are interested in semi-infinite cohomology of g. Besides, for n small, all modules of tensor fields are of this form, anyway. For the superalgebras g of series Dect and m e c t , the situation is the opposite one: the adjoint representation V = g is of the form V = I ( i d * ) . 3.4. The choice of the cocycle. Now, let us fix the cocycle that determines the nontrivial central extensions of the distinguished stringy superalgebras. One choice comes from.the study of semi-infinite cohomology; it is very interesting and reasonable for Dir. Another choice is to directly calculate the bracket [ei,e-i] in the spinor or oscillator representation. We get
[ei,e-i]
=
(4)
Nowadays such infinite sums seem meaningless (or equal to 00) to most, but less than a century ago every student who took Calculus knew a way to compute (4): evaluate the Riemann <-function ((s) = at s = -1. n>l
For other distinguished stringy superalgebras g, physicists traditionally consider the cocycles whose restriction to the subalgebra mitt c g coincides with the fixed cocycle that determines Dir. The trouble is that there is no canonical embedding mitt c g for g 9 tL(1) or tM(1).
Problems. Which cocycle to choose? I n the following Theorem, can one consider Osc(g) (so CDs are positive) instead of Spin(g) ? 3.5. Theorem. For ~ e c t ~ ( 1 1 2CD ) , is equal to -1; for each saecti(ll2), CD is equal to -2.
Acknowledgements. We are thankful to MPIMiS and (Ch.S.) to the International Max Planck Research School affiliated to it, and the Tschira Foundation for financial support and most creative environment. D.L. is most thankful to A. D. R. Choudary and A. Qadir for hospitality. References 1. B. Feigin, D. Leites, in Group-theoretical methods in physics, eds. M.Markov, et. al. v. 1, Nauka, Moscow, 1983, 269-273, (English translation: Harwood Academic Publ., Chur, 1985, Vol. bf 1-3, 623-629). 2. A.N. Rudakov, Izv. Akad. Nauk SSSR Ser. Mat. 38, 835 (1974).
38 3. B. Feigin, D. Fuchs, in Representation of Lie groups and related topics, eds. A. Vershik, D. Zhelobenko, Adv. Stud. Contemp. Math. 7,Gordon and Breach, New York, 1990, 465-554. 4. M. Green, J. Schwarz, E. Witten Superstring theory, (Cambridge Univ. Press, Cambridge, 1987) 5. M.S. Marinov, Soviet Physics Uspekhi, 20,no. 3, 179 (1977); translated from Uspehi Fiz. Nauk, 121 no. 3, 377 (1977). 6. B. Feigin, Usp. Mat. Nauk 39,no. 2 (236), 195 (1984) (English translation: Russian Math. Surveys, 39,155-156). 7. B. Feigin, in Proceedings of the International Congress of Mathematicians, Vol. I, I1 (Kyoto, 1990), 71-85, (Math. SOC.Japan, Tokyo, 1991). 8. B. Feigin, E. Frenkel, Commun. Math. Phys 137,617 (1991); ibid.; erratum: Comm. Math. Phys. 137 , no. 3, 617 (1991); MR 92h:17028. Comm. Math. Phys. 147,no. 3, 647 (1992) 9. I. Frenkel, H. Garland, G. Zuckerman, Proc. Natl. Acad. Sci. U.S.A. 83,8442 (1986). 10. P. Grozman, D. Leites, I. Shchepochkina, Lie superalgebras of string theories. Acta Math. Vietnam. 26 , no. 1, 27 (2001); hep-th/9702120 11. E. Ivanov, S. Krivonos, F. Toppan, Phys. Lett. B 405, 85 (1997). 12. S. Krivonos, A. Sorin, in Supersymmetries and quantum symmetries eds. J. Wess, E. Ivanov, (Dubna, 1997), 261-269, Lecture Notes in Phys. 524, (Springer, Berlin, 1999) 13. D. Leites, in: Operator Methods in Ordinary and Partial Differential Equations, S. Kovalevski Symposium, Univ. of Stockholm, June 2000, eds. S . Albeverio, N. Elander, W. N. Everitt and P. Kurasov (Birkhauser, BaselBoston-Berlin, 2002) (Operator Methods: Advances and Applications 132, 267 (2002)). 14. D. Leites, J . Nonlinear Math. Phys. 7,263 (2000); hep-th/0007256 15. P. Grozman, D. Leites, J . Nonlinear Math. Phys. 8, 220 (2001); math.QA/0104287 16. D. Leites, I. Shchepochkina, Classification of simple vectorial Lie superalgebras, preprint MPIM-Bonn-2003-28 (www.mpim-bonn.mpg.de) 17. V. Kac, A. Rudakov, Modules of Laurent differential forms on a superline (http://www.math.ntnu.no/-rudakov/preprint .html) 18. J. Bernstein, D. Leites, Selecta Math. Sou. 1, 143 (1981). 19. A. Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163 (Cambridge University Press, Cambridge, 2005). 20. M. Gorelik, Shapovalov determinants of Q-type Lie superalgebras, math.RT/0511623. 21. D. Leites, I. Shchepochkina, Theor. and Math. Physics 126, 339 (2001); preprint ESI-875 (www.esi.ac.at); math-ph/0510048. 22. E. Poletaeva, J . Math. Phys. 42,526 (2001) 23. A. Dzhumadildaev, C. R . Acad. Sci. Paris, Ser. Mathem. 324,497 (1997). 24. Seminar on Supennanifolds, Reports of Stockholm University, 1987-1 992, ed. D. Leites vv. 1-33 (1997).
TRANSITION AMPLITUDES FOR TIME-DEPENDENT LINEAR HARMONIC OSCILLATOR WITH LINEAR TIME-DEPENDENT TERMS ADDED TO THE HAMILTONIAN M. A. FlASHID Centre for Advanced Mathematics and Physics (CAMP), National University of Sciences €4 Technology, EME Campus, Peshawar Road, Rawalpindi, Pakistan E-mail address: [email protected] Techniques developed earlier t o obtain the transition amplitudes for a general timedependent linear harmonic oscillator using standard operator techniques, are used to calculate transition amplitudes incorporating a linear timedependent term in the Hamiltonian. This allows for odd t o even and even to odd transitions which were not allowed when these linear terms were absent.
1. Introduction Time-dependent harmonic oscillators have been considered by many authors [1- 41. From the point of view of a physical application, Parker [5] applied the alpha and beta coefficients of the problem to the cosmological creation of particles in an expanding universe. Earlier, Kanai [a] had considered a simple form of the time-dependent linear oscillator. Though this model was criticized by Brittin [6] and Senitzky [7] for various reasons, Landovitz et al, ignoring the criticism, proceeded to calculate the Green’s function [8] for the general form of Kanai’s model and used it to calculate the corresponding transition amplitudes [9]. Their calculations are very difficult to comprehend. Recently we used standard operators to calculate the transition amplitudes for the general time-dependent linear harmonic oscillator in a transparent manner [lo]. This approach is expected to be relevant to other physical problems including Senitzky’s [7] complex model for the dissipative quantum mechanical oscillator. Here these manifest operator techniques are used to obtain the transition amplitudes after adding linear time-dependent
39
40
terms to the Hamiltonian which allow for transitions from odd to even and even to odd states. These transitions were not allowed under the even-parity Hamiltonian which did not include the linear terms. This paper is organised as follows. In Section 2, we introduce the modified Hamiltonian and obtain the transformed operator x i ( t ) , p + ( t ) in terms of the non-transformed time-dependent operators x and p and the coeficients in the"transition matrix". In Section 3, we calculate the corresponding transformed creation and annihilation operators. In Section 4, we derive the recursion relations satisfied by these transition amplitudes. These recursion relations are used in Section 5 to calculate a generating function which gives the transition amplitudes in term of the initial one. This initial one is then evaluated using its relationship with a known identity to complete the calculation. 2.
The Transformed Operators z-J-, p*
The Hamiltonian for a time-dependent linear harmonic oscillator with linear time-dependent terms added is given by
In the above, the functions f ( t ) , g ( t ) , u(t) and v ( t ) are all real continuous functions to make the Hamiltonian hermitian. Also f ( t ) = g ( t ) = 1, u ( t )= w ( t ) = 0 gives the usual time-independent linear harmonic oscillator Hamiltonian whereas putting u ( t ) = w ( t ) = 0 reduces to our earlier work [lo]. The wave-functions at an arbitrary time t are related to the ones at time t = 0 through a time-dependent unitary transformation U ( t ) by
Q ( x ,t ) = U ( t ) Q ( X O), ,
(2)
where U ( t ) satisfies the Schrodinger equation, i h & U ( t ) = H ( t ) U ( t ) . We define the operators O+(t)corresponding to any operator O ( t ) (which may have a manifest time-dependence) by
O*(t) = U t ( t ) O ( t ) U ( t ) , O _ ( t )= U ( t ) O ( t ) U t ( t ) .
(3)
These operators satisfy
d 1 -O-(t) = - [H(t),O-(t)] dt ah
+
(5)
41 The operators a+@), p + ( t ) are related to z and p by
+
+
z+(t)= u+(t)zu(t) = a ( t ) z b ( t ) p y1(t), p + ( t ) = u+(t)Pu(t) = c ( t ) z+ d ( t ) P + y2(t), where
a ( 0 ) = d(0) = 1, b(0) = c(0) = 0 , a ( t ) d ( t )- b(t)c(t)= 1 Using 1 dz+ [ z + ( t ) , H + ( t )=] f ( t )P+, ( 4 + u(t) at = ih and the expressions for z+(t),p + ( t ) in equation (6) we arrive at
Similarly from
we obtain
Yz(t) = - g ( t ) m w 2 y l ( t ) - v ( t ) . Equations (8) and (11)yield second order differential equations
Z ( t )- f ' O ( t ) + f (t)g ( t )w"(t) f (t) for the functions a ( t ) , b ( t ) and for c ( t ) ,d ( t ) we get .'
g'(t)
'
Z ( t ) - ---Z(t) 9 (4
=0
+ f ( t )g ( t )W 2Z ( t)= 0.
Equations ( 8 , l l ) have unique solutions, given the initial conditions, a ( 0 ) = d ( 0 ) = 1,b(0) = c(0) = 0. For the time independent case (f ( t )= g ( t )= 1), these are 1 a ( t )= d ( t )= cos w t , b ( t )= -sin w t , c ( t )= - mw sinwt. (15) mw
42
For completeness, we note that y1 ( t ),y2 ( t ) satisfy the non-homogeneous second order differential equations
3. Calculation of the transformed creation and annihilation
operators The non-Hermitian creation and annihilation operators A+ and A are related to the Hermitian operators z and p through
At
=
(p
J5aG
-i + i m w z ) , A = d%zz
( p- imws)
(18)
In terms of the energy eigenstates of the Hamiltonian H = H ( 0 ) given by
the operators A and At have matrix elements
< m IA In >= &S,,-1,
< m IAt In >= &&,+I.
(20)
Next we compute the transformed operatores A+ ( t )and A 1 ( t )in terms of the functions u ( t ) , b ( t ) , c ( t ) ,d(t),yl(t), y 2 ( t ) appearing in the transformed operators z+ ( t )and p+ ( t )in eq. (6). Indeed using equations (4,18b, 6) we have
A+ ( t ) = U t ( t ) A U ( t )=
i ~
d5aG
~t ( t )( p - i m w z ) u ( t )
where
i a ( t )= -( c ( t )- i m w (u ( t ) d ( t ) )- m2w2b( t ) ) 2mw
+
(22a)
43 i
p ( t )= (c ( t )- imw ( a ( t )- d ( t ) )+ m2w2b( t ) ) 2mw
(22b)
Similarly,
A; ( t )= U+ ( t )A+ ( t )U ( t )= p* ( t )A
+ a* ( t )A’ + y*.
(23)
Note that using the earlier initial conditions we get y1 (0) = y2 (0) = 0, a (0) = 1,p (0) = y (0) = 0.
(24)
For completness, we give below the expression for A- ( t )and A+ ( t )
A- ( t ) = a* ( t ) A - P ( t ) A ++ y ’ ( t ) , A t ( t )= -p* ( t ) A + a ( t ) A + + y ’ *( t ) (25) where
As in eq. (24), y’(0) = 0 4. Recursion relation for the transition amplitudes amn ( t ) = ( mIU (t)l n) Following the methods of our earlier paper
[lo] and using eqs. (21,25)
1
a m n ( t ) = (mlU(t)ln) = -(mlU(t)A+In-l)
fi
=-[fi 1
-ff
(4 P* ( t )J n - i ( m
( t )fib
+ff
-
Transposing the term
=
1
-(ATIU(t)ln-l)
fi
IU (t)ln - 2) - IPI2fib IU (t)I4 1IU (t)l n - 1) + (-P* ( t )Y ( t )+ 7’’ ( t ) )( mIU (41n - 1)
1
(26) (m IU (t)ln) and noting that unitarity yields
IaI2 - IpI2 = 1 and using equations (32,33) , we arrive a t
(27)
44
A similar procedure results in another recursion relation
5. Calculation of the transition amplitude We define coefficient Bmn(t)by means of the equation
In terms of Bmn(t)the recursion relation in eq. (28) becomes
n Bmn(t) = Bm-zn(t)
1 + -Bm-ln-l(t) + A(t)Bm-ln(t), IPI
where we have used
(31) The recursion relations in eq. (30) are sufficient to determine all Bmn(t) and hence the transition amplitudes up to a (in general) complex constant. To determine Bmn(t),we define a generating function M
m,n=O
where we shall assume that any Bmn(t)with any index taking negative integral value is zero. Then the recursion relation in eqn. (30) gives
which can be easily solved to give
45
Thus,
From equation (32) Bmn(t)is the coefficient of xm, yn in the above expansion and fix k and 1 as
k=
m-p-q 2
,1=
n-p-r 2
,
(37)
This results in
(38) where the summations over p , q , r are restricted by the argument of any factorial present in equation (38) to be a non-negative integer. For example, if m is an even integer, then p+q have to be even and m - p - q 2 0. We note, in particular, that when m = n = 0 , p = q = r = 0 and Boo(t) = aoo(t), which is consistent equation (35). From eqs. (29) and (38), we have
which gives amn(t) upto the function aoo(t). To determine this function, we use the standard normalization
But from the eq. (39)
x(-l)T(') 1 n--r- P
ao,(t)
= aoo(t)Jnr
From the above equations,
B*(t)". (y)!r!
(41)
46
where the expression in the bracket is obviously non-negative. Also the summation over n though infinite is convergent as
We can choose the phase such that aoo(t) = This reduces to the case of linear harmonic oscillator when f ( t ) = g ( t ) = 1, u ( t )= v ( t ) = 0.
References 1. P. Camiz, A. Garaldi, C. Marchioro, E. Presulti and E. Scacciatelli, J . Math. Physics 12,2040 (1971). 2. E.Kanai, Prog. Theor. Phys. Kyoto 3, 440 (1948). 3. E.H. Kerner, Can. J . Phys 36, 3719 (1958). 4. W.K.H. Stevens, Proc. Phys. SOC.Lond. 72, 1027 (1958). 5. L. Parker, Phys. Rev. 183,1057 (1969), (see in particular equation (13)). 6. E. Brittin, Phys. Rev. 77, 396 (1950). 7. I.R. Senitzky, Phys. Rev. 119,670 (1960). 8. L.F. Londonvitz L. F, A.M. Levine and W.M. Schreiber, Phys. Rev. A 20, 1162 (1979). 9. L.F. Londonvitz, A.M. Levine and W.M. Schreiber, 1980 J. Math. Phys 21, 2159 (1980). 10. M.A. Rashid and A. Mahmood J . Phys. A : Math. Gen 34, 8185 (2001).
ENTROPY SOLUTIONS TO A GENUINELY NONLINEAR ULTRAPARABOLIC KOLMOGOROV-TYPE EQUATION S. A. SAZHENKOV' Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Peshawar Road, Rawalpindi, Pakistan E-mail: [email protected] We consider a non-isotropic convection-diffusion-reaction equation of a very general form, in which the diffusion matrix is nonnegative and may change its rank depending on temporal and spatial variables, and convection and reaction terms may be discontinuous. This equation arises in astrophysics and plasma physics, in fluid dynamics, mathematical biology and financial mathematics. We assume that the equation a priori admits the maximum principle and is genuinely nonlinear, and we prove that there exists at least one entropy solution and that the genuinely nonlinear structure of the equation rules out fine oscillatory regimes in entropy solutions. The proofs rely on the method of kinetic equation and on theory of H-measures. Keywords: Ultra-parabolic equation; Vlasov-Fokker-Planck equation; Entropy solution; Genuine nonlinearity; Non-isotropic diffusion.
1. Introduction
In a space-time layer II := IW: x (O,T),T = const > 0, we consider the Cauchy problem for the quasilinear equation with partial diffusion and discontinuous convection and reaction terms
ut+&ai(x,t,u) - a 3 c , ( a i j ( ~ , t ) d , j b ( u+ ) )r ( x , t , u )= 0, (la) endowed with periodic initial data belonging to Lw(IWd) and periodicity conditions ult=o = uo(x), x E
u(x
IWd ,
+ ei,t ) = u(x,t ) , (x,t ) E II.
(1b) (1c)
*Permanent affiliation and address is : Lavrentiev Institute of Hydrodynamics, Prospekt Lavrentieva 15, Novosibirsk 630090, Russia.
47
48
In (1)ei (i = 1,...,d ) are standard basis vectors in Rd, u ( x ,t ) is an unknown function; the flux vector a := ( a i ) , the diffusion matrix A := ( a i j ) , the diffusion function b, and the reaction function r are given and satisfy the conditions
ai,Dx,ai,r E ~ ~ o ~ ( I R C ~ o c ( Raij u ) ) , C;oc(n), b E C,2,,-(R), (2) ai, aij, r are 1-periodic in x , A is symmetric, (3) aij(x,t)
In particular, the derivatives a,, and Dxi are connected via identity
&,g(x,t , u)= Dz&,
t ,).
+ a U g ( x , t ,.)az%u.
It is supposed that the rank of the diffusion matrix A may be less than the dimension of the space R : and may vary depending on x and t. Therefore Eq. (la) is an ultraparabolic equation. Equations of the form (la) arise in particle physics, fluid dynamics, combustion theory, mathematical biology, and financial mathematics. They are called Kolmogorov-type equations in line with the works of A. N. Kolmogorov relating to problems on stochastic diffusive processes modeling Brownian motion (see in survey Ref. 2). Particular forms of Eq. (la) have also other names: in problems about nonlinear convection-diffusion-reactionin anisotropic continuous media they are called Graetz-Nusselt equations and in studies of the transport of cosmic-rays they are named Fokker-Planck equations. They describe, in particular, non-stationary transport of energy or matter in cases, when effects of diffusion in some spatial directions are negligible as compared to convection and reaction. Considerably recently Kolmogorov-type equations have been applied to astrophysical problems: in solar physics with the investigation of acceleration of fast electrons in the solar corona and in space physics with ion acceleration at the solar termination shock and with particle acceleration at astrophysical shocks, including the possibility of second-order Fermi a~celeration.~
49
2. Notion of entropy solutions
We are interested in developing the existence and qualitative theory for the Cauchy problem (1) under conditions (2)-(4) and under the assumption that the maximum principle is a priori guaranteed. From the physical point of view, the most appropriate concept of solution to problem (1) is the notion of entropy solutions, since it is consistent with the fundamental fact that in diffusive processes entropy does not d e ~ r e a s e . ~ ? ~ In order to define an entropy solution of problem (l),let us introduce some notation. By Q we denote i2 x (O,T),where R stands for the unit cube [0, l)d. By LP c Lroc(Rd)and HS9p c H&dZ(Rd) we denote the Banach spaces, which consist of 1-periodic functions and are supplemented with the norms llull~p= I I U I I ~ - , ~ ( ~ and I I U I I H ~ , ~ = I I U I I H ~ , ~ ( ~ . For 1 L 0, let C‘ be the closed subspace of u E C’(Rd) such that u is 1-periodic with respect to xi, 1 5 i 5 d. Note that since Eq. (la) is degenerate, then the gradient V,u of a possible solution u E L”(II) may be understood merely in the distributions sense. However, also note that since the matrix A is symmetric and nonnegative, then there is a unique square root A1/2 = { a i j } ,which is a symmetric and nonnegative matrix, as well. This and the standard energy estimate’ yield that a possible solution u of problem (1) should a priori satisfy the bound IIA1/2V,p(u)llL2(Q)5 c, where the constant c does not depend on u and p(u) := m d s . This means that, although a particular derivative d Z i u may not be measurable on II, the differential expressions of the form cxijd,,p(u) involving these derivatives are measurable in II and integrable with the square in Q, i.e., they belong to L&,(II). Therefore the demand of partial integrability of V,u should be introduced into a notion of entropy solution. Now we are in a position to define an entropy solution of problem (1).
s”
Definition 2.1. Function u = u ( x ,t ) is an entropy solution of problem (l), if it satisfies the regularity and periodicity conditions u E LM(O,T ;L’) and cxijdZjp(u)E L2(0,T ;L 2 ) ,1 5 i 5 d; the entropy inequality
dtcp(u) + dzaqi(X, 4 u)- DZ,Qi(X,t ,).
+ cp’(u)D,iai(x, t ,u)
in the distributions sense for all functions cp, qi and w such that cp E C~oc(R), cp”(u) 2 0 , duqi(x, t ,u)= cp’(u)duai(x,t ,u ) , and w’(u)= cp’(u)b’(u);and
50
the initial data (lb) in the weak sense, i.e., in the sense of the limiting relation u ( - , 7 )4 uo(.)weakly* in L”, as T \ 0. Note that taking p(u) = f u in (5) we see that entropy solution satisfies Eq. ( l a ) in the sense of distributions. 3. Formulation of the main results Besides conditions (2)-(4), the following genuine nonlinearity condition is imposed on the functions ai, a i j , and h. Condition 3.1. For a.e. ( x , t )E Il the following demand is fulfilled: for all ( 6 , ~ E ) Rd+’ such that I6I2 72 = 1 the intersection of the sets {A E R I h’(A)aij(x,t)Ci& = 0 ) and {A E R I T (dAai(x,t,X) (l/2)h’(A)dz, a i j ( x , t ) ) [ i = 0 ) has zero Lebesgue measure.
+
+
+
The following new existence theorem is the first main result of the paper. Theorem 3.1. Assume that Eq. (la) is genuinely nonlinear, satisfies conditions (2)-(4) and a priori admits the maximum principle. Then problem (1) has at least one entropy solution for any given data uo E Loo. Also we establish a qualitative property of genuine nonlinearity to rule out fine oscillations developing from initial data, which is the second main result of the paper. Theorem 3.2. Assume that Eq. (la) is genuinely nonlinear, satisfies conditions (2)-(4), a priori admits the maximum principle, and is provided with highly oscillatory initial data uk E L““, k = 1 , 2 , .. . such that uk 4 uo weakly* in L” as k 703. Then there exists a subsequence of entropy solutions u k , corresponding ; ) as k /” 03 to an to initial data uk, which tends strongly in L 2 ( 0 , T L2 entropy solution u, corresponding to initial data U O . 4. Method of justification of Theorems 3.1 and 3.2
Proofs of Theorems 3.1 and 3.2 rely upon the method of kinetic e q ~ a t i o n which allows to reduce quasilinear equations and systems to linear scalar equations on “distribution” functions involving additional “kinetic” variables. Alongside this method, the theory of H - m e a s u r e ~is~implemented. ~~ In this final section we give a brief explanation of the methodology of the proofs of Theorems 3.1 and 3.2.
51
First, the kinetic formulation of problem (1) is introduced in the form proposed in Ref. 6. It is linear with respect to the desired function, which is the distribution function f (x,t , A) = f z , t ( X ) of the parametrized Dirac measure on concentrated at the point X = u(x, t ) , where u(x,t ) is the entropy solution of problem (1).Notion of entropy solution and the kinetic formulation are equivalent to each other. Linearity of the kinetic formulation becomes possible due to appearance of an additional kinetic variable A. Second, the proof of Theorem 3.2 is fulfilled by virtue of the toolbox of the theory of H-measures. The construction of H-measures associated with a weakly convergent subsequence of the distribution functions fk(x,t , A), Ic = 1 , 2 , . . . is introduced in the form which was proposed in Ref. 8 for studying scalar conservation laws. By their nature, H-measures are microlocal defect measures that allow to track evolution of fine oscillatory regimes in the space of time t , positions x and frequencies E . More precisely, for any fixed X E R they indicate where in the physical space of time and positions, and at which frequencies in the Fourier space, weakly convergent in Lfo, sequences fail to converge strongly. One of the main properties of H-measures is that they are zero measures if and only if their generating subsequence is strongly convergent. Using the technics of Ref. 10, it is possible to establish a localization principle for the H-measures, i.e., to define a set in the (x,t,<)-spacesuch that the H-measures vanish in its complement. This principle combined with the genuine nonlinearity condition immediately imply that the H-measures are equal to zero measure everywhere for almost all X E R.Hence their generating subsequence is compact, which finishes the proof of Theorem 3.2. Finally, in order to prove Theorem 3.1, we introduce the well-posed approximation of problem (1), which incorporates a regularizing ‘‘small viscosity” coefficient. Availability of such approximation is guaranteed by the well-known theory of parabolic equations.’ After this, it is sufficient to remark that the approximate problem admits the kinetic formulation of exactly the same form as problem (1). Thus the rest of the proof of Theorem 3.1 is merely the byproduct of the proof of Theorem 3.2.
Acknowledgment The work was partially supported by the grant of Higher Education Commission of Pakistan under National Research Program for Universities (project title: Modern Mathematical Analysis for Phenomenon of Anisotropic Diffusion and Acoustic Wave Propagation in Porous Media).
52
References 1. 0. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type (Amer. Math. SOC.,Providence, RI, 1968). 2. E. Lanconelli, A. Pascucci, S. Polidoro, in Nonlinear Problems in Mathematical Physics and Related Topics 11, I n Honor of Professor O.A. Ladyzhenskaya, International Mathematical Series (Kluwer Academic, Dordrecht, 2002). 3. A. Marcowith, J.G. Kirk, Astron. Astrophys. 347,391 (1999). 4. L. D. Landau, E. M. Lifshitz, Statistical Physics, Part I, vol. 5 of Course of Theoretical Physics (Butterworth-Heinemann, Oxford, 1999). 5. L. D. Landau, E. M. Lifshitz, Fluid Mechanics, vol. 6 of Course of Theoretical Physics (Pergamon Press, Oxford, 1959). 6. G.-Q. Chen, B. Perthame, Ann. Inst. H. Poincare' Anal. Non Line'aire 2 0 , 645 (2003). 7. P. I. Plotnikov, S. A. Sazhenkov, J. Math. Anal. Appl. 304,703 (2005). 8. E. Yu. Panov, Sbornik Math. 81,211 (1995). 9. L. Tartar, Proc. R. SOC.Edinb. 115A,193 (1990). 10. S. A. Sazhenkov, Siberian Math. J. 47,355 (2006).
FOKKER-PLANCK-KOLMOGOROV EQUATION FOR fBm: DERIVATION AND ANALYTICAL SOLUTIONS. GAZANFERUNAL Faculty of Sciences, Istanbul Technical University, Maslak 34469. Turkey E-mail: gunalOitu.edu. tr The Fokker-Planck-Kolmogorov (FPK) equation for It6 systems with fractional Brownian motion (fE3m) has been derived. The generalized Liouville theorem has been proved. Analytic solutions to the FPK have been obtained via conserved quantities of the deterministic part.
1. Introduction
F'ractional Brownian motion (fBm) was introduced by Kolmogorov to study turbulence in an incompressible fluid flow'. Long range dependence and selfsimilarity properties of fBm announced itself in the Nile river level studies of Hurst2. The parameter H E ( 0 , l ) which scales fBm BF (t 2 0) is called the Hurst parameter. Mandelbrot and van Ness studied properties of the fBm3. Standard Brownian motion becomes a special case of the fBm with the Hurst parameter H = 1/2. In contrast with the standard Brownian motion, increments of BF (t 2 0) are no longer statistically independent for nonoverlapping intervals of t. The correlation function becomes negative for H E (0,1/2) and it is known as antipersistent behaviour3. This is also known as intermittency in the study of turbulent fluid flows'.The correlation between the increments of BY (t 2 0) for non-overlapping intervals become positive for H E (1/2,1). It shows long-range dependence i.e. it becomes persistent. We now witness the foot prints of the fBm BF (t 2 0) for H E (1/2,1) ranging from the characteristics of solar activity to weather derivatives in mathematical finance4. An fl3m BF (t 2 0) is neither a semi-martingale6 nor a Markov process7. Therefore, a new stochastic calculus is needed for its treatment. It6 calculus for H E (0,1/2) has been given in reference [5] and for H E (1/2,1) in references [6] and [7]. Recently, Bender' has developed
53
54
a new calculus which is valid for H E ( 0 , l ) (see also reference [9] for an alternative approach). Here we rely on Bender's It6 formula in this paper. Here we consider It6 stochastic ordinary differential equation (SODE) of the form
+
dxi = fi(x,t)dt gia(x,t ) d B f , 1 I iI n;1 I aI T,
(1)
where f i ( x ,t ) is a drift vector and gi,(x, t ) is a diffusion matrix and, dB," is increment of fBm (summation convention applies to repeated indices and drops for indices in paranthesis hereafter). Based on an It6 lemmas~g which is valid for H E ( 0 , l ) we derive the FPK equation. Furthermore the Liouville theorem has been extended to It6 SODE given in equation (1). Finally we have obtained analytic solutions to the FPK equation via conserved quantities of the deterministic. This result leads to the fluctuationdissipation theorem
'.
1.1. Derivation of the FPK equation for f B m It6's formula for fBm for a scalar function h ( x ) reads as
The expectation value of Eq. (2) is
H t Z H - ' g j , gd2h k , ~ ] dt+E [gja%dBf1 dh
Therefore, we have
We now recall
This leads to
55
Let us treat each term on the right hand side separately.
where circumflex d i j denotes the omission of the variable. Integration by parts of the integral yields
The first term on the right hand side must vanish since p = 0 a t x j = f m . Hence,
Similarly we now treat the second integral in equation (7) to obtain
where
aij = g j a g k a .
Rendering back Eq. (10) and (11) into Eq. (7) yields
from which we deduce that
This is the celebrated FPK equation for It6 systems driven by fBm. Notice that when H = 1/2 Eq.(13) reduces to the celebrated Fokker-Planck equation for It6 system driven by Brownian motion. 2. Generalized Liouville theorem The Liouville theorem has been extended to the Langevin equations with standard Brownian motion by ChandrasekharlO. Here we extend it to It6 SODEs with tBm. We first define the following vector fields
H=hj-
d
dXj
and
d G,=gj,---, dXj
56
where
We now calculate i)
a
% ( p ( x , t ) R ) ; i i ) ~ H ( p ( xt , ) ~ ) iii) ; ~ 2 , ( p ( xt ), R ) ;
(15)
where p ( x , t ) is a smooth function, R = d x l A . . . A d x , is a volume form, C,is a Lie derivative with respect to the vector field given in the subscript and
~ 2 =, cG,cG,. Straightforward calculations for i) and ii) in Equation (15) lead to i)
zd( p ( x , t ) R ) = ( azP) R ,
ii) &(PO) =
In order to calculate the term iii) in Equation (15), we first evaluate CG, (Po):
We then calculate the Lie derivative of Equation (17) with respect to the vector field G, and sum over Q to obtain
Equations (16-18) are combined to give
where
Notice that 0 is an expansion of the term
57
We now rewrite Equation (19) as
Suppose that the function p ( x , t ) satisfies the FPK equation (13). Then the RHS of Equation (20) vanishes. If the LHS of Equation (20) vanishes, the function p ( x , t ) should satisfy the FPK equation given in (13). Hence, we have proven the following theorem.
Theorem 2.1. A sufficiently smooth function p ( x ,t ) satisfies
(g+LH--L 2 G2 , p R = O iff it satisfies the FPK equation given by Equation (13). Remark 2.1. Theorem 2.1 reduces to
for a deterministic dynamical system with the vector field
F=
fj-.
a
dXj
This is, indeed, the celebrated Liouville’s theorem which can be found in references [ll]and [12]. Therefore, Theorem 2.1 can be considered as an extension of Liouville’s theorem to stochastic systems. 3. Analytical solutions to the modified FPK equation
In this section we will make use of the generalized Liouville theorem to obtain the exact solutions to the modified FPK equation.
Proposition 3.1. Let the It6 S O D E with fBm be given as
dI dxi = f i ( x , t ) d t - d ( i ) ( x l , ...,&, ...,z,,t)-dt+g(i)(x1, dXi
..., &, . . . , X n , t ) d B y ,
(22) where caret denotes omission, I ( x , t ) = c r = l ckIrc(x,t ) , f j , j = Cons and C k s are constants, and
58
Then p ( x , t ) = Ae-(b'Sf3Jt), A and b are constants
is a solution to modified FPK equation (13) provided that di(x1, . . . , ~ i , . . . ,~ , , t )= bHt 2H-1 gi(xl,...,~i-i,...,x,,t) 2 .
(24) (25)
Proof. Let us define the following vector fields
Notice that div C =Cons , diw G,=O and
gkagia,k = 0. (26) This leads to H = F . We now appeal to the Liouville theorem which was proved earlier. To achieve this goal we calculate
i) zd( p ( x ,t ) R ) = ( azP) R , ii) , C F ( P ~ = ) ( L C P ) ~p(divC)R+ ( L D P ) R
+
()::
+p(divD)R, iii) L&(pR)= g i 7
R.
Rendering them back into (21) yields
The terms in the paranthesis must be zero for this relation to hold. Let p = 7r(I)e-(di"c)t where m
k=l
Here I k ( x ,t ) satisfy the condition given in (23). This leads to
Let us now use the condition given in (25). We now have
d7T
dl = -bI.
Solution to this equation leads to (24) thereby proving the proposition.
Remark 3.1. Constant A will be determined from the normalization condition i.e..
59
77 .. .
A
-w
e-('CEI Ck'k(X,t)f(divC)t)dlC1. . . d s ,
1.
:
-w
Therefore the constants ck must be chosen such that the function p(x,t ) is normalizable, so that it can represent a probability density function.
Corollary 3.1. Let the It6 SODE with fBm be given as
+ g ( i ) ( q , P l , ...,ljil
...IP,, t)dB?
(27)
where ' H ( p , q ) is the Hamiltonian and (i = l...n). The associated modified FPK equation (13) has a stationay solution of the form p ( p , q) = Ae-bx7 A and b are constants
(28)
provided that di(q,~l,...,lji,...,~,lt) = bHt
2H-1
2
gi(q,pl,...,lji,...,~,,t) *
(29)
Proof: Notice that the diffusion matrix enjoys the property (26). Letting
c =--ax a a ~ aqj j
-
ax a
--
aqj
a~j
in the proof of the proposition leads to the required result.
Remark 3.2. Notice that when the Hurst parameter H = 1/2 and, g i and di are constants in (29) leads to the celebrated fluctuation-dissipation relation
Acknowledgments Financial support provided by the conference organizers is gratefully acknowledged.
References 1. A. N. Kolmogorov, c. R. (Doklady) Acad. Sci. URSS (N.S.). 26, 115 (1940). 2. H.E. Hurst et al, Constable, London (1965). 3. B. B. Mandelbrot and J.W. Van Ness, S I A M Review 10, 422 (1968).
60
4. A. Shiryaev, in Workshop on Mathematical Finance, eds. A Shiryaev and A Sulem ( INRIA, Paris, 1998). 5. E. Al'os, 0. Mazet and D. Nualart, Stochastic Processes and their Applications 86,121 (2000). T. E. Duncan, Nonlinear Analysis 47, 4775 (2001). E. Al'os and D. Nualart, Stochastics and Stochastics Reports 75, 129 (2003). C. Bender, Stochastic Processes and their Applications 104,81 (2003). F. Biagini, B. Oksendal, B., A. Sulem and N. Wallner, Proc. R. SOC.Lond. A 460, 347 (2004). 10. S. Chandrasekhar, Rev. Mod. Phys. 15,2 (1943). 11. W.H. Steeb, Foundations of Physics 10,485 (1985). 12. G. Unal, Phys. Lett. A 233, 193 (1997). 6. 7. 8. 9.
Particle Physics and String Theory
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HOLOGRAPHY AND DE SITTER SPACE MOHSEN ALISHAHIHA
Institute for Studies in Theoretical Physics and Mathematics (IPM), P . 0 . Box 19395-5531, Tehran, Iran E-mail: alishahOtheory.ipm.ac.ir In this talk I will review the simplest derivation of the holomorphic principle and the role of AdS/CFT correspondence in understanding it. I will then generalize it for de Sitter space where I present a holographic dual of gravity on de Sitter space in the static patch.
1. Holographic principle and AdS/CFT correspondence Let us start with one of our favorite equations which has been at the center of attention in theoretical physics for decades, namely the BekensteinHawking area-entropy law formula
s = 4G -A
'
This means that the entropy of a black hole, with area of the horizon A , is given by quarter of the area in natural units. One interesting feature of this formula is its universality and in fact it applies to all kinds of black holes. It is universal in the sense that it is independent of the specific characteristics and composition of the matter system. More interestingly it even applies for cosmological horizons like de Sitter space. We note, however, that its validity is not truly universal, namely it applies only when there is weakly coupled gravity. In particular if we consider higher order corrections to the Einstein-Hilbert action we will get a deviation from the area law. In this case one needs to use the Wald formula for the entropy l. Now the question we would like to ask is whether this equation could put a bound on the entropy of a system with gravitational interaction. To do this we closely follow the notation of papers by 't Hooft2 and Susskind3. To proceed we consider a spherically symmetric weakly coupled gravitational system. We will also assume that the space time is asymptotically flat.
63
64
Now with these assumptions consider an isolated matter system with mass M and entropy S,. Let us denote by A the area of the smallest sphere that fits around the system. To get a stable system M must be less than the mass of a black hole of the same area. Now consider a shell with mass m such that adding it to the system converts it to a black hole. Therefore the initial entropy of the system is initial Stotal -
sm + Sshi
(2)
s&
where is the entropy of the shell. Since the system is a black hole after adding the shell, the final entropy of the system is given by final Stotai = SBH =
A
-. 4G
(3)
On the other hand we note that the initial entropy must not exceed the final entropy so that S, s& I Taking into account that s s h > 0 leads one to the “spherical entropy bound” as follows
+
&.
s
A 4G
<-.
-
(4)
From a thermodynamical point of view entropy has a statistical interpretation. To be precise es is the number of independent quantum states on the volume surrounded by A. Therefore the entropy bound will put a bound on the number of degrees of freedom. In fact it seems that degrees of freedom are sufficient to fully describe any stable region enclosed by a sphere of area A. As a conclusion we are led to the holographic principle which states: A region with boundary of area A is f i l l y described by n o more than degrees of freedom or about 1 bit of information per Planclc area. Most of our intuitions about the holographic principle have come from AdS/CFT correspondence. Roughly speaking one may summarize the statement of AdS/CFT correspondence as follows. A d 1 dimensional theory which includes gravity is “dual” to a d dimensional theory without gravity. The prototype example of this correspondence comes from type IIB string theory on an A d s 5 background. More precisely the conjecture says: Type IIB string theory on A d s 5 x S5 with N units of five-form flux on S5 is dual to = 4 SYM theory an four dimensions with gauge group
6
+
U(N). Let us explain different points which are involved in this conjecture (1) Ads5 is a maximally symmetric solution of the Einstein action with
65
negative cosmological constant whose metric is given by
- + -dU2 R2
U2 ds2 = - d X 2 r2
u2
in Poincar6 coordinates and
ds2 = R2(cosh2pdt2
+ dp2 + sinh2 pdR;)
in global coordinates.
(2) The dual gauge theory lives on the boundary of the AdS5. Therefore if we take the Poincar6 coordinates the dual gauge theory is defined on R4 while in the global coordinate it is defined on R1 x S3. (3) There is one to one correspondence between objects of gauge theory and those of gravity which we summarize in table 1.
Table 1. Correspondence between objects of gauge theory and gravity ~
String theory side
Gauge theory side
String coupling Curvature S0(2,4) x SO(6) fields, strings, ... Partition functions
N 't Hooft coupling SC-group x R-symmetry Gauge inv. operators Generating functions
To see how the correspondence works consider for example a scalar field q5 with mass m such that dlboundary = $0. According to this correspondence one has
and the conformal dimension of the corresponding operator 0 , A, is given in terms of d and m: A = $ ( d d -).
+
2. Holography for de Sitter space In this section we would like t o see how one can find a holographic dual for gravity on de Sitter space. In this section we will present two different holographic descriptions for gravity on de Sitter space in two different coordinates. So let us first review a few facts about de Sitter space.
66
2.1.
What is de Sitter space?
de Sitter space is the maximally symmetric solution of the Einstein equations with positive cosmological constant. In fact the d-dimensional de Sitter space (dsd) can be realized as a hypersurface described by the following algebraic equation in flat d 1 dimensional Minkowski space
+
The dS metric is the induced metric from the flat space
Comparing the corresponding metric with the Einstein equations which are written in terms of the cosmological constant one finds a relation between the cosmological constant and the radius of the dS space as follows
A=
(d - l ) ( d- 2 ) 212
+
Therefore the dSd in the flat d 1 dimensional Minkowski space is a hyperboloid. The constraint ( 5 ) can be solved in different ways leading to different coordinate systems for dS space. Different coordinate systems are good for different purpose. The dS metric in global coordinates is given by
d S 2 = -dr2
+ l 2 cosh2 71 d R i - ,
As we see in these coordinates dsd space looks like a d - 1 sphere which starts out infinitely large at r = -00 then shrinks to its minimum finite size that r = 0 and then grows again to infinite size at r = 00. 2 . 2 . W h y de Sitter space? Having had enough information about dS space it is natural to ask why we are interested in dS space? Actually the recent astronomical observations indicate that the cosmological constant in our universe is not zero. In fact not only it is not zero but also its contribution is quite important and in fact it is responsible for almost 73% of the energy of the universe ( dark energy). If this interpretation of the observational data is correct, it means our universe might currently be in the de Sitter phase. Therefore it is worthwhile to study (quantum) gravity in de Sitter space.
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Another interesting feature of dS space is that it has a cosmological horizon in which one can associate a temperature and an entropy which behaves very similar to the black hole area law
s = 4G -A ’
(7)
where A is the area of the cosmological horizon. This is the same as for a black hole and we would like to understand the origin of this entropy for dS as well.
2.3. d S / C F T correspondence From what we have learned in AdS/CFT correspondence one may hope that some kind of holography can also be applied here and could help us to understand quantum gravity on dS. In fact there is a naive observation. Namely consider an Ads space with radius R. Under R -+ iR one gets
A Ads SO(2,d)
+
-
+
-A dS SO(l,d+l)
Therefore one may conclude that gravity on dS is dual to a Euclidean CFT. Although this is just a naive observation and indeed it does not mean that we can study dS space by just an analytic continuationa, one can still think about it as an insight and try to make this statement as precise as we can. In fact this is the content of what is known as dS/CFT correspondence5. To make this correspondence precise let us consider the Brown-York stress tensor6, TP”,for a space time which is asymptotically dS7. One may ask the following questions: ( 1 ) What would be the boundary conditions for the metric if we want the stress tensor to be finite? ( 2 ) What is the most general diffeomorphism which preserves these boundary conditions?
The first question can be answered by perturbing dS space and computing T”” and then one may answer the second question. The answer to the second question is that the most general diffeomorphism is the conformal group of the ( d - 1)-dimensional Euclidean space. aIn particular we note that under this transformation there is no one to one correspondence between representations of SO(2,d ) and SO(1, d 1).
+
68
This is one of the main hints of the dS/CFT correspondence which says5:Quantum gravity o n dSd is dual t o a ( d - 1)-dimensional Euclidean conformal field theory residing o n the past boundary I- of dsd. We note that this CFT may be non-unitary! We note also that in studying this correspondence one uses the dS in the planar metric. So it is natural to ask: How would the holography work if we picked another coordinate system? In the next subsection we are going t o answer this question by studying gravity on dS space in the static coordinates. 2.4. d S / d S correspondence
In this subsection we would like to give a new holographic picture for gravity on dS in the static coordinate^'^^. The static patch in d dimensional de Sitter space of radius R can be foliated by dSd-1 slices:
ds2
= sin2
(i)
ds2sd-l
+ dW2.
The resulting metric (8) has a warp factor which is maximal with finite value at a central slice w = 1rR/2,dropping monotonically on each side until it reaches zero at the horizon w = 0, nR (see Fig. 1).The region near dS, static patch (spatial)
Localized graviton
E=O
1 R
E= -
E=O
Fig. 1. dSd-1 slicing of dsd.
the horizon, which corresponds to low energies in the static coordinates, is
69
isomorphic to that of d dimensional Ads space foliated by dsd-1 slices (for which the warp factor is sinh2(w/R) rather than sin2(w/R)) and hence constitutes a CFT on dsd-1 at low energies. Correspondingly, D-brane probes of this region exhibit the same rich dynamics as a strongly coupled CFT on its approximate Coulomb branch. Meanwhile, probes constructed from bulk gravitons range from energy 0 up to energy 1 / R at the central slice, and upon dimensional reduction their spectrum exhibits the mass gap expected of d - 1 dimensional conformal field theory on de Sitter space. Dimensionally reducing to the d - 1 dimensional effective field theory also yields a finite d - 1 dimensional Planck mass, so the lower dimensional theory itself includes dynamical gravity. Altogether, the geometry and energy scales are as summarized in Fig. 1. This leads to the following statement of de Sitter holography: The dsd static patch is dual to two conformal field theories on dSd-1 (hence thermal with temperature T = l/R), cut off at an energy scale 1 / R and coupled to each other as well as to (d - 1)-dimensional gravity. As it stands, restricting ourselves to times short compared to the decay time, and codimension one, this type of duality is analogous to the holographically dual description of the Randall-Sundrum or warped compactification geometries or, more precisely, the multi-throated versions. That is, the d-dimensional gravity theory is dual to a d - 1 dimensional cut off quantum field theory coupled to gravity. Both sides of the correspondence involve gravity (in different dimensions); nonetheless the holographic relation is useful because the bulk of the entropy is carried by the field theoretic degrees of freedom. This analogous RS case is reviewed in Fig. 2. At higher
"~
Localized graviton
Fig. 2.
&,
RS picture of dS/dS correspondence
energies E + Ads and dS differ. In AdSd-1, the warp factor diverges towards the UV region of the geometry (far away from the horizon) and d - 1 dimensional gravity decouples. In our case, as in Randall-Sundrum, the warp factor is bounded in the solution and one finds a dynamical d - 1 dimensional graviton. In the Randall-Sundrum construction, one truncates
70
the warp factor at a finite value of the radial coordinate by including a brane source (or a compactification manifold) with extra degrees of freedom. In the dS case, the additional brane source is unnecessary; a smooth UV brane at which the warp factor turns around is built in to the geometry. As reviewed above, the d - 1dimensional holographic dual of dSd is only a pair of CFTs up to the energy scale 1/R. On the d dimensional gravity side of the correspondence, one has a local effective field theory description good up to the d-dimensional Planck scale Md >> 1/R (or perhaps the bulk string scale in a stringy construction). At scales above Md quantum gravity effects become important in the bulk and one has to appropriately UV complete the system, for example by embedding it as a metastable dS into string theory following one of the constructions Most of our analysis now will be concerned with using the gravity side of the correspondence to determine the behavior of the d - 1 theory in the range of energies 1 / R < E < hfd. In the d-1 description, the Planck mass is dominated by an induced contribution of order MtIf N S/Rd-3, where S N (MdR)d-2 N ( M ~ - I R ) is the Gibbons-Hawking entropy of dSd and the effective species number (central charge) of our dual low energy CFTs. These scales of interest are summarized in Fig.3. One can use AdS/CFT correspondence to study some
'.
t
BRANE .,.
.. ....................,..,,.
,, ,,,,,,,, ,,
Quantum Gravity
_ _ _ _ _ _ - _ - _ - - - -_- _ - - - - - - - - BULK Classical Gravity in dSd
Md-1
Md
?
_ _ _ _ _ - - - - _ _ _l/L -Gravity + 2 CFTs in dS d-l
Fig. 3.
The correspondence in different energy scales
properties of this duality. In dS slicing the metric of dSd reads
71
while Ad& can be written as
Therefore the two can be related by a simple conformal transformation. 1 d S i d S d = tanh2(z)d 4 S d We can use this to map the physics in dS to dynamics in Ads, albeit with unusual actions. Namely the conformal map yields scalars with position dependent masses and gravity with a position dependent Newton constant. By applying the AdS/CFT dictionary to the resulting system, this allows us to make a direct comparison of the UV behavior of the d - 1 dual of dsd to the UV behavior of a strongly 't Hooft coupled CFT. The results are as follows. Scalar field: We get a conformally coupled scalar in Ads, independent of what values of the parameters M and E we started with in dS! The corresponding UV dimension of the dual operator is d
This ensures that the < 00 > two point function for the second choice behavior of a scalar field in d dimensions. reduces to the usual Gravity: We see that the dS/dS graviton corresponds to a varying Newton's constant in Ads, Md(z) = f (2)Md. As we approach the boundary the gravitational coupling increases. This allows for the localized graviton. Close to the boundary the graviton looks like a flat space graviton! In particular, the possible boundary behaviors are zo and z' , as opposed to zo and z4 for ordinary Ads gravity. This seems to yield two possibilities of the dimension of the dual operator, d - 1 and d - 2. Of course we expect the graviton to couple to the energy-momentum tensor with dimension d - 1. Conformal anomaly: One may also study the conformal anomaly for both dS slicing of Ads and dS slicing of dS. In the first case we get conformal anomaly for dS2, dS4 and dS6, while for the latter case the conformal anomaly is zero. One possible interpretation is that lower dimensional gravity screens the central charge to be zero, just as is well known from 2d gravity on string theory world sheets. In this scenario one does not even need a conformal field theory beyond scales 1 / R since the gravitational dressing will also make any FT a CFT. In the same spirit the universal UV dimension of the scalar fields can be understood as gravitational dressing.
72
3. Conclusions Let us conclude the talk by a few remarks.
(1) Holography might lead to a fundamental theory which includes quantum gravity.
(2) String theory has provided us an explicit example of holography: AdS/CFT correspondence. (3) This not only might help us t o understand quantum gravity and black holes but also it might provide a framework to study non-perturbative gauge theory (QCD). (4) One may also study quantum gravity on dS using holography. ( 5 ) So far there are two proposed holographies for dS space: dS/CFT and dS/dS.
References 1. R. M. Wald, “Black hole entropy in the Noether charge,” Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038]. 2. G. ’t Hooft, “Dimensional reduction in quantum gravity,” gr-qc/9310026. 3. L. Susskind, “The world as a hologram,” hep-th/9409089 4. S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240]. 5. A. Strominger, JHEP 0110,034 (2001) [arXiv:hep-th/0106113]. 6. J. D. Brown and J. W. York, Phys. Rev. D 47, 1407 (1993). 7. V. Balasubramanian,J. de Boer and D. Minic, Phys. Rev. D 6 5 , 123508 (2002) [arxiv:hep-t h/0110108]. 8. M. Alishahiha, A. Karch and E. Silverstein, JHEP 0506, 028 (2005) [arXiv:hep-th/0504056]. 9. M. Alishahiha, A. Karch, E. Silverstein and D. Tong, AIP Conf. Proc. 743, 393 (2005) [arXiv:hep-th/0407125].
BRANE COSMOLOGY WITH STRING ANTISYMMETRIC FIELD F. ARDALAN Physics Department, Sharif University of Technology, Tehran, Iran School of Physics, I w t . for Studies in Theoretical Phys. and Math. (IPM), Tehran, P.O. Box 19395-5531,Iran E-mail: [email protected] physics.ipm.ac.ir The equations of a string inspired noncommutative gravity model on a brane are shown to admit a first integral, generalizing the five dimensional Friedman equation for the FRW Hubble parameter. Keywords: Gravity, Cosmology and Noncommutativity
In spite of extensive work on noncommutative gravity in the past few years, there is very little known about the noncommutative gravity theory induced on a brane, from string theory in the low energy, in the presence of a nonzero antisymmetric B field.” The one result in this area is Ref. [2] whose action is studied here. To present the result a short review of the string-induced gravity of Ref. [2] is in order: Closed strings generically have a massless spectrum consisting of the states
aThere are references to earlier work in Ref. 111 where the failure of the latest proposed noncommutative gravity theory to agree with string low energy is confirmed.
73
74
which give, the g,,
Graviton, symmetric,
the I?,,
Kalb-Ramond, antisymmetric,
and cp
Dilaton.
On the other hand open strings with Neumann boundary conditions give gauge fields in low energy and with Dirichlet boundary conditions lead to D-branes. Existence of branes provides for a simple description of our world, the brane world, where gauge fields live on 3-branes, and gravitons, cp, and B,, live in the bulk. It is known that, generally, in the low energy theory of string theory there are
1- Yang-Mills fields, with the action
C = FpyFPu , F,,
= a,A, - &A,
+ [A,, A,]
and 2- Gravity metric g,,
field with Einstein-Hilbert Lagrangian
R = R,V~pgpAgVP
C = &R,
In the presence of a constant nonzero B,, become noncommutative .
.
[d,53]
-
coordinates on the brain
B
and the low energy theory is modified as:
1’- Noncommutative Yang-Mills
LCg = F,,
* F’”,
F,,
= a,A,
-
&A,
+ [A, * A,
-
A,
* A,]
75
the product appearing in the scattering of gauge bosons; and 2’- An as yet unknown modification of Einstein gravity theory. In Ref. [2] by considering graviton scattering, it was found that the gravity model induced on a p-brane in the presence of a nonzero covariantly constant B field (Bab;c= 0), has the following form
/CB
= /dd’d-Rabcd(-)ac(-)bd,
1 g+B
1 g+B
a,b,c,d=O,l,..., p.
(1) It is to be noted that the form of the action in [2] is slightly different from equation (1);in particular B is taken to be constant there, which is in conflict with general covariance. The point of this work is to try to do cosmology with CB.There are a number of points in this regard: 1- B a b # 0 on 3-branes has been studied, independent of the action L B in eq. (l),in some detail, and has produced bounds on B, and naturally incorporates anisotropy in R4 cosmology. 2- B a b # 0 in the bulk may stabilize brane moduli, a point to be addressed in a separate work.
Here a simple situation on a 5-brane is considered where in the 5 and 6 directions.
Bab #
0 only
76
The FRW ansatz in 6 dimensions is
ds2 = -n2(t,y)dt2
+ a 2 ( t ,y)d2Ci + gABdyAdyB;
(3)
is the maximally symmetric homogeneous where A , B = 5,6, and isotropic 3 dimensional manifold with k = 0, ~1 The covariant constancy of B,
BAB;C= 0, leads to
BAB = Bod-
EAB
(4)
with Bo a constant; the two dimensional metric g ( 2 ) is taken as
The Einstein tensors of this action with the above metric are rather tedius and will be reported elsewhere. Here only a simple 5-dimensional limit is presented:
a2 6 ci iz GiB = ~ i { - [ - ( -+ 2-) n b2 a a
-
1 6 7i --(f2b n
ci
1 2a'l a
+ 2-)a + -(f2
n" + -)] n
77
where
b - bl
,
a,
E
=y5
,
a*
E
y
a -a av
, etc.
a
, etc.
and f2 =
+ Bg.
1
In the case of B = 0, a remarkable solution, Ref.[3], was obtained by noting that the Einstein tensor can be written as
with
F
ah
E (-)
b
aa - (-)2
n
- ka2;
and Einstein eqs. with
where the discontinuity across the brane has been used,
h ab
[-]
n2
= --p
3
b.
(13)
78
When B # 0 it is easy to verify that a generalization of the above construction exists. Then it can be shown that the relevant Einstein tensors may be written in terms of two functions F and F B ,
F = a2Fo,
a’
FB = a2f2Fo, Fo = (,)2
a - (--)2
-
Ic,
(19)
as
Then the Friedman eq. becomes
where now
&
from the dark This is a matter content modified by the factor of energy relation Ps = - p of eq. (16). Note also that a constant of integration C, as in equation (17), is not allowed, i.e. ”dark radiation” is not permitted when B # 0 Thus turning on a B field in two extra dimensions modifies the 4 dimensional evolutions in a simple controllable manner.
References 1. L. Alvarez-Gaume, F. Meyer and A. Vazquez-Mozo, Comments on noncom-
mutative gravity, [arXiv: hep-th/0605113]. 2. F. Ardalan, H. Arfaei, M.R. Garousi, and A. Ghodsi, Int. J . Mod. Phys. A18, 1051 (2003), [arXiv: hep-th/020411]. 3. P. Binetruy, C. Deffayet, U. Ellwanger and D. Langlois, Phys. Lett. B 477, 285 (2000), [arXiv: hep-th/9910219].
BOSONIZATION OF A FINITE NUMBER OF NON-RELATIVISTIC FERMIONS AND APPLICATION AVINASH DHAR Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India [email protected]?.res. in In this talk, I review our recent work on exact bosonization of a finite number of non-relativistic fermions moving in an arbitrary potential in one space dimension. I then discuss applications to problems in string theory and field theory, including to Tomonaga’s half-century old problem of bosonization of free fermions on a circle. The main results outlined here can be found in the papers hepth/0509164, hepth/0512312 and hepth/0603154.
Keywords: Bosonization; String Theory; Quantum Field Theory
1. Introduction The idea of bosonization, namely finding a bosonic system equivalent to a given fermionic system, is almost as old as quantum mechanics itself. The earliest observation for the existence of quantized collective bose excitations - sound waves - in a gas of fermions in 3-dimensions is due to Bloch’. Bohm and Pines* discussed charge density waves - plasma oscillations - in a gas of electrons. The first important technical breakthrough in treating a large system of interacting fermions is due to Tomonaga3 who showed, in a rigorously defined simple 1-dimensional model, that interactions between fermions can mediate new collective bosonic degrees of freedom. Historically, the Tomonaga problem has played an important role in the development of tools for treating a system of large number of fermionst interacting with long-range forces like Coulomb force, which is a recurrent problem one encounters in condensed matter systems. In his toy model *See Ref. 2 and references therein. tSee, for example, Ref. 4 for an account of this and for many of the original works cited below.
79
80
of a l-dimensional system of non-relativistic fermions on a circle, Tomonaga made essential use of the observation that a long-range (in real space) force like Coulomb interaction becomes short range in momentum space, so particle-hole pairs in a low-energy band around the Fermi surface do not get “scattered out” of the band. As a result, the interacting ground state, as well as excited states with low excitation energy compared to the Fermi energy, involve only particle-hole pairs in a small band around the Fermi surface. When the number of fermions is large, there is a finite band around the Fermi surface in which the excited states satisfy this requirement. For excitations in this band, the quadratic dispersion relation of non-relativistic particles approximately linearizes and becomes relativistic. Taking a cue fom this, Luttinger later used a strictly linear dispersion relation. Other works on the Luttinger problem, notably by Mattis and Lieb, Luther and Peschel and Haldane, which gave the exact solution in this case, have led to the class of systems now known as Tomonaga-Luttinger liquids, which provide an important paradigm4 in condensed matter physics. In the field theory context, bosonization of relativistic fermions was achieved by Mandelstam and by Coleman5. Besides condensed matter systems, non-relativistic fermions appear in many interesting situations in string theory and quantum field theory, like non-critical strings in (1 1)-dimensions, half-BPS sector of N = 4 super Yang-Mills theory in Cdimensions (which is dual to string theory on Ads5 x S5background), pure Yang-Mills theory on a cylinder in 2-dimensions , etc. In all the examples cited above, fermions arise from an underlying matrix quantum mechanics problem whose dynamics is controlled by the action
+
s=
s: dt
[ l W -V(M)].
(1)
Here A4 is an N x N hermitian matrix. In the U(N)-invariant sector, this matrix model can be shown to be equivalent to a system of N nonrelativistic fermions in l-dimension6. Jevicki and Sakita7 exploited this equivalence and used the method of collective variables to provide an alternative approach to approximate bosonization of non-relativistic fermions. Like Tomonaga’s approach, this method also works only in the low-energy approximation for a large number of fermions. Motivated by its application to non-perturbative formulation of (1 1)-dimensional string theory, an exact bosonization of a finite number of non-relativistic fermions, moving in an arbitrary potential in one space dimension, was developed in Ref. 8; see also Ref. 9. The basic object, in terms of which the bosonization is carried out, is the Wigner phase space
+
81
density u ( p ,q , t ) =
/
c N
dx e-ipz
+;(q
-
x/2, t ) + i ( q
+ x/2, t ) .
(2)
i= 1
Here + i ( q , t ) , i = 1,2,. f . , N are the single-particle states occupied by the fermions. The phase space density satisfies two constraints, viz.
/
F u ( p , q , t )= N, u * u = u.
(3)
The first of these is obvious. The second constraint, which equates to u the Moyal star product of u with itself, has a simple semiclassical limit. In this limit it reads u2 = u , which has the solutions u = 0 , l corresponding respectively to an unoccupied phase space cell or a cell occupied by a fermion. This constraint shows that a description of the bosoized system in terms of u is highly redundant. In fact, in the semiclassical limit, the only dynamical variables are the boundaries of the occupied regions of phase space. Although a semiclassical treatment of these constraints is relatively easy, an exact solution in terms of independent bosonic degrees of freedom has never been obtained. Because of this reason, applications of this bosonization formalism to non-perturbative situations have not been possible. This motivated us to look for and discoverlo an alternative exact bosonization scheme in terms of independent degrees of freedom, which is described below. 2. Exact Operator Bosonization Consider a system of N fermions each of which can occupy a state in an infinite-dimensional Hilbert space 'Flf. Suppose there is a countable basis of 7-lf : { Im),m = 0 , 1 , . . . , 00). For example, this could be the eigenbasis of a single-particle hamiltonian, hlm) = E(m)lm),but other choices of basis would do equally well, as long as it is a countable basis. In the second quantized notation we introduce creation (annihilation) operators +k (+m) which create (destroy) particles in the state Im). These satisfy the anticommutation relations {+my
+A>
= 6mn
(4)
The N-fermion states are given by (linear combinations of) Ifl,'"
,f N ) =
""$~,lo)F
(5)
where f m are arbitrary integers satisfying 0 5 f l < f 2 < . . . < f N , and 10)~ is the usual Fock vacuum annihilated by qm,m = 0 , 1 , . . . , 00.
82
It is clear that one can span the entire space of N-fermion states, starting from a given state I f i , . . . , f N ) , by repeated applications of the fermion bilinear operators Qmn = $k qn. However, the problem with Qmn’s is that they are not independent; this is reflected in the W, algebra that they satisfy, [Qmn,Qm/nl]= bm/nQmn/- ~ m n ~ Q m ~ n .
(6)
This is the operator version of the noncommutative constraint u * u = u satisfied by the Wigner density u. A new set of unconstrained bosonic operators was introduced in Ref. 10, N of them for N fermions. In effect, this set of bosonic operators provides the independent degrees of freedom in terms of which the above constraint is solved. Let us denote these operators by U k , k = 1 , 2 , . . . , N and their conjugates by, u:, Ic = 1 , 2 , .. . , N . The action of on a given N-fermion state I f 1 , . . . , f N ) is stated simply. It just takes each of the fermions in the top lc occupied levels up by one step. One starts from the fermion in the topmost occupied level, f N , and moves it up by one step to ( f N I), then the one below it up by one step, etc. proceeding in this order, all the way down to the kth fermion from top, which is occupying the level f N - k + l and is taken to the level ( f N - k + l 1). For the conjugate operation, U k , one takes fermions in the top occupied Ic levels down by one step, reversing the order of the moves. Thus, one starts by moving the fermion at the level f N - k + l to the next level below at ( f N - k + l - I), and so on. Clearly, in this case the answer is nonzero only if the (k 1)th fermion from the top is not occupying the level immediately below the kth fermion , i.e. only if ( f N - k + l - f N - k - 1) > 0. If k = N this condition must be replaced by f l > 0. These operations are necessary and sufficient to move to any desired fermion state starting from a given state. This can be argued as follows. First, consider the operator U k - 1 utk .Acting on a generic fermion state this
UL
+
+
+
operator moves only the lcth fermion from top up by one level. In other by words, U k - 1 U: = $ : N - k + l + l $ f N - k + l - Q f N - k + l f l , f N - k + l ’ In this composing together different f l k operations we can move individual fermions around. Clearly, all the N g k operations are necessary in order to move each of the N fermions indvidually. It is easy to see that by applying sufficient number of such fermion bilinears one can move to any desired fermion state starting from a given state. By definition, u:, fsk operators satisfy the following relations: c k 0:
= 1, g kt U k = e ( T k
-
[(Tl1a:] = 0,
1 # k,
(7)
83
where ( f N - k + l - f N - k - 1) = T k and O(m) = 1 if m 2 0, otherwise it vanishes. Moreover, all the (Tk'S annihilate the Fermi vacuum. Consider now a system of bosons each of which can occupy a state in an N-dimensional Hilbert space ' H N . Suppose we choose a basis {I k) , k = 1,. . . , N} of 7 - t ~In . the second quantized notation we introduce creation (annihilation) operators ul ( a h ) which create (destroy) particles in the state Ik). These satisfy the commutation relations [ ~ k , a != ] 6kl,
(8)
k,l = I , . . . ,N.
A state of this bosonic system is given by (a linear combination of)
It can be easily verified that equations (7) are satisfied if we make the following identifications (Tk =
1 1 tJGUk' JG' t
O k - ak
(10)
together with the map Tk
= f N - k + l - f N - k - 1, k = 1, 2, * .
'
N - 1;
TN
=fl.
(11)
This identification is consistent with the Fermi vacuum being the ground state of the bosonic system. The first of these arises from the identification (10) of (Tk'S in terms of the oscillator modes, while the second follows from the fact that (TN annihilates any state in which f 1 vanishes. Using the above bosonization formulae, any fermion bilinear operator can be expressed in terms of the bosons. For example, the hamiltonian can be rewritten as follows. Let E ( r n ) , m = 0 , 1 , 2 , . . . be the exact single-particle spectrum of the non-interacting part H of the fermion hamiltonian. That is, H = CzToE(m) +k Qm. Its eigenvalues are E = ~ f =E ('f= k ) . ,Using f k = X L N - k f l ri k - 1, which is easily derived from (ll),these can be rewritten in terms of the bosonic occupation numbers, N = c k = l E ( z L N - k + l r i + k - 1).These are the eigenvalues of the bosonic hamiltonian I
+
N H = X E ( A k ) , k=l
N
fi.lc
3
xaiai
f N-k.
(12)
i=k
This bosonic hamiltonian is, of course, completely equivalent to the fermionic hamiltonian we started with.
84
It should be clear from the above discussion that our bosonization technique does not depend on any specific choice of fermionic hamiltonian. Moreover, an interaction term, for example a four-fermi interaction, can also be bosonized following similar methods. A generic four-fermi interaction term will involve a generic polynomial interaction in the bosonic variables, the degree of the polynomial depending on the range of interaction. 3. Applications
Our bosonization techniques can be usefully applied to various problems of interest in many areas of physics, like non-critical strings, half-BPS sector of N = 4 super Yang-Mills theory, the Tomonaga problem, Yang-Mills theory on a cylinder, etc. In the following we briefly discuss some of these applications.
3.1. Half-BPSsector of h/ = 4 SYM and LLM geometries It is widely believed today that Einstein's gravity action, small quantum fluctuations around a classical solution to it described by gravitons, and even space-time itself are low-energy emergent properties of an underlying microscopic dynamics. Since string theory is a consistent theory of quantum gravity, we should be able to test these ideas within this framework. The AdS/CFT correspondence" provides a precise setting in which to explore these ideas. The classic example is N = 4 SYM and its dual string theory on Ads5 x S5. The duality states that weakly coupled low-energy type IIB gravity on Ads5 x S5and strongly coupled N = 4 SYM theory in the largeN limit have exactly the same physical content. The amazing thing is that there is no hint of either a 10-d space-time or gravitons in the SYM theory! In the boundary SYM theory, half-BPS states are described by a holomorphic sector of quantum mechanics of an N x N complex matrix Z in a harmonic potential12. This sector can be shown to be equivalent to the quantum mechanics of an N x N hermitian matrix in a harmonic potential. Gauge invariance implies that physical operators in the boundary theory are U(N)-invariant traces: trZk, k = 1 , 2 , . . . ,N . By the operator-state correspondence, then, the physical states in this sector are in one-to-one correspondence with the operators (trZkl)ll(trZkz)l2. . . . The total number of 2's is a conserved RR charge Q = Ci kili, with E = Q because of the tSee the last of Ref. 12.
85
BPS condition. At large N there is a semiclassical picture of the states of this system in terms of droplets of Fermi fluid in phase space. The ground state is a filled disc of radius centered at the origin in phase space. Small fluctuations of the boundary of the disc give low-energy excitations, while an arbitrary distribution of filled droplets corresponds to a generic state. By explicitly solving equations of type IIB gravity, Lin, Lunin and Maldacena13 (LLM) showed that there is a similar structure in the classical geometries in the half-BPS sector! In the LLM solutions, two of the space coordinates are identified with the phase space of a single fermion, leading to noncommutativity in two space directions in the semiclassical de~cription’~. Small fluctuations around AdS space, i.e low-energy graviton excitation^'^ are equivalent to low-energy fluctuations of the Fermi vacuumlG. Since the free fermi system on the boundary can be quantized exactly, one might hope to learn about aspects of quantum gravity by using the duality. At finite N , only the low-energy excitations on the boundary can be identified with low-energy (<< N ) gravitons in the bulk. These excitation on the boundary are created from the Fermi vacuum by fermion bilinears, which can be related to our bosonic stated7, using the bosonization rules. We have
One can compute exactly the correlation functions < p:,/?$, . . . >. The result of a computation of the 3-point function can be summarized as follows: (i) At low energies, perturbation theory is good and reproduces supergravity answers18. The low-energy interactions are described by an effective cubic hamiltonian. (ii) Perturbation theory breaks down for P’s with energy of order In the dual gravity description this is reflected in the fact that gravitons with energies larger than fl have a size smaller than 10-dim planck scale. Furthermore, this breakdown of perturbative calcualtions coincides with the existence, in the bulk gravity theory, of nonlocal solitonic excitations, giant gravitons 19, whose size is larger than 10-dim Planck scale precisely for energies larger than (iii) At energies of order N , the p interactions grow exponentially with N . At such energies, therefore, 0’s cease to provide a meaningful semiclassical description. On the boundary, single-particle states dual to bulk giant graviton states map to specific linear combinations of states dual to multi-graviton states
a.
n.
86
in the bulk12. These states are linear combinations of multi-/3 states and hence are related to our bosonic states. In fact, one can show17 that the boundary states corresponding to single-particle bulk giant graviton states are just the single-particle bosonic states! lgiant graviton of energy Ic) = aklO) t
(14) The bosonized hamiltonian for fermions in the harmonic potential is given by N
n
k=l
Thus, our bosonic states, which we have seen are dual to the giant graviton states in the bulk, exactly diagonalize the hamiltonian in the half-BPS sector. They provide a microscopic description of gravity in this sector, which is valid at all energies! 3.2. Free fermions on a circle
This is the classic Tomonaga problem. Note that interactions between fermions can be taken into account once the bosonization of the free part has been dealt with properly. The hamiltonian for free fermions on a circle is
s.
where Fourier modes are used in the second equality and w E Let us set x + = ~ +Zn and x - ~= +Zn-1. Then, the fermionic hamiltonian becomes H
=
wRC;==,
(
+;gn,
where e ( n ) = 1 if n is odd and
)2
vanishes otherwise. The corresponding bosonized hamiltonian2' is N
fik
+ e(%)
N i= k
k= 1
In the large-N, low-energy limit, H can be rewritten as a sum of O ( N ) and 0(1)pieces, H = H F HO H I , where H F is the fermi energy and
Ho
+ + k alak + C ) ,
(5
=TwN
2
k=l
N
C = c ( e ( f i k )- e ( N - 14)).
(18)
k= 1
The operator i. measures the number of excess fermions in negative momentum states over and above the number in fermi vacuum. Since it commutes
87
with the number operator aiak for all Ic, eigenstates of HO can be labeled by eigenvalues of 6 . Also, H1 evaluates to O(1) on excited states whose energy is low compared to N . The eigenstates of Ho have a simple interpretation; for large N , these represents a massless compact scalar in (1 1)-dimensions! It is the second term in (18) that effects this transmutation from a bunch of oscillators to a massless scalar, as can be seen from a computation of the partition function for Ho. In the limit N -i 00, the partition function turns out to be
+
The sum over v is the sum over eigenvalues of the momentum zero-mode of the compact scalar, but the sum over winding modes is missing because of the restriction to a fixed number of fermions, N . The O(1) piece in the hamiltonian, H I , leads to cubic interactions of the massless scalar. A systematic large-N expansion can be carried out to obtain the effective low-energy cubic theory. Our bosonization, of course, is exact and goes beyond the large-N limit. We should, therefore, be able to discuss even non-perturbative effects in this formalism. The following example illustrates precisely such an application.
3.3. 2 0 Yang-Mills on a cylinder
Non-relativistic fermions appear in 2D YM on a cylinder with U(N) gauge group21. They also appear, via this connection with 2D YM, in the physics of certain black Because of this, our bosonization has applications to these problems as well. Below we will briefly elaborate on the connection with black holes and baby universes. Type IIA string theory compactified on a Calabi-Yau manifold supports supersymmetric configurations of D4, D2 and DO branes. The back-reacted geometry is a black hole in the remaining four non-compact directions, labeled by the D-brane charges. The counting of the number of bound states of the branes maps to the partition function of pure two-dimensional YangMills theory on a cylinder22, and hence to the partition function for the Tomonaga problem. Also, as pointed out in Ref. 24, the partition function with a given asymptotic charge must necessarily include multi-centered black holes, corresponding to configurations with multiple filled bands of fermion energy levels. The existence of such multiple configrations gives rise
88
to nonperturbative corrections to the OSV relation23; schematically zBH
=
1$12
+ O(e-N).
(20)
The uncorrected equation is valid for a single black hole, and corresponds in the fermion theory to two decoupled Fermi surfaces (at the top and at the bottom) which is the correct description in the N -+ 00 limit. In case of the fermion theory, the 0 ( e c N )corrections signify the fact that a t finite N , the approximation of the Fermi sea as having two infintely separated Fermi surfaces is not valid and includes in the partition function many more states than actually exist in the system; the corrections subtract those states iteratively. In our bosonized theory, the structure of equation (20) can be recognized in the finite N partition function20 for Ho:
We see that there are two types of O ( e - N ) corrections compared to (19); one arises from the truncated sum, for finite N , over v and the other from the truncated product factors. Thus, the hamiltonian HO provides a simple example of the two types of nonperturbative corrections discussed in Ref. 24.
References 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11.
F. Bloch, Z. Phys. 81,363 (1933) ; Helv. Phys. Acta 7,385 (1934). D. Bohm and D. Pines, Phys. Rev. 92,609 (1953). S. Tomonaga, Progr. Theor. Phys. 5 , 304 (1950). M. Stone, Editor, reprint volume entitled Bosonization (World Scientific, Singapore, 1994). S. Mandelstam, Phys. Rev. D 11,3026 (1975);S.Coleman, Phys. Rev. D 11, 2088 (1975). E. Brezin, C. Itzykson, G. parisi and J. B. Zuber, Comm. Math. Phys. 59, 35 (1978). A. Jevicki and B. Sakita, Nucl. Phys. B 165,511 (1980). A. Dhar, G. Mandal and S. Wadia, Mod. Phys. Lett. A 7,3129 (1992), hepth/9207011. B. Sakita, Phys. Lett. B 387,118 (1996), hep-th/9607047. A. Dhar, G. Mandal and N. Suryanarayana, J . High Energy Phys. 0601,118 (2006), hep-th/0509164 0. Aharony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000), hep-th/9905111.
89 12. V. Balasubramanian, M. Berkooz, A. Naqvi and M. Strassler, Giant Gravitons in Conformal Field Theory, hep-th/0107119; S. Corley, A. Jevicki and S. Ramgoolam, Adv. Theor. Math. Phys. 5,809 (2002), hep-th/0111222; D. Berenstein, J . High Energy Phys. 0407,018 (2004), hep-th/0403110; Y . Takayarna and A. Tsuchiya, J. High Energy Phys. 0510,004 (2005), hepth/0507070. 13. H. Lin, 0. Lunin and J. Maldacena, J . High Energy Phys. 0410,025 (2004), hep-t h/0409174. 14. G. Mandal, J . High Energy Phys. 0508, 052 (2005), hep-th/0502104. 15. L. Grant, L. Mmz, J. Marsano, K. Papadodimas and V. Rychkov, J . High Energy Phys. 0508,025 (2005) , hep-th/0505079; L. Maoz and V. Rychkov, J. High Energy Phys. 0508,096 (2005) , hep-th/0508059. 16. A. Dhar, J. High Energy Phys. 0507,064 (2005), hep-th/0505084. 17. A. Dhar, G. Mandal and M. Srnedbkk, J. High Energy Phys. 0603, 031 (2006), hepth/0512312. 18. S. Lee, S. Minwalla, M. Rangarnani and N. Seiberg, Adv. Theor. Math. Phys. 2,697 (1998), hep-th/9806074. 19. J. McGreevy, L. Susskind and N. Tournbas, J. High Energy Phys. 0006, 008 (2000), hepth/0003075; M. Grisaru, R. Myers and 0. Tafjord, J . High Energy Phys. 0008,040 (2000), hep-th/0008015; A. Hashimoto, S. Hirano and N. Itzhaki, J . High Energy Phys. 0008,051 (2000), hep-th/0008016. 20. A. Dhar and G. Mandal, Bosonization of non-relativistic fermions on a circle: Tomonaga’s problem revisited, hep-th/0603154. 21. J. A. Minahan and A. P. Polychronakos, Phys. Lett. B 312, 155 (1993), hep-th/9303153. 22. C. Vafa, Two dimensional Yang-Mills, black holes and topological strings, hep-th/0406058 23. H. Ooguri, A. Strominger and C. Vafa, Phys. Rev. D 70, 106007 (2004), hep-t h/0405146 24. R. Dijkgraaf, R. Gopakumar, H. Ooguri and C. Vafa, Baby universes in string theory, hep-th/0504221
RECENT APPLICATIONS OF THE WEYL ANOMALY M.J. DUFF
Blackett LabOTatOTy, Imperial College London, Prince Consort Road, London SW7 ZAZ , U.K. E-mail: m.duffOimperia1. ac.uk We review some recent applications of the Weyl anomaly: (1)The holographic Weyl anomaly, (2) The Weyl anomaly and corrections to Newton’s law in the Randall-Sundrum braneworld, (3) The revival of the original (Starobinsky) model of inflation driven by the Weyl anomaly, (4)The Weyl anomaly and the mass of the graviton in the Karch-Randall braneworld.
1. Thirty three years of the Weyl anomaly In 1973 Derek Capper and the author (two Salam prot6g6s on our first postdocs in Trieste) discovered that the conformal invariance under Weyl rescalings of the metric tensor gpu(x) 4 f12(x)gpu(z) displayed by classical massless field systems in interaction with gravity no longer survives in the quantum theory’. As a consequence, the energy-momentum tensor develops a non-vanishing trace A233: 1 A = gpu(TpV) = -(cF (4r)2 where F is the square of the Weyl tensor:
F
-
aG + dV2R),
1 + -R2, 3
= CpuPuCpVPuR P V W Rpupu- 2RPU Rpu
(2)
G is proportional to the Euler density: G = RpvpuRpuPu - 4RPV RpuiR2,
(3)
and a , c and d are constants. Since then these Weyl anomalies have found a variety of applications in quantum gravity, black hole physics, cosmology, string theory and statistical mechanics. On the occasion of the Salamfest in l’rieste in 1993, the author gave a review entitled T w e n t y years of the
90
91
Weyl unomuly4. Since this conference is, in some sense, another Salamfest, I wish to review some of what has happened since. Weyl anomalies appear in their most pristine form when conformal field theories are coupled to an external gravitational field. They also have a role to play when gravity is coupled to non-conformally invariant theories and when gravity is itself quantized but their significance is less clear cut. Fortunately, all of the recent applications I shall focus on in the present paper fall into the first category. In this case the constants u , c and d are given in terms of the field content of the CFT by3
where N, are the number of fields of spin s. The coefficients a and c are independent of the renormalization scheme but d is not. We have quoted the result given by dimensional regularization; the result given by zeta-function regularization or point-splitting has -18 instead of +12 as the coefficient of N1 5 . In fact, d can be adjusted to any desired value by adding the finite counter term2
SCt /d4q/ijR2.
(5)
N
A particularly important example of a CFT is provided by Yang-Mills with gauge group U(N), for which
(NilN1/2, No) = (N2’4N2,6N2).
N
= 4 super
(6)
Then
and hence
The first application of the Weyl anomaly, discussed in section 2, is the holographic Weyl a n 0 m a l y ~ 3that ~ appears in the AdS/CFT correspondence’-’’. The next three sections are devoted to the Weyl anomaly and braneworld: the corrections to Newton’s law” in the RandallSundrum model12 in section 3, the recent revival l3>l4of the original Starobinsky model15 of inflation driven by the Weyl anomaly in section 4,and the mass of the graviton16 in the Karch-Randall rnodell7 in section 5.
92
2. The holographic Weyl anomaly, 1998 Here we follow the work of Henningson and Skenderis‘ who recall that a theory containing gravity and defined on an open (d+ 1)-manifold X can, in some cases, be equivalent to a d-dimensional conformal field theory defined on the boundary M of X8-l0. The partition function is then a functional of the boundary data: zgrav[4(0)1=
S,
~exP(-s[+l>,
(0)
where the subscript on the integration sign indicates that the functional integral is over field configurations 4 that satisfy the boundary condition given by 4(0). There is a one-to-one correspondence between the fields 4 on X and the primary operators 0 on M . The set of correlation functions of the latter are conveniently summarized by a generating functional:
where q+o) is now regarded as a formal expansion parameter. The partition function of the gravity theory on X and the generating functional on M are then equal, regarded as functionals of +(o): z g r a v [ + ( ~ )= l ZCFT[+(O)].
A field of particular importance in a theory containing gravity is of course the metric G M N The . corresponding operator in the boundary conformal field theory is the stress-energy tensor T,”. The boundary data for the metric G,, is not a boundary metric g(o),v, but only a conformal structure [g(o)].(This is defined as an equivalence class of boundary metrics when two metrics that differ by a local rescaling are considered equivalent , i.e. g(o) exp 2 c ~ ( z ) g ( Ofor ) an arbitrary positive function ~ ( z ) . ) Henningson and Skenderis‘ consider the effective action N
s [ g ( O ) ]= - 1% ZCFT[g(O)I
(9)
of the theory on M . A priori, this is a functional of the metric g(0) on M , but by conformal invariance, it should actually only depend on the conformal equivalence class [g(o)].However, this invariance is sometimes broken by the Weyl anomaly. This means that S [ g ( o ) ]is not invariant under a conformal rescaling bg(o) = 2 b c ~ g ( of ~ )the metric, but transforms as
93
where A is the anomaly. On general grounds, the gravitational part of the Weyl anomaly vanishes when the dimension d of M is odd but when d is even there can be an anomaly. In d = 4, for example, it is of the form (1). For pure gravity the theory in the bulk is described by the EinsteinHilbert action plus a cosmological constant term:
A choice of conformal structure [g(o)]on the boundary M determines a unique metric G M Nin the bulk of X that solves the equations of motion 1 RMN- -RGMN = R G M N . (11) 2 (This is of course true only up to diffeomorphisms). However, the bulk action diverges when evaluated for such a field configuration because of the second order pole in G,, on the boundary. Furthermore, the boundary terms in the action are ill-defined, since G M Ndoes not induce a finite metric ljp, on M . To regulate the theory in a manner consistent with general covariance, we pick a specific representative g(o) of the boundary conformal structure [g(o)].This determines a distinguished coordinate system ( p , d’), in which the metric on X takes the formlg Ld+lL
GMNdxMdxN = 4 p-’dpdp
+ p-’gp,dx,dx”.
(12)
Here the tensor g has the limit g(0) as one approaches the boundary represented by p = 0. The length scale Ld+l is related to the cosmological constant A. Einstein’s equations for G,, can then be solved order by order in p with the result that g is of the form = g(0)
+ Pg(2) + . . + pd/’g(d) + pdl2 logph(d) + O(pd/2+1). *
(13)
The regularization procedure now amounts to restricting the bulk integral in the action to the domain p > E for some cutoff E > 0 and evaluating the boundary integrals at p = E . The regulated action evaluated for this field configuration is then of the form
The coefficients a ( o ) ,a(’), . . . , a ( d ) are all given by covariant expressions in g(o) and its curvature tensor R P v p V . The divergences as E goes to zero can thus be canceled by adding local counterterms to the action, so that
94
we are left with a finite effective action Wfin[g(0)]. To find the variation of S f i n [ g ( o ) ]under a conformal rescaling of the boundary metric g(o), we note that the regulated action W[g(o)] is invariant under the combined transformation Sg(0) = 2 6 a g ( o ) and 6~ = 2 6 a ~for a constant parameter 6u. The terms proportional to negative powers of E are separately invariant, so the variation of Wfin[g(0)]must therefore equal minus the variation of the logarithmically divergent term. The latter is given by 1
since log 6 transforms with a shift whereas a ( d ) itself is invariant. Although this formula was derived under the assumption that 60 is a constant, it follows from the general form of the conformal anomaly that it can be applied also for a non-constant 60. If we now evaluate the anomaly explicitly for the case d = 4, we get6
A = “ (L 3 64~G5
RP”R,,
1
-
?Rz> .
Inserting for example the values of L5 and G5 appropriate for the ads5 x S5 geometry of N coincident D3-branes in type IIB string theory,
we get the same anomaly as that of an N = 4 U(N) supermultiplet as given in (8). The agreement between the strong coupling calculation and this weak coupling result indicates that there is a non-renormalization theorem for the Weyl anomaly. 3. The Weyl anomaly and corrections to Newton’s law, 2000
In his 1972 PhD thesis under Abdus Salam, the author showed that, when one-loop quantum corrections to the graviton propagator are taken into account, the inverse square r - 2 behavior of Newton’s gravity force law receives an r-4 correction whose coefficient depends on the number and spins of the elementary particles20)21.Specifically, the potential looks like
where G4 is the four-dimensional Newton’s constant, h = c = 1 and (Y is a purely numerical coefficient proportional to the c coefficient in the Weyl
95
anomaly
a=
8 37r
-C.
(19)
From (7), (18) and (19) we see that the contribution of a single N = 4 U ( N )Yang-Mills CFT is
Now fast-forward to 1999 when Randall and Sundrum proposed that our four-dimensional world is a 3-brane embedded in an infinite five-dimensional universe. Gravity reaches out into the five-dimensional bulk but the other forces are confined to the four-dimensional brane. Contrary to expectation, they showed that a n inverse square T - ~law for gravity is still possible but with an rP4correction coming from the massive Kaluza-Klein modes whose coefficient depends on the bulk cosmological constant. Their potential looks like T
where L5 is the radius of AdS5. Since (20) was the result of a four-dimensional quantum calculation and (21) the result of a five-dimensional classical calculation, they seem superficially completely unrelated. However, if we now invoke the miracle of the AdS/CFT correspondence of Maldacena we can demonstrate that the two are in fact completely equivalent ways of describing the same physics". If we use the AdS/CFT relation (17) and the one-sided brane-world relation 2G5 G* = -, L5
where G5 is the five-dimensional Newton's constant, then (20) reproduces exactly the Randall-Sundrum result (21). Experimental tests of deviations from Newton's inverse square law are currently under way. 4. Weyl anomaly driven inflation, 2000
In a recent paper, Hawking, Hertog and Reall14 considered a model in which inflation is driven by the Weyl anomaly of a large number N of matter fields. In the large N approximation, they perform the path integral
96
over the matter fields in a given background to obtain an effective action that is a functional of the background metric: exp(-W[gl) = J441 exp(-S[4; gl).
(23)
They neglect graviton loops, and look for a stationary point of the combined gravitational action and the effective action for the matter fields. This is equivalent to solving the Einstein equations with the source being the expectation value of the matter energy momentum tensor:
where
Matter fields might be expected to become effectively conformally invariant if their masses are negligible compared to the spacetime curvature. The Weyl anomaly,
# 0,
gpLy(TpY)
(26)
is entirely geometrical in origin and therefore independent of the quantum state. In a maximally symmetric spacetime, the symmetry of the vacuum implies that the expectation value of the energy momentum tensor can be expressed in terms of its trace
Thus the trace anomaly acts just like a cosmological constant for these spacetimes. Hence a positive trace anomaly permits a de Sitter solution to the Einstein equations22. This is very interesting from the point of view of cosmology, as pointed out by Starobinsky15. Starobinsky showed that the de Sitter solution is unstable, but could be long-lived, and decays into a matter dominated Friedman-Robertson-Walker (FRW) universe. The purpose of Starobinsky’s work was to demonstrate that quantum effects of matter fields might resolve the Big Bang singularity. From a modern perspective, it is more interesting that the conformal anomaly might have been the source of a finite but significant period of inflation in the early universe. This inflation would be followed by particle production and (p)reheating during the subsequent matter dominated phase. Starobinsky’s work is reviewed and extended by Vilenkin in [23].
97
In order to test the Starobinsky model, it is necessary to compare its predictions for the fluctuations in the cosmic microwave background (CMB) with observation. This was partly addressed by Vilenkin23.Vilenkin showed that the amplitude of long wavelength gravitational waves could be brought within observational limits at the expense of some fine-tuning of the coefficients parameterizing the Weyl anomaly. The analysis of Starobinsky and Vilenkin was complicated by the fact that tensor perturbations destroy the conformal flatness of a FRW background, making the effective action for matter fields hard to calculate. However, we now have a way of calculating the effective action for a particular theory, namely N = 4 U ( N ) super Yang-Mills theory, using the AdS/CFT correspondence'. In their paper Hawking, Hertog and Reall14 calculate the effective action for this theory in a perturbed de Sitter background. This enables them to calculate the correlation function for metric perturbations around the de Sitter background. They can then compare their results with observations. The fact that they are using the N = 4 Yang-Mills theory is probably not significant, and we expect their results to be valid for any theory that is approximately massless during the de Sitter phase. They include in their action higher derivative counterterms, which arise naturally in the renormalization of the Yang-Mills theory. There are three independent terms that are quadratic in the curvature tensors: the Euler density, the square of the Ricci scalar and the square of the Weyl tensor. The former just contributes a multiple of the Euler number to the action. Metric perturbations do not change the Euler number, so this term has no effect. The square of the Ricci scalar has the important effect of adjusting the coefficient of the V 2 Rterm in the trace anomaly, as shown in eq. (5). It is precisely this term that is responsible for the Starobinsky instability, so by varying the coefficient of the R2 counter term we can adjust the duration of inflation. The Weyl-squared counterterm does not affect the trace anomaly but it can contribute to suppression of tensor perturbations. The effects of this term were neglected by Starobinsky and Vilenkin. They also neglected the effects of the non-local part of the matter effective action. Hawking et. al. take full account of all these effects. Vilenkin showed that the initial de Sitter phase is followed by a phase of slow-roll inflation before inflation ends and the matter-dominated phase begins. Since the horizon size grows significantly during this slow-roll phase, it is important to investigate whether modes we observe today left the horizon during the de Sitter phase or during the slow-roll phase. If the present horizon size left during the de Sitter phase, Hawking et. al. find that
98
the amplitude of metric fluctuations can be brought within observational bounds if N , the number of colours, is of order lo5. Such a large value for N is rather worrying, which leads us to the second possibility, that the present horizon size left during the slow-roll phase. Their results then suggest that the coefficient of the R2 term must be at most of order lo’, and maybe much lower, but N is unconstrained (except by the requirement that the large N approximation is valid so that AdS/CFT can be used). They also find that the tensor perturbations can be suppressed independently of the scalar perturbations by adjusting the coefficient of the Weyl-squared counterterm in the action.
5. The Weyl anomaly and the graviton mass, 2002 An old question is whether the graviton could have a small but non-zero rest mass. If so, it is unlikely to be described by the explicit breaking of general covariance that results from the addition of a Pauli-Fierz mass term to the Einstein Lagrangian. This gives rise to the well-known Van DamV e l t m a n - Z a k h a r ~ vdiscontinuity ~~?~~ problems in the massless limit, that come about by jumping from five degrees of freedom to two. Moreover, recent a t t e m p t ~ to ~ ~circumvent l~~ the discontinuity in the presence of a non-zero cosmological constant work only at tree level and the discontinuity re-surfaces at one loop2’. (A similar quantum discontinuity arises in the “partially massless” limit as a result of jumping from five degrees of freedom to four2’ and also for spin 3 / 2 where the jump is from four degrees of freedom to two30). On the other hand, in analogy with spontaneously broken gauge theories, one might therefore prefer a dynamical breaking of general covariance, which would be expected to yield a smooth limit. However, a conventional Higgs mechanism, in which a scalar field acquires a non-zero expectation value, does not yield a mass for the graviton. The remaining possibility is that the graviton acquires a mass dynamically and that the would-be Goldstone boson is a spin one bound state. Just such a possibility was suggested in 197531. Interestingly enough, the idea of a massive graviton arising from a spin one bound state Goldstone boson has recently been revived by P ~ r r a t in i~~ the context of the Karch-Randall brane-world17 whereby our universe is an AdS4 brane embedded in an Ads5 bulk. This model predicts a small but finite four-dimensional graviton mass
99
in the limit L4 3 00, where L4 and L5 are the 'radii' of Ads4 and AdS5, respectively. From the Karch-Randall point of view, the massive graviton bound to the brane arises from solving the classical D = 5 linearized gravity equations in the brane background17. The Randall-Sundrum complementarity of the previous section can be generalized to the Karch-Randall AdS braneworld picture. From an AdS/CFT point of view, one may equally well foliate a Poincar6 patch of AdSs in AdS4 slices. The Karch-Randall brane is then such a slice that cuts off the Ad& bulk. However, unlike for the Minkowski braneworld, this cutoff is not complete, and part of the original AdSs boundary remains. Starting with a maximally supersymmetric gauged N = 8 supergravity in the five dimensional bulk, the result is a gauged N = 4 supergravity on the brane coupled to a n/ = 4 super-Yang-Mills CFT with gauge group U(N), however with unusual boundary conditions on the CFT fields As was demonstrated in ref. [32], the CFT on Ads4 provides a natural origin for the bound state Goldstone boson which turns out to correspond to a massive representation of S0(3,2). However, while [32] considers the case of coupling to a single conformal scalar, in ref. [16]we provided a crucial test of the complementarity by computing the dynamically generated graviton mass induced by a complete N = 4 super-Yang-Mills CFT on the brane and showing that this quantum computation correctly reproduces the KarchRandall result, eq. (28). The result again depends on the c coefficient of the Weyl anomaly:
This expression is our main result, and generalizes that obtained in [32]. For the Karch-Randall braneworld17, where the CFT fields are that of N = 4 U(N) super-Yang-Mills we use (7) and find simply
which reproduces exactly the Karch-Randall result (28) on using (17) and (22). Although we focused on t h e N = 4 SCFT, the result is universal, being independent of which particular CFT appears 'in the AdS/CFT correspondence. This suggests that c plays a universal r61e in both the Minkowski and AdS braneworlds, as indicated in (30) and (18), and that our result is robust at strong coupling. This presumably explains why our one-loop computation gives the exact Karch-Randall result. However, we do not know for certain whether this persists beyond one loop.
100
6. Conclusions Thirty-three years on, t h e Weyl anonaly continues t o find new a n d interesting applications. Some others that space did not allow include t h e Weyl anomaly as an infrared diagnostic33 a n d t h e Weyl anomaly and the volume of Sasaki-Einstein manifolds34 that arise in string compactifications.
7. Acknowledgements My thanks t o Faheem Hussain and Riazuddin for their kind invitation t o Islamabad.
References 1. 2. 3. 4. 5. 6.
7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
D. M. Capper and M. J. Duff, Nuovo Cim. A 23, 173 (1974). S. Deser, M. J. Duff and C. J. Isham, Nucl. Phys. B 111,45 (1976). M. J. Duff, Nucl. Phys. B 125,334 (1977). M. J. Duff, Class. Quant. Grav. 11,1387 (1994), [arXiv:hep-th/9308075]. N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, (Cambridge University Press, 1982). M. Henningson and K. Skenderis, JHEP 9807, 023 (1998), [arXiv:hept h/9806087]. M. Henningson and K. Skenderis, Fortsch. Phys. 48,125 (2000), [arXiv:hepth/9812032]. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38,1113 (1999)], [arXiv:hep-th/9711200]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998), [arXiv:hep-th/9802109]. E. Witten, Adv. Theor. Math. Phys. 2,253 (1998), [arXiv:hep-th/9802150]. M. J. Duff and J. T. Liu, Phys. Rev. Lett. 85, 2052 (2000) [Class. Quant. Grav. 18,3207 (2001)], [arXiv:hep-th/0003237]. L. Randall and R. Sundrum, Phys. Rev. Lett. 83,4690 (1999), [arXiv:hepth/9906064]. S. W. Hawking, T. Hertog and H. S. Reall, Phys. Rev. D 62,043501 (2000), [arXiv:hep-th/0003052]. S. W. Hawking, T. Hertog and H. S. Reall, Phys. Rev. D 63,083504 (2001), [arXiv:hep-th/0010232]. A. A. Starobinsky, Phys. Lett. B 91,99 (1980). M. J. Duff, J. T. Liu and H. Sati, Phys. Rev. D 69,085012 (2004), [arXiv:hepth/0207003]. A. Karch and L. Randall, JHEP 0105,008 (2001), [arXiv:hep-th/0011156]. C.R. Graham and J.M. Lee, Adv. Math. 87,186 (1991). C. Fefferman and C.R. Graham, in Elie Cartan et les Mathematiques d’aujourd’hui (Asterisque, 1985) p. 95. M. J. Duff, Problems in the classical and quantum theories of gravitation, Ph. D. thesis, Imperial College, London (1972).
101 21. 22. 23. 24. 25.
M. J. Duff, Phys. Rev. D 9, 1837 (1974). J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224 (1976). A. Vilenkin, Phys. Rev. D 32, 2511 (1985). H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22, 397 (1970). V. I. Zakharov, JETP Lett. 12, 312 (1970) [Pisma Zh. Eksp. Teor. Fiz. 12,
447 (1970)]. 26. M. Porrati, Phys. Lett. B 498, 92 (2001), [arXiv:hep-th/0011152]. 27. I. I. Kogan, S. Mouslopoulos and A. Papazoglou, Phys. Lett. B 503, 173 (2001), [arXiv:hep-th/0011138]. 28. F. A. Dilkes, M. J. Duff, J. T. Liu and H. Sati, Phys. Rev. Lett. 87, 041301 (2001), [arXiv:hep-th/0102093]. 29. M. J. Duff, J. T. Liu and H. Sati, Phys. Lett. B 516, 156 (2001), [arXiv:hepth/0105008]. 30. M. J. Duff, J. T. Liu and H. Sati, Nucl. Phys. B 680, 117 (2004), [arXiv:hepth/0211183]. 31. M. J. Duff, Phys. Rev. D12, 3969 (1975). 32. M. Porrati, JHEP 0204, 058 (2002), [arXiv:hep-th/Oll2166]. 33. K. Intriligator, Nucl. Phys. B 730, 239 (2005), [arXiv:hep-th/0509085]. 34. D. Martelli, J. Sparks and S. T. Yau, “Sasaki-Einstein manifolds and volume minimisation,” arXiv:hep-th/0603021.
HIGHER DIMENSIONAL PERSPECTIVE ON N=2 BLACK HOLES ANSAR FAYYAZUDDIN Department of Natural Sciences, Baruch College, CUNY, New York, NY 10010, USA E-mail: [email protected]. edu We present 11 and 10 dimensional supergravity descriptions of N=2 4 dimensional black holes. The problem is formulated in the context of type IIA string theory compactified on R(3!1)~CY3.The black holes are bound states of D4-branes wrapped on 4-cycles and DO-branes. We reduce the problem to a set of differential equations which describe black hole solutions in the lowest derivative supergravity theory. We argue that the full space-time is inherently 10 (or 11) dimensional. We then proceed to solve these equations in the nearhorizon regime, which we define using ideas of ”geometrical transitions” - where localized branes are replxed by smooth flux. Keywords: black holes, supergravity
1. Introduction In these proceedings I summarize some results obtained in [l].The problem, generally speaking, is to find ways of treating string theory black holes in a manifestly 10 (or 11) dimensional setting. In this paper we will work in the low-energy approximation of string theory known as supergravity. Much is known about supersymmetry preserving black holes that arise as solutions in 4- and 5-dimensional supergravity. These solutions have provided us with the setting for a number of interesting results, including the crowning achievement of the Strominger-Vafa microscopic counting of black hole entropy’. In this short article we will be concerned with a class of black holes that arise in the context of Calabi-Yau compactifications of type IIA string theory down to 4-dimensions. The approximation that one usually employs is one where the energy scales for the theory are restricted such that the internal Calabi-Yau manifold cannot be resolved geometrically. In other words, the Calabi-Yau is treated geometrically as a point. The low-energy
102
103
fluctuations of the Calabi-Yau provide us with fields in the d=4 theory. For instance, there are low-energy fluctuations of the Calabi-Yau metric which preserve the complex structure. These excitations appear as massless fields in the d=4 effective theory, known as the Kahler moduli fields. In the d=4 type IIA supergravity theory there are a number of U(l) gauge fields. These gauge fields can be understood as follows. In ten dimensions there are charged 2- and 4-branes, they couple to anti-symmetric tensor fields and are magnetic duals of each other. In the 4-d theory there are point particles which arise from D2-branes wrapping 2-cycles and D4branes wrapping 4-cycles. These point particles are also charged, but now with respect to 1-forms. The number of independent 1-forms is simply given by the number of 2-cycles or 4-cycles, i.e. the Betti numbers bz(= b 4 ) . There is an additional U(l) gauge field, known as the "graviphoton". The graviphoton is a U(l) gauge field in d=10 which couples electrically to DObranes and magnetically to D6-branes. In the d=4 theory this U(l) field continues to couple electrically to the DO-branes and magnetically to D6branes wrapping the entire Calabi-Yau. Thus there are bz 1 = h(l?') 1 gauge fields. Given these gauge fields one can try to construct Reissner-Nordstrom black holes. It is known that a large class of these black holes preserve supersymmetry - these are extremal black holes where the charge and mass are equal. These black holes have some very interesting properties. The geometry interpolates between two regimes: R(3,1)at infinity and Ad& x S2 close to the horizon. A general feature of these black holes is that there is a so-called "attractor mechanism" at play. The attractor mechanism ensures that the near-horizon geometry depends only on the charges of the black holes; it is independent of the values of any fields at infinity. In addition to these general results, it is also known that black holes made up of wrapped Dbbranes and DO-branes can have a non-zero horizon area in the classical lowest derivative supergravity theory. In the usual conventions, these black holes are magnetically charged with respect to the b2 U(l) fields and electrically charged with respect to the graviphoton. The area of the horizon is given by the magnetic charges pz,i = 1,...,b2, electric charge 40 with respect to the graviphoton of the black hole as well as the triple intersection number of the underlying Calabi-Yau, cijk, which appears as a coupling in the effective d=4, N=2 supergravity action. The area of the horizon at leading order is given by 4-6:
+
+
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2. 11-dimensional solutions for 4-d black holes In this section we outline some results on how to treat a class of N=2 black holes in d=4 from an 11-dimensional point of view. The class of black holes we are interested in are the ones mentioned at the end of the previous section. They are magnetically charged with respect to the b2 U( 1) fields and electrically charged with respect to the graviphoton. These black holes have a well understood origin in 10-d, as mentioned above. They descend from D4-branes wrapping 4-cycles and DO-branes. It is also well understood how to lift this configuration to l l - d i m e n ~ i o n s Under ~ ~ ~ ~ this ~. lift the background geometry lifts to the Calabi-Yau geometry times a circle, CY3xS'. The D4-branes lift to M5-branes wrapping both a 4-cycle in the Calabi-Yau and the S1. The DO-branes lift to momentum along the S1. We would now like to find the supergravity solution for this configuration directly in d = l l . In [l]we took the supergravity solution for M5-branes wrapping a holomorphic 4-cycle as found in [8] (see also [9]) and added momentum along the circle. The final metric is given by:
+ f ( d y + (f-' l ) d t ) 2 ) + H2/3(dr2+ r2dR;) + 2H-1/3gm,dzmdz",
ds2 = H-1/3(-f-1dt2
-
(2)
where g is a Kahler metric dR; is the metric on a unit 2-sphere, and H is a function. H is related to the determinant of the metric g through det(g) = aH21hI2,where h is holomorphic in z m (allowing for holomorphic changes of coordinates') and a is a constant. The 4-form field strength of 11-d supergravity is given by: 928,
F
i i --r2dTgmfidzm A dz" A dvoZ(S2) -r2dmHdzm A dr A dvoZ(S2) 2 2 i - --r2dmHdzmA dr A dvoZ(S2). (3) 2
+
=
In the above expression, dvoZ(S2)is the volume form on the unit two sphere. The functions f and H , and the metric g satisfy the differential equations: 1 H-l--dT(r2dTf) 2g""dmdfif = 0, r2 1 -dT(r2dTgmii) 2dmdfiH = 0. (4) r2 Once we solve these two differential equations we can completely specify the metric and 4-form. In general the differential equations (4) are difficult to solve. In fact, the differential equation for g is non-linear because of the presence of H .
+ +
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Nevertheless, there are some important lessons we can learn even at this stage. First, it should be clear that in general the metric does not have a product form. This is because H , f and g depend on T , zm, z m , and, except for the extremes of the black hole space time, as we shall see, the metric is irreducibly 11-dimensional. Second, the metric g, which approaches a Calabi-Yau metric as one moves far away from the black hole, is not Calabi-Yau or even conformal to Calabi-Yau throughout the remainder of the spacetime. Thus the “compactifying” manifold in the interpolation between the asymptotic and near-horizon regimes does not have a simple description except as a Kahler manifold. If we compute the “6-d” Ricci tensor of the Kahler metric g, it can be expressed as derivatives of H with respect to zms and does not vanish generically. 3. The near-horizon regime and geometrical transitions
Black holes are often thought of as space-times interpolating between the asymptotically far and the near-horizon regions a. These two regimes are useful for studying different aspects of black holes. The far regime is where one defines the ADM mass, but it is the near-horizon geometry that determines its Bekenstein-Hawking entropy. In the case at hand we know that at infinity we should approach R3t1xCY3 - the ”vacuum” manifold. In this limit, g approaches the metric on the Calabi-Yau and H , f + 1. We would like to say something about the near-horizon behavior of the geometry and other supergravity fields. Generally speaking the near-horizon geometry can be thought of as a truncation of the full space-time geometry to the desired region. Since we don’t know the entire geometry of the black hole, we might wonder if there are simplifications that occur in the near-horizon region which allow one to determine it even while the full geometry remains unknown. We will argue that there is a sense in which one might hope to isolate the near-horizon region without attempting to find the entire space-time of the black hole. The central idea was presented inlo where it was argued that as one hones in on the region of space-time close to the branes composing the black hole, the localized branes dissolve and we are left with a smooth region with flux. This flux computes the charge of the dissolved branes. aIt sometimes makes sense to continue the space-time beyond the horizon a s well. This will not be relevant to anything we are interested in here.
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This transition is called a "geometric transition" in [lo].We will use this idea to develop an ansatz for the supergravity solution in the near-horizon region and solve the differential equations that we wrote down earlier. To develop the ansatz it is easier to think about the system in type IIA string theory. In this theory our black holes are bound states of D4 and DObranes. The DCbranes couple magnetically to the tensor field whose field strength is given in (3). To compute the charge of DCbranes we should compute the flux produced by F through an appropriate cycle. The DObranes, on the other hand, couple to a R-R field C1 = (f-' - 1)dt. The DO brane charge is computed by calculating the flux of the 8-form Gg = *dC1. We take as an ansatz':
F
t
F' = ~2 A dvoZ(S2),
Gg + GL = WG A
dvol(S2).
(5)
Here w2 and wg are 2- and 6-forms defined on the complex %fold over which the metric g is defined. We will assume that these forms do not explicitly depend on r , since they compute the number of branes which is independent of r . This ansatz implies, by a quick comparison to (3) and the definition of Gg, that 8,H = 0 and 8, f = 0, or H = H ( r ) ,f = f ( r ) . As explained in [ l ] this , ansatz leads one finally to the supergravity solution:
ds?,
=
r2 R2(--ddt2 R2
R2 r + -dr2) + R P 4 ( d y + (- - l ) d t ) 2 r2 a0
+ R2R2dOi+ 2k,fid~md~fi,
(6)
with
IC,~
=
-KPR-~(W~),~
(7)
the metric on the complex %fold. R and R are constant parameters which count the number of D4 and DO-branes. The metric Ic is Calabi-Yau (Ricci flat and Kahler). If we define Jk = ilCmAdzm A dz" to be the Kahler form on the complex 3-fold, then
F = ~2 A dvoZ(S2) = R R J k A dvoZ(S2), G2 = dC1 = d( f-' - 1 ) = R 3 R - l d t A dr.
(8)
This is a completely smooth solution. A particular solution of this kind (0 = 1 ) was also found in [ l l ]We . list the main features of the solution: 0
The metric is an AdS2xS2xCY3 space with a U ( l ) bundle over the Ads2 factor. The U ( l ) bundle is such that the space-time is in
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0
0
0
fact Ads3 x S2 x CY3. The radius of curvature of the Ads3 space is f i R R and the radius of curvature of the S2 is OR. The field strengths F and G2 are non-singular. In the presence of localized branes these field strengths would blow up at the location of the branes. This is not the case here, the flux is completely smooth. The metric k is a Calabi-Yau metric. This Calabi-Yau space is not the same manifold as the one that appears in the asymptotic region. There are many ways to see this but perhaps the easiest way to see it is by noting that certain cycles of the Calabi-Yau in the near-horizon region have non-zero size due to the presence of flux, while the asymptotically far away Calabi-Yau does not have this property since there is no flux present as we move infinitely far away from the black hole (T -+00). More explicitly
The first of these integrals counts the number of D4-branes wrapping a 4-cycle dual to & while the second counts the number of DO-branes. We assume that both integrals are non-zero - i.e. that the number of D4 and DO branes are non-zero. On the right hand side the volumes of appropriate cycles appear. We have thus related the volume of cycles to the flux and therefore to the number of branes. There is no such relation in the T ---t 00 limit. From the previous point one can see that the near-horizon supergravity solution is determined in terms of the charges of the branes - the number of D4, DO branes and specifying the cycles through which the F-flux should be computed. This is the manifestation of the attractor mechanism in this higher dimensional setting.
4. Acknowledgments
I am grateful to the organizers, Faheem Hussain and Riazuddin, for the invitation to attend and for organizing a stimulating conference. I would also like to thank the participants at the workshop who came from diverse countries to attend for many interesting discussions. It was particularly inspiring to have a large number of physicists from India there.
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References 1. A. Fayyazuddin, Class. Quantum Grav. 23,5279 (ZOOS), hep-th/0603141. 2. A. Strominger and C. Vafa, Phys. Lett. B379 , 99 (1996), hep-th/9601029. 3. S. Ferrara, R. Kallosh, and A. Strominger, N=2 eztremal black holes, hepth/9508072 A. Strominger, Macroscopic entropy of N=2 extremal black holes, hepth/9602111 S . Ferrara and R. Kallosh, Supersymmetry and attractors, hep-thl9602136. 4. K. Behrndt, G. Cardoso, B. de Witt, R. Kalosh, D. Lust, T. Mohaupt, Classical and quantum N=2 supersymmetric black holes, hep-th/9610105. 5. M. Shmakova, Calabi- Yau black holes, hep-th19612076. 6. J. Maldacena, A. Strominger, and E. Witten, Black Hole Entropy an M-theory, hep-th/9711053. 7. C. Vafa, Black holes and Calabi- Yau threefolds, hep-th/9711067. 8. H. Cho, M. Emam, D. Kastor and J. H. Traschen, Calibrations and Fayyazuddin-Smith Space-times hep-th/0009062 9. T. Z. Husain, That’s a wrap! hep-th/0302071. 10. C. Vafa, Superstrings and Topological Strings at Large N , hep-th/0008142. 11. A. Simons, A. Strominger, D. Thompson, X. Yin, Supersymmetric Branes in Ad& x S 2 x CY3, hep-th/0406121.
pSTRINGS vs. STRINGS DEBASHIS GHOSHAL Harish-Chandra Research Institute, Chhatnag Road, Allahabad 21 1019, India E-mail: [email protected] The amplitudes for the tree-level scattering of the open string tachyons, generalised to the field of padic numbers, define padic string theory. We briefly review some properties of this pstring theory and take a fresh look at the p + 1 limit, where it is known to approximate the usual string theory. We argue that it should be thought of as a continuum limit.
In this talk, we will consider the relation between p-strings and strings. However, while the latter is well known, the former is perhaps not so. It may not be out of place, therefore, to review some aspects of p-string theory before we revisit the p -+ 1 limit, where p-strings are known (empirically) to approximate ordinary strings”. Recall that the tree-level scattering amplitude of N on-shell open-string tachyons of momenta Ici (i = 1,.. . ,N ) , Ic: = 2, C Ici = 0 is: N J
45i<j
i=4
The integrals are over the real line R and the integrand involves absolute values of real numbers. The 4-point amplitude can be computed exactly, but d N for N 2 5 cannot be evaluated analytically. Perhaps with a mathematical motivation, Ref. [l]considered the above problem over the local field of p-adic numbers, to which it admits a ready extension. In order to describe it, let us digress briefly. On the field of rational numbers Q, we are familiar with the absolute value norm. The field R of real numbers arise as its completion when we put in the limit points of Cauchy sequences, in which convergence is decided by the absolute value norm. However, it is possible to define other ~
would like to suggest that the p in the beginning of ‘p-string’ is silent, making it phonetically same as ‘string’.
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norms on Q consistently. To this end, fix a prime number p and determine the highest powers n1 and 722 of p that divides the numerator z1 and denominator z2 respectively, in a rational number z1/z2, ( 2 1 , z2 coprime). The p-adic n o m of z1/z2, defined as: 121/221, = p"2-n1, satisfies all the required properties, indeed even a stronger version of the triangle inequality 1x yl, 5 max(lxlp, /yip). In fact, apart from the absolute value norm, the p-adic norms are the only possible ones (upto a natural notion of equivalence). If we require completion using the notion of the padic norm, we get the field Q p . Any padic number E E Q, has a representation as a Laurent-like series in p:
+
E = PN (Eo + t1P + E2P2 + . . . ) , where, N is an integer, En E {0,1, . , p - 1) and 60 # 0. A review of these properties may be found in Ref. [3]. Coming back to the Koba-Nielsen amplitudes (l),Freund et. al. proposed to modify these by replacing the absolute values in the integrand by the p-adic norms and the real integrals by integrals over the field Q,. These are, by definition, the amplitudes for the scattering of N open pstring tachyons. An amazing consequence is that all these integrals over Q, can be evaluated analytically. This means that the tree level effective action of the open p-string tachyon T is known exactly. It is best summarised as2l4 +
.
(3)
+
in terms of the rescaled and shifted field 'p = 1 g,T/p. The potential has a local minimum and two (respectively one) local maxima for odd (respectively even) integer p (see Fig. 1).We note parenthetically that the spacetime action can be extrapolated to all integers. Moreover, in this modification, the boundary of the the open p-string worldsheet is valued in p-adic numbers, but the target spacetime is the usual one.
Fig. 1. Exact tachyon potential in pstring theory for p = 2 , 3 .
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Apart from the trivial constant solutions, 'p = 0'1, to the equation of motion from (3), there are soliton solutions. In fact, the equation separates in the space-time arguments and for any (spatial) direction, there is a localised gaussian lump, whose amplitude and spread are correlated2. More recently p-string theory has come in to focus through the realisation that the exact spacetime theory of its tachyon allows one to study nonperturbative aspects of string dynamics, such as the process of tachyon condensation. In Ref. [5], the solitons of the effective theory of the p-adic were identified with the D-branes and shown that the dynamics is according to the behaviour conjectured by Sen7. A rather unexpected relation emerges with the usual bosonic open string in the p 4 1 limit8 (see also the prescient comments in Ref. [9]). In this limit, the effective action (3) of the tachyon of p-string theory turns out to approximate that obtained from the boundary string field theorylO>" (BSFT) of usual strings. The formalism of the latter was useful in proving the Sen conjecture^^^^^^^^. This correspondence remains true even after a noncommutative deformation of the p-string effective a c t i ~ n ' ~In ? ~fact, ~. it can be used to find exact noncommutative solitons of the usual string theory for all values of the deformation parameter14. The relation to the usual strings is all the more surprising and counterintuitive from the point of view of the p-string 'worldsheet'. In fact, the 'worldsheet' description16 of the p-string itself is not in the least obvious. The interior of the worldsheet, analogous to the unit disc or the upper halfplane of the usual theory, is an infinite lattice with no closed loops, i.e., a tree B, in which p + 1edges meet at each vertex (see Fig. 2). This is known to mathematicians as the Bruhat-Tits tree and is familiar to physicists as the Bethe lattice. The boundary of the tree B,, defined as the union of all infinitely remote vertices, can be identified with the p-adic field Q,. One may use e.g., the representation (2) in which case, the integer N chooses a branch along the dotted path (in Fig. 2) and the infinite set of coefficients determine the path to the boundary. On the other hand, the tree 17, is the (discrete) homogeneous space PGL(2 ,Q,) /PGL( 2,Z,) : the coset obtained by modding PGL(2,Q,) by its maximal compact subgroup PGL(2,ZP). This construction parallels the case of the usual string theory, in which the UHP is the homogeneous coset PSL(2,R) modulo its maximal compact subgroup SO(2). The action of PGL(2,Q,) on Qp extends naturally to B,. The Polyakov action on the 'worldsheet' B, is the natural discrete lattice action for the free massless fields X p . Zabrodin16 showed that starting with a finite Bethe lattice and inserting the tachyon vertex operators on the
en
112
boundary, one recovers the prescription of Refs. [l,21 in the thermodynamic limit. If we set p = 1, we get a one dimensional lattice. However, the relation to the usual string is through the limit p -+ 1 and naively it is not apparent how to make sense of this. This is the problem we will address in the following. First, we claim that the p-string ‘worldsheet’, i.e., the Bethe lattice B,, gives a discretisation of the worldsheet of the usual string, the disc. But in B,, the number of sites upto some generation n from an origin C (say):
grows exponentially for large n. Therefore, the formal dimension of the Bethe lattice is infinite, if embedded in an Euclidean space. It is for this reason that these lattices are useful in the study of statistical and field theory models: being infinite dimensional, they give the results of mean field theory and coincide with the result in the upper critical dimension. For example, for a free scalar field theory with arbitrary interactions (that would come from, say, vertex operators) the upper critical dimension is two. One would expect to get the results of the two dimensional scalar field theory with arbitrary interactions from the computation on a Bethe lattice.
Fig. 2. A finite part of the ‘worldsheet’ of the 3-adic string: the tree B, for p = 3, 2 3 3 = Q3. A pat.h from the points 0 and 00 on the boundary is shown by a dotted line.
On the other hand, in a d dimensional hyperbolic space with the metric Ri sinh2 d f l i - , the volume of a ball of radius R (R >> Ro, the radius of curvature)
ds$ = dr2
+
(k)
vold(R)
N
exp
(x R) d-1
(5)
also grows exponentially for large R. This suggests a natural embedding of
113
Bethe lattices in hyperbolic spaces. If we parametrise a p=l+-(d-l),
Ro
and consider the limit a -+ 0 so that p 4 1, the formulas (4)and (5) agree with the identification n-m lim n a = R, from which we see that a is the lattice a-+O
spacing. Thus we see that a Bethe lattice Bp can be used to discretise a hyperbolic space of constant negative curvature. Moreover, it provides a natural continuum limit when p 4 1. In particular, this is true when the dimension d = 2, the case of our interest. In fact the embedding of B, in to the unit disc/UHP equipped with a metric of constant negative curvature (say, the Poincard metric) is isometric - it is related to the hyperbolic tessellation of the disc/UHP and often has rather interesting connections with the fundamental domains of the modular functions of SL(2,C) and its subgroups17. The standard way to obtain a continuum limit from a lattice regularisation is to go to ever finer lattices with smaller lattice spacings and eventually consider the limit in which the lattice spacing becomes vanishingly small. In Fig.3, the ‘black’ lattice with lattice spacing a = 3a’ gives a coarser approximation compared to that with spacing a‘. Suppose we start with the ‘black’ lattice, the boundary of which is isomorphic to Q3. In comparing this sublattice to the full lattice, we see that between two neighbouring ‘black’ nodes there are two (in general rn) new nodes, which in turn branch further so that the new lattice is similar to the old one. What, if any, is the relation of the new lattice to Q3? The answer to this question is interesting and takes us in to the second digression in arithmetic.
Fig. 3. A finer lattice with with lattice spacing a’ leads to a coarse grained lattice with lattice spacing a = ma’ ( m = 3 here), when the ‘grey’ branches are integrated out.
The field Q p (just like R) is not closed algebraically. That is, not all roots of polynomials with coefficients in the field belong to the field. In the
114
case of R, one can adjoin a root of x’ -t 1 = 0 and get the algebraically closed (and complete) field of complex numbers C. It is said to be an index two extension: this means that C is a two dimensional vector space over R. The story is more complex for Q p . One starts with an algebraic, say quadratic, extension; but this field is not closed. In fact, none of the finite algebraic extensions of Q, are closed, so one needs to consider the union of all such extensionsb. Now consider an extension -(m) Q p of index m. Actually there are several such extensions, but among those there are some which are called totally ramified with which one can associate a Bruhat-Tits treeC -(m)
B,
that can be obtained from the original one of Q p through the process described above. Namely, to get the tree for a totally ramified start with the ‘black’ tree for Qp and introduce (m- 1) new nodes between the exisiting ones. Connect (infinite) ‘grey’ branches to these so that the tree is uniform with coordination number p as before. In there is a special element 7r that plays the role that p does for Q p . Specifically, any element of the extended field can be expressed as a Laurent series in terms of 7r (just like (2)), and the norms of its elements -(m) are powers of 7 r . In particular, for the element p E Q p , we have
QLm),
qLm’,
P E Tm *
(7)
Parametrising both p and T as in (6), we see that the lattice spacing a’ of Ekm) is related to a of Bp as a mu’, a fact that is apparent from the construction. We see from (7) that when we consider larger and larger extensions 7~ N p l l m approaches the value 1 for any p . The corresponding lattices provide a finer discretisation and a passage to the continuum limit. This suggests a way to understand the limit p 4 1 through string theories based on the extensions of Q,. However, not everything is in place yet. If we compute the tachyon amplitudes given by (1) based on the field -(m) Q p , the answer we get is exactly the same as that for Q p ! This is because the coefficients in the Laurent expansansions of both are the same, the trees are similar, therefore, the measures that affect the integrals work out to be identical’. The resolution comes from the following. In taking a continuum limit, we are not really interested in the exact answer, but rather turns out that it is not complete. Thankfully, after completion, the resulting field is still closed. We will not worry about it at this point, although this may turn out to be the right setting for the closed p-strings. CThere are Bruhat-Tits trees for all algebraic extensions, and our argument can be repeated in those cases. The case we discuss is the most transparent.
115
in comparing the degrees of freedom of the coarse-grained lattice from the fine one from a (real space) renormalisation group perspective. In order to do this, we should integrate out only the degrees of freedom on the new nodes and new branches (‘grey’ in Fig. 3) so that we are left with those in the ‘black’ sublattice with some effective interaction between these residual degrees of freedom. A rescaling of the lattice so that the spacing a 4 ba = a’ completes the RG transformation. Let us see the effect of these on the Poisson kernel on the Bethe lattice. It is more transparent for the Dirichlet problem for which the Green’s function is16 V ( z , w )= where d ( z , w)is the number of steps in lattice units between the sites z , w. Since the spacing in t3, is m times that in -(m) B, , da = mdfic,, z md, after f i p - d ( Z l w ) ,
integrating out the intermediate sites we have V e = ~ & p - m d ( Z ~ W ) . When we rescale the lattice, the original form of the kernel is recovered with the substitution p -+ 7~ = p 1 I m . The Green’s functionN(z,w) for the Neumann problem is roughly the logarithm of D ( z , w ) ,so the same argument holds there. Thus we conclude that the RG procedure would make sense if its effect on the tachyon action (3) is to replace p -+ r = PI/”. The action for the usual bosonic string is obtained in the limit m -+ 00, which is a continuum limit in the sense of RG. Related results in the literature lend support to this argument. Ref. [18] finds an exact solution to the problem of a random walk on the Bethe lattice. In the limit p -+ 1 (continuum limit) this gives the solution of the Brownian motion on a hyperbolic space of constant negative curvature. This proves that the Green’s function for the diffusion equation on the disc/UHP with a metric of constant negative curvature can be obtained as a continuum limit from the Bethe lattice. Using the well known relation between the kernel of the diffusion equation and the Green’s function of a free scalar field theory, the latter kernel for the disc/UHP with a metric of constant negative can be obtained from the Bethe lattice in the p -+ 1 limit. We are interested in a diffeomorphism and Weyl invariant free scalar field theory coupled to the metric on the disc/UHP. There are also marked points corresponding to asymptotic states given by vertex operators on its boundary. Only hyperbolic metrics can be consistently defined on such a surface. Further with the freedom from diffeomorphism and Weyl invariance, the metric can be made one of constant negative curvature. In the worldsheet functional integral, therefore, the contribution is from such a surface and the sum over surfaces is equivalent to integrating over curvature. The continuum limit of a scalar field theory on a Bethe 1,atticewould seem to give a good approximation.
116
Acknowledgments
I a m grateful t o the organisers of this very interesting conference for the invitation. I cannot thank Faheem Hussain, Riazuddin a n d all t h e others enough for the trouble they took t o make everything, especially the visa procedure, so easy for us. T h e warmth of the hospitality of t h e people of Pakistan will always remain a cherished memory. Discussion with Chandan Dalawat, Peter F'reund a n d Stefan Theisen and hospitality at the Albert Einstein Institute during the process of writing are also acknowledged gratefully.
References 1. P. Freund and M. Olson, Phys. Lett. B 199,186 (1987);
P. Freund and E. Witten, Phys. Lett. B 199,191 (1987). 2. L. Brekke, P. Freund, M. Olson and E. Witten, Nucl. Phys. B 302, 365 (1988). 3. L. Brekke and P. Freund, Phys. Rept. 233,1 (1993). 4. P. Frampton and Y . Okada, Phys. Rev. D 37,3077 (1988). 5. D. Ghoshal and A. Sen, Nucl. Phys. B 584, 300 (2000), [arXiv:hepth/0003278]. 6. P. Frampton and H. Nishino, Phys. Lett. B 242,354 (1990). 7. A. Sen, Non-BPS states and branes in string theory, arXiv:hep-th/9904207. 8. A. Gerasimov and S. Shatashvili, JHEP 0010, 034 (2000), [arXiv:hepth/0009103]. 9. B. Spokoiny, Phys. Lett. B 207,401 (1988). 10. E. Witten, Phys. Rev. D 46, 5467 (1992), [arXiv:hep-th/9208027]. 11. S. Shatashvili, Phys. Lett. B 311,83 (1993), [arXiv:hep-th/9303143]; S. Shatashvili, Alg. Anal. 6 ,215 (1994), [arXiv:hep-th/9311177]. 12. D. Kutasov, M. Marino and G. Moore, JHEP 0010,045 (2000), [arXiv:hepth/0009148]. 13. D. Ghoshal and A. Sen, JHEP 0011,021 (2000), [arXiv:hep-th/0009191]; D. Ghoshal, in Proceedings of Strings 2001, eds. A. Dabholkar et. al., [arXiv:hep-th/0106231]. 14. D. Ghoshal, JHEP 0409,041 (2004), [arXiv:hep-th/0406259]. 15. D. Ghoshal and T. Kawano, Nucl. Phys. B 710, 577 (2005), [arXiv:hepth/0409311]. 16. A. Zabrodin, Commun. Math. Phys. 123,463 (1989); L. Chekhov, A. Mironov and A. Zabrodin, Commun. Math. Phys. 125,675 (1989). 17. S. Nechaev and 0. Vasilyev, J. Phys. A 37, 3783 (2004), [arXiv:condmat/0310079]. 18. C. Monthus and C. Texier, J. Phys. A 29, 2399 (1996), [arXiv:condmat/9509067].
DETECTING TWO-PHOTON EXCHANGE EFFECTS IN HARD SCATTERING FROM NUCLEON TARGETS PERVEZ HOODBHOY Department of Physics, Quaid-e-Azam University, Islamabad 45320, Pakistan. Following the discovery of a discrepancy between nucleon formfactors extracted from polarization transfer methods and the traditional (single photon) h s e n bluth formula, two-photon exchange processes have come under scrutiny. We consider here the production of a lepton-antilepton pair by real photons off a hadronic target. The interference of one and two photon exchange amplitudes leads to a charge asymmetry term that may be calculated explicitly in the large-t limit in terms of hadronic distribution amplitudes. A rather compact expression emerges for the leading order asymmetry at fixed angle in the centre-of-mass of the lepton pair. The magnitude appears sizeable and is approximately independent of the pair mass in the asymptotic limit.
Measurement of the nucleon's electric and magnetic form factors has traditionally been based upon the well-known Rosenbluth formula' which assumes that scattering occurs through the exchange of a single photon. A recent extraction of G$/GG in the Q2 range from 0.5 to 5.6 GeV2 has used the polarization-transfer technique exploited at Jefferson Laboratory2. This method, which does not use the Rosenbluth separation, revealed a large discrepancy with previously published form factors. Subsequently, a close scrutiny was made of two-photon exchange effects in elastic electron-proton scattering. In this process a virtual photon knocks the incident proton into an excited state, and a second one de-excites it back into the ground state. Both photons may be hard, and hence they probe nucleon structure. The effects were found to exist at a few percent level and are capable of resolving the observed discrepancy3. Model dependence is inevitable. Chen et al.4 have also calculated the two-photon exchange contribution and related it to the generalized parton distribution~~ that occur in various other hard processes as well. A clear exposition of experimental techniques and
117
118
two-photon physics may be found in a review by Wright and Jager6. In this note, I shall consider two-photon effects in the production of a lepton-antilepton pair from a hadronic target by real photons (see Fig. 1) at high centre of mass energy s.
2-
Fig. 1.
Exclusive photoproduction of a lepton pair off a proton target.
The process predominantly occurs through the exchange of one photon, with a two-photon admixture. Both amplitudes are for the same final state and so interfere with one another. These exchanges come from diagrams with opposite charge conjugation7. This fact enables one to isolate the interference term by counting the difference in the number of produced antileptons and leptons. In the limit of large momentum transfer t , they can be explicitly computed from the leading Fock-state in terms of lowest order hadronic distribution amplitudes. We find a rather compact expression for the leading order asymmetry at fixed angle in the centre-of-mass of the lepton pair. The inputs to this calculation, apart from the hadron distribution amplitudes, are experimentally determined hadronic form factors at sufficiently large t. We note that Berger, Diehl, and Pire had analyzed and estimated exclusive lepton pair photoproduction (at small t ) as a means of studying generalized parton distributions in the nucleon'. Exclusive electroproduction of J/psi mesons (which then decay into lepton pairs) has been studied at HERAS for fairly small Q 2 values. Before considering the more complex case of the proton, we shall first consider a pion target. It is convenient to work in the rest frame of the produced lepton-antilepton pair. The mass of the quarks, and of the hadronic target, have been ignored in this preliminary calculation, i.e. P 2 = PI2 = 0 where P p and P I P are the momenta of the initial and final hadron. The squared invariant mass of the produced leptons (also assumed massless) is
119
M 2 . This may be selected at will and should be chosen far away from a re+ onance. Simple expressions emerge only for s >> -t, M 2 >> A&D where t is defined, as usual, from t = ( P - P ' ) 2 .It is negative in the physical region. The incoming real photon (g2 = 0) is taken along the z-axis, and the x - z scattering plane is defined by the incoming vectors g p and Pi', + where P makes angle .1c, with the z-axis. The outgoing, elastically scattered, hadron is in the x - z plane. The sum of the lepton-antilepton momentum vectors is Kfi = Zp 1'". A convenient parametrization of the scattering kinematics is then provided by
+
where the incoming hadron energy E , incoming photon energy w , and angle .II, are
s+t 2M M 2-t w=2M c=-
cos$=l--.
S
2WE
The outgoing lepton and antilepton, also taken to be massless, have spinors u(Z)and v(Z'). The lepton and anti-lepton momentum vectors are
M
2 p = -(1,sin 8 cos 4, sin 8 sin 4, cos 8) , 2 t M Z p = -(l, - sin8 cos 4, - sin 8 sin 4, - cos 8 ) . 2
(8) (9) 4
The azimuthal angle 4 is measured relative to the plane formed by P and (which defines the z-axis and hence 8). In the centre-of-mass frame 4 used here P also lies in this plane. By angular momentum conservation, the two massless leptons have opposite helicities. Since we shall work a t the amplitude level, we need simple, covariant expressions for the matrix v(Z')E(Z) in the helicity basis. The method developed by Vega and Wudka" is especially convenient when used in the cm frame:
7
120
An overall phase has been set to zero in the above matrix since it will not enter the cross-sections. The auxiliary vector q$ is defined as, T&
= ( 0 , c o s ~ c o s $ f i s i n ~ , c o s ~ c o s $ ~ z c-sine). os$,
Here f refers to the lepton helicity.
QP
(12)
satisfies,
Consider now lepton pair production from a pion via the form factor diagram (Fig.2). This, together with its crossed counterpart, is easily calculated in the large s limit.
Fig. 2.
Form-factor contribution to lepton pair production at lowest order.
With the pion factor normalized such that F,(O) = 1, the amplitude for producing a positive helicity lepton from a positive helicity real photon is
Ail
= AI(B,$,yT1I t l rl)
For convenience we have defined the dimensionless momentum transfer,
t
p = --
M2.
By examining the y-matrix structure, the other helicity amplitudes are
121
For momentum transfers much higher than the invariant mass of the produced lepton pair
Note that the $ dependence entirely disappears in this limit. The singular behaviour for 0 4 0 comes from the lepton propagator in Fig. 2 and disappears upon including the lepton mass. However, for purposes of comparing with the other amplitudes to be computed below, where including the mass would make the formulae less transparent, this mass will be kept a t zero in this preliminary calculation. The squared amplitude from Fig. 2, summed over lepton polarizations, and averaged over photon polarizations, is:
The coefficients a, are a0 a1
,
= -(p2
+ +
4p 1) 2(p2 = 4&(p - 1) cot 8, -
+ 1) csc26 ,
(23) (24)
2p. (25) Under spatial inversion (i.e. 0 + B 7r and $ -+ $ ) the outgoing lepton and anti-lepton are exchanged. But , computed from Eq. 22 suffers no change. In other words the charge asymmetry a t leading order is zero. A fermion line attached to three vector vertices has the opposite charge conjugation properties relative to the same line with two vertices. We shall use this property in an essential way. So, now consider scattering into the same final state but through the exchange of two photons. The three photon vertices may be connected to the Z-l+ line in six different ways, two of which have been shown in Fig. 3. In general, the scattering from any hadronic state is extremely complicated and, in the language of Fock states, involves an infinite number of proton wavefunction components. However, in the large-t limit, all higher components beyond the minimal one are strongly suppressed in a purely exclusive process. The reason is straightforward: every quark and gluon present a2 =
+
122
Fig. 3.
Typical two-photon contributions to lepton photoproduction from a pion target.
in the initial state must be "turned around" and its momentum components redirected into the final state. This simple fact allows for calculation of hadronic f o r m - f a ~ t o r s ~ as~ ,well as generalized parton distributions1*, etc. In the calculation reported below we assume that large enough values of t can be selected for the minimal 2-quark component of the pion to dominate the scattering. Integrating over the transverse momentum of quarks, one may then approximately represent the pion state by
(26) where the integration measure is [ds] = ds1dz26(1 - z1 - 4.
(27)
Here a = 1 , 2 , 3 denotes colour. Asymptotically, as is well known, &(x) approaches fT&s(l- x). The calculation of diagrams in Fig. 3 is straightforward. Typically one has a denominator that, expanded out in the large s limit, looks like 1 s(xl - yl) [I
+ p + 2ficos4sinO + ( p - 1)
+ if'
This is singular at 51 = y1, and so has a principal part in addition to an imaginary (delta function) piece. The principal parts cancel when taking the sum of all six diagrams for the upper diagrams. The numerator is of 0(s2),and so the overall contribution is proportional to s.
123
After a tedious calculation, the amplitude for producing a positive helicity lepton from a positive helicity real photon via two-photon exchange reads,
In the above, 1:
= 1- X I and the integrand depends upon
8 and 4 :
The other amplitudes can be expressed in terms of A;' :
The negative signs in the above amplitudes will be responsible for the charge asymmetry in the crossection, as we shall see shortly. But first, let us remark on the apparent problem in the integral,
For 4 = 0 the integrand is real and has a singularity inside the integration range. However, reinstating the lepton mass removes the singularity by introducing a term of order m2/M2. This still leaves a very strong 4 dependence, thereby distinguishing the two-photon exchange term from that with a single photon. A proper calculation must, of course, include lepton masses. One might wonder about other diagrams, also of O(e3), such as those in Fig. 4. However, these can be shown to be of O(l/s). Now consider the interference term in IAl A2I2,
+
124
Fig. 4. A typical diagram for lepton pairproduction from a single quark in the target. The sum of all such diagrams is suppressed in the large - s limit for real photons.
After simplication, this becomes,
(35) In the above, C = 1 - x = 1 - XI and e, = 2/3, ed = -1/3. We define the charge asymmetry as:
+
Obviously E(e,$) = -E(O 7r,+). This quantity is proportional to the difference in count rates between antileptons and leptons. As is apparent from the definition of A, the integral over x asymptotically goes to a constant, finite value as p -+ m. A leading order c a l ~ u l a t i o nfor~ ~ the ~~~ pion form factor gives (-t)F, = 12f,27rC~asat large t or, equivalently, pF, = 1 2 ( f , 2 / M 2 ) 7 r C ~ a sNote . that M , the lepton pair invariant mass, has disappeared from the final formula for E. In Fig. 5 , E is plotted as a function of p for fixed angles. Note that we have assumed collinear quarks and so the formula is valid only for M greater than the typical kT of quarks inside the pion.
125
0.4
0.3 0.2 0.1
Fig. 5 .
Lepton pair asymmetry from a pion target.
With the above case for the pion as a warm-up, we now proceed to lepton pair production from a %quark target. For calculational purposes, it is useful to introduce a 4-vector for the scattered hadronlO,
tP= (0, cos$, -i,
-
sin$),
P.E = 0,
(37)
(38)
in terms of which the proton spinor matrix can be expressed as,
The form-factor contribution to lepton photoproduction off a proton is identical to that from a pion in the limit where the spin-flip term (Pauli form factor) is set to zero. Indeed for massless quarks and no transverse momentum, this is strictly true. The minimal state of the proton is,
126
Fig. 6. Typical diagrams for lepton pair production from a 3-quark proton.
where [dz] = dz1dz2dz36(1- 2 1 - z 2 - z3).Asymptotically, @(XI, 22,z3) 120 ~ 1 2 2 x 3but more realistic wavefunctions have been constructed and can be found in refs. [11,121. The proton case requires more work but it follows the calculation detailed above for the pion. With some typical diagrams shown in Fig. 6, we imagine that 3 collinear quarks with charges el,e2,e3 enter from the left with momentum fractions z1,52,23 and emerge to the right with fractions y1,y2,y3. An extra gluon is required to transfer the hard momentum on to the remaining quark, and this brings an additional factor of g2 into the amplitude. The incoming proton is taken to be in a definite (positive) helicity state. In the absence of quark transverse momentum, as well as quark mass, the helicity of the final proton is that of the incoming one. Equivalently, in this approximation, the Pauli form factor is zero. Summing over the twelve different ways of connecting two photons to the 3 quark lines gives, after a tedious calculation: N
(41) In the above (as well as in the equation below), it is implicitly understood that the complex conjugate, and exchange of labels (1,3)% (3,l)
127
Fig. 7.
Lepton pair asymmetry from a proton target.
are to be added on. The charges ei are in units of the electron charge e (ae = e2/47r). Inserting this and the single-photon results into Eq. 34 yields,
J XI idyl el(e222y2 +
Y3)216(!/1- 21) [ Y l E f A1 E2
e323!/3)@(21,22,23)@(!/1, YZ, 212223YlY2Y3
(42)
As a check on the correctness of the calculations for scattering amplitudes
-
for both the meson and baryon cases, we have verified that the Ward iden-
tity is satisfied in all different ways. Perturbatively ( - t ) ’ G M ( t ) constant at large t , and so again one has approximate scale invariance. In Fig. 7 we have plotted the asymmetry off a proton target. In this exploratory calculation we have used the Chernyak-Zhitnitsky wavefunction in ref. [9]. In this paper we have calculated asymmetries, not cross-sections because the latter involve a 3-body phase space. Since J-Psi photoproduction has
128
been measured in exclusive reactionsg, lepton pair production should also be possible. Finally we remark that Sudakov effects, which arise from the bremstrahlung of widely separated quarks that undergo large changes in momentum, will lead to a weakening of the effective coupling. Thus, although the angular structure of the amplitude will probably be similar, one must investigate diagrams that are of one order higher in a,. For the proton case this will involve a very large number of diagrams that will require a machine computation. We have not attempted this calculation.
Acknowledgments The author thanks Stanley J. Brodsky and Xiangdong J i for valuable comments and encouragement.
References 1. M.N. Rosenbluth, Phys. Rev. 79, 615 1950. 2. M.K. Jones, et al., Phys. Rev. Lett. 84, 1398 (2000); 0. Gayou, et al., Phys. Rev. Lett. 88, 092301 (2002); V. Punjabi, et al. Phys. Rev. C71,055202 (2005). 3. P.G. Blunden, W. Melnitchouk, J.A. Tjon, Phys. Rev. C72, 034612 (2005), nucLth/0506039. 4. Yu-Chun Chen, Andrei V. Afanasev, Stanley J. Brodsky, Carl E. Carlson, Marc Vanderhaeghen, Phys. Rev. D72, 013008 (2005), hep-ph/0502013. 5. X. Ji, Phys. Rev. Lett. 78, 610 (1997); ibid., Phys. Rev. D 5 , 7114 (1997); A. V. Radyushkin, Phys. Lett. B380, 417 (1996); ibid., Phys. Lett. B 3 8 5 , 333 (1996); ibid., Phys. Rev. D 5 6 , 5524 (1997). 6. C.E.H Wright and K. Jager, Ann. Rev. Nucl. Part. Sci. 54, 217 (2004). 7. S.J.Brodsky and J.R.Gillespie, Phys. Rev. D173, 1011 (1968). This paper discusses photoproduction of lepton pairs from a nuclear target and also uses interference with the Born amplitude to generate asymmetries. 8. E.R. Berger, M. Diehl, B. Pire, Eur. Phys. J. C23, 675 (2002), hepph/0110062. 9. ZEUS Collaboration, Nucl. Phys. B 6 9 5 , 3 (2004), hep-ex/0404008. 10. R. Vega and J. Wudka, Phys. Rev. D 5 3 , 5286 (1996), erratum: ibid. Phys. Rev. D 5 6 , 6037 (1997). 11. G.R. Farrar, H. Zhang, A.A. Ogloblin, and I.R. Zhitnitsky, Nucl. Phys. B311, 585 (1988). 12. V. Braun, R.J. Fries, N. Mahnke, E. Stein, Nucl. Phys. B 5 8 9 , 381 (2000), erratum: ibid. Nucl. Phys. B607, 433 (2001). 13. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).
129 14. P. Hoodbhoy, X. Ji, and F. Yuan, Phys. Rev. Lett. 92, 012003 (2004), hepph/0309085. 15. A.V.Efremov and A.V.Radyushkin, Phys. Lett. 94B,245 (1980).
BPS M-BRANE GEOMETRIES TASNEEM ZEHRA HUSAIN*
Jefferson Physical Laboratory, Harvard University, Cambridge, M A 02138, U . S . A *E-mail: [email protected] In the search for a classification of BPS backgrounds with flux, we look at geometries that arise when M-branes wrap supersymmetric cycles in CalabiYau manifolds. We find constraints on the differential forms in the back-reacted manifolds and discover that the calibration corresponding to the (background generating) M-brane is a co-closed form.
Keywords: Supergravity, M-Branes, Flux Backgrounds.
1. Introduction Ever since it was found that compactifications of String/M-Theory on special holonomy manifolds preserve supersymmetry, such manifolds have been widely studied. If Euclidean they must also be Ricci-flat and according to Berger's classification their holonomy groups are then dictated by their dimension d ; we have S U ( n ) holonomy (Calabi-Yau manifolds) in d = 2 n dimensions, S p ( n ) holonomy (hyper-Kahler manifolds) if d = 4 n , Gz holonomy in d = 7 and S p i n ( 7 ) in d = 8. This neat categorization however, applies only in the absence of flux; as we will now see, supersymmetric backgrounds are no longer quite so simple once space-time contains flux! 2. D = l l Supergravity
We will study the geometry of BPS M-branes using 11-d supergravity'. While admittedly just an approximation to M-theory, supergravity is useful when considering BPS states since these are protected from quantum corrections by supersymmetry and hence guaranteed to survive the transition to strong coupling/M-Theory. The bosonic fields of d = l l supergravity are the metric G I J and a threeform gauge potential A (with associated field strength F = dA) which cou-
130
131
ples electrically to M2-branes and magnetically to M5-branes. The bosonic actiona of this theory
S
=
1
/ 2t9
d ” s a
1 2
1 6
R - - F A *F - -AA F A F
leads to the equation of motion 1
- -GIJF 1 KLMN FKLMN 12 144 together with the Bianchi Identity dF = 0 and equation of motion for the field strength d * F f F A F = 0. In BPS backgrounds the supersymmetric variations of all fields vanish when the variation parameter is a Killing Spinor. Since we are restricting ourselves to purely bosonic backgrounds, we have set the gravitino 9 to zero, hence the supersymmetric variation of the bosonic fields vanishes trivially. The variation of the gravitino however, is proportional to the bosonic fields and hence not zero a priori. In order t o guarantee supersymmetry, we impose
R I J = - F I K L M F JK L M
+
+
6 x 9 1 = [VI - -r J K L F I J K L -rI J K L M F J K L M ] X = 0. (3) 18 144 When F = 0, the Ricci tensor vanishes and supersymmetry is preserved only if the background admits covariantly constant spinors. From V I X= Ob it follows that [VI, v J ] x= 0 and the identitiy [VI, v J ] x= ~ R I can then be used to show that a Killing spinor in such a background must be a singlet of the Spzn(1,lO) subgroup 7-l generated by R I J K L r K L . In other words, ‘FI has to be a special holonomy group. This is the logic that lies behind the oft-quoted statement that String/M-Theory compactified on special holonomy manifolds is supersymmetric. Note however that this entire argument depends crucially on the fact that F = 0. As it turns out, flux is an intrinsic part of most realistic backgrounds, for instance those generated by BPS M-branes. Being charged gravitating objects, these branes modify any space into which they are placed, warping the geometry and giving rise to a field strength flux. The resulting ‘back-reacted’ manifold is no longer Ricci-flat nor does it admit covariantly constant spinors, but since the brane is BPS, the background remains supersymmetric. aThe field content of 11-dimensional supergravity also contains a fermion - the gravitino q.However, since we are considering purely bosonic solutions, we are not concerned with how the gravitino appears in the action or its equations of motion. bHere, VrX = [El1 $~Yrij]x.
+
132
The work reviewed here is part of a scheme to characterize the geometries of BPS M-branes and arrive at an exhaustive classification of these flux-filled, yet still supersymmetric, backgrounds.
3. The Geometry of BPS M-branes Before we discuss M-branes wrapping more complicated supersymmetric cycles, we look at a planar M5-branec in order to build an intuition for the general features exhibited by supergravity solutions. Consider an M5-brane placed in Minkowski space. Spacetime, after being curved by the brane, is described by the metric ds2 = H-1/3~ p u d X ” d X u H2/36,pdXQdXP and field strength F,p-,a = ;caprap 8,H. Poincare invariance on the worldvolume implies that H be independent of coordinates X” tangent to the brane. Rotational invariance in the transverse directions X a says that H can depend only on the radial coordinate T = and the metric is diagonal in this subspace. The conditions dF = 0 and d * F = 0 fix H to be a harmonic function in the space transverse to the brane, i.e, H = 1+ 5 . More complicated BPS configurations can be generated by wrapping M-branesd on supersymmetric cycles in a compactification manifold M . The very presence of the brane deforms space-time such that M no longer has special holonomy. Regardless of the particular cycle it wraps, there are some univeral features exhibited by the geometry of a BPS brane wrapped on a supersymmetric cycle C embedded in a manifold M . Space-time splits naturally into three subspaces; unwrapped directions X” transverse to M but tangent to the brane, M itself, spanned by dX’ and the X a directions transverse to both brane and manifold. The flat worldvolume directions exhibit Poincare invariance so nothing can depend on X”. Along directions X,, the brane appears point-like and the configuration is rotationally symmetric. This leads to a diagonal metric in this subspace and further dictates that the functional dependence of any physical quantity only involve the radial coordinate r. Fayyazuddin and Smith incorporated these isometriese
+
Jm’
=We are mentioning only M5-branes explicitly but of course a parallel analysis can - and in fact has been - carried out for M2-branes as well. dAn M2-brane along the 012 directions preserves the 16 supersymmetries which survive ro12x = x. The metric and field strength are specified by ds2 = H - 2 / 3 q p v d X p d X V H’/3b,gdXadXP and F012, = where H = 1 + p = 0,1,2 spans the worldvolume and T is the radial coordinate in the transverse space spanned by a = 3 , . . . l o . eThe metric in M is left completely general, since we have made no assumptions about either this manifold or the supersymmetric cycle yet.
+,
+
133
into the metric ansatz
+
ds2 = H ~ ~ p u d X c ” d XGIjdX’dXJ u
+ Hi6,pdX“dXp.
(4)
They then began their search for a supersymmetric supergravity solution by imposing d X Q = 0. This lead to a set of equations which can be solved by expressing the field strength and the metric in terms of a single function H. If, in addition dF = 0 and d * F = 0, we are guaranteed that Einstein’s equations are satisfied. While the former condition is trivially satisfied, the latter leads to a non-linear differential equationf for H. Strictly speaking, a supergravity solution is not found until this equation is solved; though possible in theory, this proves very difficult in practise. However, the process we have just described yields an unexpected boon. In addition t o relating the metric to the field strength, dxQ = 0 also imposes constraints on certain differential forms in M . It is well known that special holonomy manifolds can be defined through conditions on the differential forms they admit. Proceeding in analogy we can begin to characterize M using the constraints obtained by the above analysis. 4. How BPS M-branes deform Calabi-Yaw
In the following, we will restrict ourselves to M-branes wrapped on supersymmetric cycles in Calabi-Yau manifolds. A Calabi-Yau n-fold is, by definition, a Ricci-flat Kahler manifold, with SU(n) holonomy. Equivalently, a Calabi-Yau is defined through the conditions dJ = dR = 0 , where J is the Kahler form and R the unique (n, 0) form on the manifold. Even when explicit metrics are not known, as is the case for most Calabi-Yaus, these conditions on the differential forms provide a wealth of information about the geometry of the manifold. The supersymmetric cycles of a Calabi-Yau are even-dimensional holomorphic cycles, calibrated by the appropriate power of J , and a Special Lagrangian n-cycle calibrated by R. Branes wrapped on these cycles saturate the BPS bound so their charges are fixed by their masses and the calibrating forms are simply the respective volume forms3. Into spacetime of the form R ( 1 > 3CY2 ) ~ xR3, we introduce an M5brane with worldvolume R(’13)x Cz where C2 is a holomorphic 2-cycle and consequently, calibrated by J. This M5-brane deforms the four-manifold such that it is no longer Calabi-Yau4 but instead satisfies the constraint dm[H1I3x J ] = 0. ‘H can hence be thought of as the appropriate generalisation (for a wrapped brane case) of the harmonic function in the solution for a planar brane.)
134
If spacetime looked like R('l3)x C Y ~ X Rthe , presence of an M5-brane with worldvolume R(133)x CZ would modify the geometry and the sixmanifold would be subject to the constraint a M [ H - 1 / 3 * J ] = 0. An M5brane wrapping a holomorphic 4-cycle (calibrated by J A J ) can only be nontrivially embedded into a CY 3-fold. The brane deforms this six dimensional space in such a way that the Calabi-Yau condition dJ = 0 is replaced by i 3 ~ [ H l* /J~A J ] = 0. When membranes wrap holomorphic two-cycles in CY n-folds, their back-reactions distort the 2n-dimensional manifolds such that d ~ [ [ H ( " - ~*)Jl ]~= 0. Recall that Calabi-Yau manifolds also admit another kind of supersymmetric cycle, the Special Lagrangian (SpelL). A SpelL n-cycle C,calibrated by Reie is a n n-dimensional real submanifold of C" on which Jlr. = 0 and ReieIr. = Vol(C).Even though a phase can be incorporated in general, for simplicity, we will consider SpelL cycles calibrated by ReR [or I m R ] . Since a SpelL is a real manifold, there is no reason to assume that a brane wrapping it will deform space in such a way that a complex structure survives on the backreacted manifold M . In fact, the Fayyazuddin-Smith analysis of these geometries5 shows us that only an almost complex structure J survives. For a M5-brane wrapping the SpelL 3-cycleg calibrated by Re R, we find that the Calabi-Yau %fold is deformed into a manifold on which a M * [Re R ] = a M [ I m = 0. The requirement of supersymmetry i3Q = 0 leads to other constraints as well on R and J . Once again, all physical quantities can be expressed in terms of a single function H, which is subject to a non-linear differential equation.
a]
5 . Conclusions
In backgrounds without flux, bosonic supersymmetric solutions of supergravity are given simply by metrics with special holonomy. The question is, how do we generalise this classification when flux is non-vanishing? As a case in point, we studied the background generated by a M-brane wrapping a holomorphic cycle in a Calabi-Yau manifold. Even though we were able to specify the metric and field strength only modulo solution of a non-linear differential equation, we found certain defining equations for the manifold M , into which the Calabi-Yau was deformed. We then proceeded to study M-branes wrapping other supersymmetric cycles, aiming not to find explicit metrics for a handful of examples but intead to characterise gSince a SpelL 2-cycle in a CY 2-fold is merely a holomorphic 2-cycle in a redefined complex structure, we will mention only SpelL 3-cycles here
135
a back-reacted manifold through constraints on its differential forms. Our approach differs from the usual in that most of the work done in this field focuses on finding explicit supergravity solutions in some approximation; these solutions will at best tell us about the local geometry near the cycle. In contrast, the statements we make are global and can be used to classify the back-reacted manifold. For all M-branes wrapping supersymmetric cycles in Calabi-Yau manifoldsh we obtain a constraint on the dual of (the appropriately rescaled) calibration on M . Such constraints, we hope, will lead to a concise and exhaustive classification of supersymmetric flux backgrounds. Acknowledgments: I am grateful to the organizers for the stimulating atmosphere they created in Islamabad this April. Faheem Hussain, in particular, went out of his way to make the conference a success. I would also like to thank Ansar Fayyazuddin for several vary enjoyable collaborations on which this review is based. References 1. D.J.Smith, Class. Quant. Grav. 20, R233 (2003). 2. A.Fayyazuddin & D.J.Smith, JHEP 9904, 030 (1999); B.Brinne, A.Fayyazuddin, T.Z.Husain & D.J.Smith, JHEP 0103,052 (2001). 3. T.Z.Husain, If I Only Had A Brane (PhD thesis), hep-th/0304143; P.K.Townsend, Class. Quant. Grav. 17,1267 (2000). 4. T.Z.Husain, JHEP 0312,037 (2003); T.Z.Husain, JHEP 0304,053 (2003). 5. A.Fayyazuddin & T.Z.Husain, Phys. Rev. D73, 126004 (2006); A.Fayyazuddin, T.Z.Husain & I.Pappa, The Geometry of M-branes Wrapping Special Lagrangian Cycles hep-th/0505182. 6. T.Z.Husain, JHEP 0308,014 (2003).
hThe only exception is the M5-brane wrapping a holomorphic 4-cycle in a CY 4-fold 6. It has been known for a while that this configuration stands apart from the crowd in a number of ways, a key reason behind this apparent discord being that it is the only M-brane geometry encountered so far which does not satisfy F A F = 0
WARD IDENTITIES AND RADIATIVE RARE LEPTONIC B-DECAYS M. JAMIL ASLAM, AMJAD HUSSAIN SHAH GILANI, MARIAM SALEH KHAN AND RIAZUDDIN National Centre for Physics, Quaid-i-Azam University Campus Islamabad, Pakistan. Form factors parametrizing radiative leptonic decays of heavy mesons (Bf -+ $+q) for photon energy are computed in the language of dispersion relations. The contributing states to the absorptive part in the dispersion relation are the multiparticle continuum, estimated by the quark triangle graph, and resonances with quantum numbers 1- and 1+ which include B* and B;I and their radial excitations, which model the higher state contributions. Constraints provided by the asymptotic behavior of the structure dependent amplitude, Ward Identities and gauge invariance are used to provide useful inforare predicted mation for parameters needed. The couplings g B B * y and if we restrict to the first radial excitation; otherwise using these as an input the radiative decay coupling constants for radial excitations are predicted. The value of the branching ratio for the process B+ -+ yp+v, is found to be in the range 0.5 x lop6. A detailed comparison is given with other approaches.
1. Introduction The radiative leptonic decay B+ 4 l+viy has received a great deal of attention in the literature' as a means of probing aspects of the strong and weak interactions of a heavy quark system. The presence of the additional photon in the final state can compensate for the helicity suppression of the decay rate present in the purely leptonic mode. As a result, the branching ratio for the radiative leptonic mode can be as large as for the ,u+ case which is in accordance with the the upper limit provided by the CLEO collaboration for the branching ratio B ( B 4 I v y ) of 2.0 x at 90% confidence level'. This would open up the possibility for directly measuring the decay constant f~ as well as provide useful information about the CKM matrix element IvUbl3. In this paper, we will study the radiative leptonic B decays,
136
137
B+ + l + y y . Our main inputs are dispersion relations, asymptotic behavior and Ward Identities, all of which have strong theoretical basis and in these aspects it differs from other approaches to which we will compare our results at the end. Our approach is close to that followed in Ref. [4] for B -+ ~ 1 ~ 1 . 2. Decay kinematics and current matrix elements
The decay amplitude for radiative leptonic decay, B+ -i Z+uly, can be written in two parts, inner bremsstrahlung ( M I B )and structure dependent ( M s D ) ,as follows:
M ( B + + Z + ~ l y= ) MIB + MSD.
(1)
They are given by5
with
Here E; denotes the polarization vector of the photon; p , p l , p , ; k are the four momenta of B+, Z+, Y , and y, respectively; s1 is the polarization vector of the Z+; f B is the B meson decay constant and FA, FV are two Lorentz invariant amplitudes (form factors) defined by
where q is the momentum transfer.
138
3. Dispersion Relations
The structure dependent part, Hfi” is given by
We note that6
so that for the real photon we can write
where k p H p y = 0 and H p ’ is parametrized as in Eq.(6). The second term in (13) is absorbed in M I B . The absorptive part is
Abs [ i H p V = ] 1/ d 4 x e i k ’ x(0 I[j&(x), J,”(O)]IB ( p ) ) 2
The S-function in the first term implies p i = k2 = 0 and since there is no real particle with zero mass, the first term does not contribute. Thus contributing to the absorptive part are all possible intermediate states that couple to By and are annihilated by the weak vertex (0 IJ,”(O)ln). These include the multiparticle continuum as well as resonances with quantum numbers 1- and 1+.Thus
The dots stand for contributions from higher states with the same quantum numbers. The couplings gBB’y and fB;iBy are defined as
139
We assume that the contributions from the radial excitations of B* and B: dominate the higher state contributions. Thus we write
FA(t) =
RA 1- t / M i ; i
RA( 1- t / M $
1 4
1,
i-
M2
ImFACont !’Ids, (17) s -t
-
ZE
where M is a cut off near the first radial excitation of MB* or Mp.2 and SO= M B mT,and
+
and Rv, and R A are ~ the corresponding quantities for the radial excitations with masses MB: and M B ; ~In~ the . next section we develop the constraints on some of the parameters appearing in the above equations. If we model the continuum contribution by the quark triangular graph (similar calculations exist in the literature7), we obtain
together with the term
which appears in Eq. (13). As is well known (see for example Ref. [8]), the pole at q2 = M i in Eq. (19) arises due to the u (6) quark propagator which forms one side of the quark triangular graph, the other sides being part of the B meson wave function. 4. Asymptotic Behavior
To get constraints on the residues Ri, it is useful to study the asymptotic behavior of form factors FV and FA. It has been argued that the behavior of form factors for very large values of It1 can be estimated reliably in perturbative QCD processes [pQCD]4>9>10. For t << 0 and for It1 much larger than the physical mass of heavy resonances, pQCD should yield a very good approximation to the form factors. First we note that by vector meson dominance
140
where
fpl
having dimensions of mass, is defined as
Then using the methods employed in [lo], it is easy to calculate [only the diagram where the gluon is emitted by the light quark in (ba) bound state and absorbed by the heavy quark contributes and is by itself gauge invariant] FpQCD: PQCD = FPQCD FV A
Here
and is governed by the tail end of the B meson wave function characterized by E . Now using the asymptotic behavior of Eq. (17), namely
RiM:
+1 7T
/
1
M2
ImFCont(s)ds ,
so
we obtain
(24)
i
where a
=
+
or (- 1) and we have defined
In the heavy quark limit, c scales as f B M B Q U / A while RvM& scales as g B B * y M B f B , where in the quark model g B B * y N Thus the right hand side of Eq. (25) goes to zero in this limit. Hence we get the convergence condition N
R M ~+
2 %,
C R ~ M :+ c = 0.
(27)
i
The convergence relation, Eq. (27), is a model-independent result and constitutes a very binding constraint for model b ~ i l d i n g Here, .~ we will study the constrained dispersive model where only the first two radial excitations are kept because as the radial excitations of B* become heavier they are
141 Table 1. €3-mesons masses in GeV
MB MB*
5.71
1.12
1.24 1.22
less relevant to the form factors”. The spectrum of radial excitations is given in Table 1, where the subindices 1 and 2 correspond to the 2s and 3s excitation of the B*, etc. Thus the convergence condition (27) now reads
RM2 + RIM;
+ R2M; + c = 0.
(28)
This condition leaves two free parameters R1 and using Eq. (24), we obtain RM2 (Mi - M2)
RIM; (Mi - Mf)
+
F ( q 2 ) = (M2 - q2)(M: - q2)+(M; - $)(Mi - q 2 )
R2
in the model and
M; - M~ (Mi - q2)(M2- q2)” (29)
where in the heavy quark limit M B = M& = M and
5 . Ward Identities Constraints
It is useful to define
(y( k ,€) Izlia’”vq,bl B ( p ) ) = -iE’Va%;kapgFl(q2),
(7( k ,6 ) Iai~’~ysqAlB ( p ) )= [(q * k) €*’
- (E*
(31)
. q ) k’] F3(q2). (32)
Now we will make use of Ward Identities and gauge invariance principle to relate different form factors. The Ward Identities* used in our process for this purpose are:
(7( k ,6 ) Iaia’vqvbl Bb)) = - ( m b
+ mq) (Y (k, €1 ICy’bl
W p ) ) , (33)
(7( k ,6 ) I.lii0’”y~qvbI B b ) ) = ( m b - mq) (7 (k,€1IW’%bI B ( p ) ). (34)
Substituting these Ward Identities in Eqs. (31) and (32) and using Eqs.(9,10) we obtain [2,. k = q . k , E* . p = E* . q]
*See ref. [I41 for a detailed derivation of these Ward Identities.
142
The results given in Eq. (35) are model independent because these are derived by using Ward Identities. In order to make use of Ward Identities to relate different form factors, we define
(Y (k, €1 liuaapbl B b ) ) = - i E a p p o E * P ( t k ) [(P + kIUg++ qU9-I -iq -2
. €*(Ic)€,&&
+ k)Pq“h
[qaEpp0TE*’
(IC)(P k)aqT - a
[(P
k),
+
EppmE*’(k)
(P
+ k)O qT
p] hl -
a
H
p] h2. (36)
Since we have a real photon, gauge invariance requires that if we replace ~ p ( k by ) k p , the matrix element should vanish. This requires g+
+ g- + 2 ( q . k ) h = 0.
(37)
Thus, finally we obtain
n
Therefore, the normalization of FV and FA at q2 = 0 is determined by a universal form factor (g+ (0) - M z h z ) . Now the form factor hz does not get any contribution from the quark triangle graph nor from the pole and therefore we shall put it equal to zero. The contribution to g+ (4’) from the quark triangle graph14 is
We expect the Ward Identities to hold at low q2 below the resonance regime and as such we use the results obtained from them at q2 = 0. Thus from Eqs. (38 and 39), we obtain (mb
+ mq) FV (0) = 2g+ (0) = (mb- mq)FA (0)
*
(41)
Further, using & . = M B - mb in the above equation (41) and neglecting terms of the order of f r n , ) / M ~ ,we obtain another constraint using Eqs. (29, 30) at q2 = 0
(a
143
Restricting to one radial excitation (M2 = M I ) ,the results are summarized in Table 2. These are the final expressions for the form factors of our process B 4 y l y , if we restrict to the one radial excitation. We also observe the approximate equality F v ( q 2 )= FA(q2) of the form factors which also occur in some other model^^^$^^. For numerical work, we shall use B-meson masses given in Table 1 and f B = 0.180 GeV. Now we study the effect of second radial excitations and the results are:
where A is a parameter which in principle can be obtained when g B * B - , and f B ; B y become known. For numerical values we shall use A = 0 [i.e., M I = M2 (one radial excitation)] and A = 3 and A = 4.8. The second value of A (= 3) corresponds to the estimate of gB*B-, from vector meson dominance
where gB'Bp- = 11fi GeV-l is obtained in [15] and f p - / m p= 205 MeV. The third value of A (= 4.8) gives more or less the width for B* By --f
144 obtained from MI transition in the non relativistic quark model (NRQM). These values give decay widths for B* -+ B y transition as 23 keV, 5.5 keV and 0.8 keV respectively while MI transition in the NRQM predicts it to be 0.9 keV. These predictions will become testable when the above decay width is experimentally measured.
Fig. 1. Differential decay rate versus photon energy z is plotted and comparison is given with various approaches. The solid line (for A = 0 ) , dashed-tripledotted line (for A = 3.0) and dotted line (for A = 4.8) are our calculation, dash-dot-dot line”, dashed line12 and dash-dotted line13. The thin-solid line is the Sudakov resummation calculation result from Ref. [12].
In Fig. 1, differential decay width of the process is plotted against z which is the photon energy spectrum and we see that for our calculations the peak is shifted to lower value of z as compared to those for Eilam et al.,17, Korchemsky et a1.l’ and Chelkov et al.13. So, for the process B -+ yZvl the branching ratios obtained are:
B ( B -+ y l ~ =) 0.5 x l o p 6 B ( B -+ ~ Z Q ) = 0.38 x B ( B 4 7 1 ~=) 0.32 x 1 0 P
(1 = p, A = 0) , (1 = p, A = 3.0)
(1 = p, A
= 4.8)
,
.
(43)
The branching ratios thus obtained are for three representative cases of A. These are not much sensitive to the values of A in contrast to the decay width of B* -+ B y . The CLEO collaboration indicates an upper limit on
145
the branching ratio B(B+ ---t yvle+) of 2.0 x at the 90% confidence level2. The predicted values are within the upper limit provided by the CLEO collaboration but differ from those predicted in refs. [12, 131, namely and 0.9 x respectively. The Monte-Carlo simulation (2 - 5) x results are given in ref. [18] where the upper limit on the branching ratio for this process is predicted to be 5.2 x Finally, the results for B 4 yvll have been reproduced by using Sudakov resummation12 and have also been shown graphically. In our calculations as well as in [17], the position of the peak of the differential decay width is shifted towards lower values of photon energy spectrum. This is due to the double pole in the form factors. The over all effect of radial excitations is to soften the q2-behavior of the differential decay distribution while in ref. [12] it is due to Sudakov resummation. The experiments at the B-factories, BaBar at SLAC and Belle at KEK (Japan) and the planned hadronic accelerators are expected to measure the branching ratios studied here to as low as l o p 8 and these measurements will thus be in a position to check the various theoretical prediction~'~.
References 1. G. Burdman, T. Goldman, and D. Wyler, Phys. Rev. D 51,111 (1995); A. Khodjamirian, G. Stoll, and D. Wyler, Phys. Lett. B 358,129 (1995) [hep-ph/9506242]; E. Lunghi, D. Pirjol, and D. Wyler, Nucl. Phys. B 649,349 (2003), [hepph/0210091]; S. Descotes-Genon, and C. T. Sachrajda, Nucl. Phys. B 650,356 (2003) [hep-ph/0209216]; P. Colangelo, F. De Fazio, and G. Nardulli, Phys. Lett. B 372,331 (1996); P. Colangelo, F. De Fazio, and G. Nardulli, Phys. Lett. B 386,328 (1996) [hep-ph/9506332]; D. Atwood, G. Eilam, and A. Soni, Mod. Phys. Lett. A 11, 1061 (1996); P.Ball, and E. Kou, JHEP 0304,029 (2003); C. Q. Geng, C. C. Lih, and Wei-Min Zhang, Phys. Rev. D 57,5697 (1998). 2. CLEO Collaboration, T. E. Browder et al., Phys. Rev. D 56,11 (1997). 3. N. Cabibbo, Phys. Rev. Lett. 10,531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49,652 (1973). 4. G. Burdman and J. Kambor, Phys. Rev. D 55, 2817 (1997). 5. S. G. Brown, and S. A. Bludman, Phys. Rev. B 136,1160 (1964); D. A. Bryman, P. Depommier, and C. Leroy, Phys. Rep. bf 88, 151 (1982); D. Yu. Bardin and E. A. Ivanov, Sou. J. Part. Nucl. 7,286 (1976); J. Bijnens, G. Ecker, and J. Gasser, Nucl. Phys. B 396,81 (1993). 6. Riazuddin, Europhys. Lett. 60, 28 (2002). 7. See for details of calculations:
146
8. 9. 10.
11.
12. 13. 14. 15.
16. 17. 18.
19.
Riazuddin, T.A. Al-Aithan and A.H.S. Gilani, Int. J. Mod. Phys. A 17,4927 (2002) [hep-ph/0007164]; N. Paver and Riazuddin, [hep-ph/0107330]. N. Isgur, Phys. Rev. D 13,129 (1976); D 23,817E (1981). G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22,2157 (1980); Phys. Rev. D 22,2157 (1980). A. Szcizepanick, E. M. Henley and S. J. Brodsky, Phys. Lett. B 243, 287 (1990); G.Burdman and J. F. Donoghue, Phys. Lett. B 270,55 (1970). E. Eichten, C. T. Hill, and C. Quigg, in Proceedings ofthe Workshop, Batavia, Illinois, 1994, edited by D. Kaplan and S. Kwan (Fermilab Report No. 94/190, Batavia, 1994) P. Korchemsky, D. Pirjol, and T.-M. Yan, Phys. Rev. D 61, 114510 (2000) [hep-ph/9911427]. G. A. Chelkov, M. I. Gostkin and Z. K. Silagadze, Phys. Rev. D 64,097503 (2001). A. H. S. Gilani, Riazuddin, and T. A. Al-Aithan, JHEP 0309,065 (2003) [hep-ph/0304183]. E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane and T.-M. Yan, Phys. Rev. D 21 203 (1980); M. Wirbel, B. Stech and M. Bauer, 2.Physik C 29,637 (1985). C. A. Dominguez and N. Paver, DESY Report No: DESY-88-063 (1988) and 2.Physik C 41,217 (1988). G. Eilam, I. Halperin, and R. R. Mendel, Phys. Lett. B 361,137 (1995) M. J. Lattery, Ph.D. thesis, University of Minnesota (1996); M. A. Shifman, Usp. Fiz. Nauk 151, 193 (1987) [Sou. Phys. Usp. 30, 91 (1987)l G. G. Devidze, The short distance contribution to the Bs + yy decay in the SM and MSSM, ICTP preprint: IC/2000/71 (2000); The lowest order short-distance contribution to the Bs + yy, [hep-ph/9905431]
CALCULATION OF NLO QCD CORRECTIONS TO POLARIZATION EFFECTS IN T O P QUARK DECAYS A. KADEER Institut fur Physik der Johannes- Gutenberg- Universitat, Staudinger Weg 7,0-55099 Mainz, Germany We discuss the polarization effects in top quark decays and their QCD radiative corrections. As an example, we give NLO QCD results to polarized top decay into a charged Higgs to highlight the details of our calculations.
Keywords: top quark, polarization, NLO QCD corrections
1. Introduction
Highly polarized top quarks will become available in singly produced top quarks at hadron colliders (see e.g.[l]) and in top quark pairs produced in future linear e+ - e--colliders (see e.g. [2, 3, 4, 5, 6, 7, 81). Then it will not be difficult to experimentally measure polarization effects such as the azimuthal correlation between the ( f i , F x b )and ( f i , $ t ) planes in the semileptonic rest frame decay of a polarized top quark. Also measurements of the decay rate and the asymmetry parameter in the decay t(t)-+ H + b will be important for future tests of the Higgs coupling in the minimal supersymmetric standard model (MSSM). Regarding the first case, the O(a,) corrections to the azimuthal correlation were calculated in [9]. As to the latter case, the O(a,) corrections were also calculated in [lo]. The two calculations have the same structure. Once one of them has been calculated, most of the integrals and techniques can be used in the other. Since our aim is to highlight the details of the calculation we choose, for our discussion, the second case where the structure is simpler. The O(a,) corrections to the unpolarized top quark decay rate t + H f +b have been calculated previously in Refs. [ll,12, 131 and in Refs. [14, 15, 16, 171, and have been found to be important. The present paper provides the calculation of the O(a,) radiative corrections in polarized top quark decay t ( r ) H + b. The decay is analyzed in the rest frame of the
+
+
147
148
top quark (see Fig. 1). In terms of the unpolarized rate ential decay rate is given by
where the asymmetry parameter degree of polarization.
r and the polarized rate rPthe differ-
CYH is
defined by
(YH =
rP/rand P is the
............ ..........* H+
Xb The definition of the polar angle Bp.
Fig. 1. quark.
P' is the
polarization vector of the top
2. Born term results
The coupling of the charged Higgs boson to the top and bottom quark in the MSSM can be expressed as a superposition of right- and left-chiral coupling factors. The Born term amplitude is thus given by
Mo
= %(a1
+ b75)ut
(2)
In the MSSM, with its two Higgs doublets, one of the charged Higgs bosons is real and the top quark can decay to ( H f b) provided mt > mH+ mb. In order to avoid flavor changing neutral currents (FCNC) the generic Higgs coupling to all quarks has to be restricted. In the notation of [18] this leads to the coupling factors
+
+
for model 1, and the coupling factors
for model 2, where Vtb is the CKM-matrix element and tan/? = W Z / W ~ is the ratio of the vacuum expectation values of the two electrically neutral
149
components of the two Higgs doublets. The weak coupling factor gw is related to the usual Fermi coupling constant GF by g: = 4 d i i m g G ~ . For simplicity we define following scaled masses:
and scaled kinetic variables
1 wo = (1- 2€ 2 1 P+ Yp= -In-, 2 P-
+ y 2 ),
w3
= p3,
w* = wo f w3,
1 w+ Yw = -In-.
2
w-
+ +
where the Kallkn function is defined as A(a, b, c) := a2 b2 c2 - 2(u b t bc+ ca). For the unpolarized and polarized Born term rates one obtains
3. Virtual corrections
The virtual oneloop corrections to the (tH + b)-vertex due to one-gluon exchange between the quark legs contain IR- and UV-singularities. The UV-singularities are regularized in D = 4 - 2 w dimensions. The IRsingularities are regularized by a small gluon mass m, in the gluon propagator. The renormalization of the UV-singularities is done in the “on-shell,’ scheme. The main Feynman diagram, shown in Fig.2, gives
From the numerator we have 4(pt .pb) - 2p/t$+ 2p/b$ - Dk2 with D = 4 - 2w. Denoting the denominators of the t-quark, b-quark, and gluon propagators
150 € I : , ,
,
,,
Fig. 2.
,,
Vertex one loop QCD correction
as D t , Db and D , respectively, the loop integrals we have to calculate are
1 ._ .- v co ( 2 4 4 Db. Dt . D ,
,
IR-div.
.._
Bo +miCo. v UV-div.
The scalar threepoint one-loop integral CO,the scalar two-point one-loop integral Bo and the scalar coefficients C1 and C2 are calculated, e.g, in Sec. 4 of Ref. [19]. Then the renormalized amplitude of the virtual corrections in the right- and left-chiral representation can be written as
where the functions hl and A2 are given by
151
Following Refs. [12, 13, 16, 201 the counter term of the vertex is given by
- 1)
1 + ,(Z,"
- 1) mb
In the "on-shell" scheme the wavefunction renormalization constant 2; and the mass renormalization constant 6 m , can be calculated from the renormalized QCD self-energy C,(p) of the quarks q = t ,b. The evaluation of the two conditions C,(p)I+m, = 0 and aC,(p)/a&,, = 0 leads to the following renormalization constants
where p is the 4-momentum and m, is the mass of the relevant quark. Now the virtual oneloop contributions to the unpolarized and polarized rates can be obtained by squaring the renormalized amplitude in equation (8) and multiplying the two-body phase space factor PS2. 4. Tree graph contributions
The one-gluon emission amplitude reads
where k is the gluon momentum and the first and second term in the first curly bracket refer to gluon emission from the t quark and the b quark, respectively. In the square of the tree graph amplitude, the terms without the gluon momentum k in the numerator lead to divergency. Separating these terms will split the tree graph amplitude squared into an IR-convergent part IM:k",l2 and an IR-divergent part as follows: IMtree12 =
lMg;',",l2iIfi01~1M1$~~
(13)
152
where the universal soft-gluon (or eikonal) factor IMliGF is given by
and it multiplies I&&12 which is the threebody Born term amplitude squared evaluated for pt = Pb q k . A soft gluon mass is used to regulate the IR-divergency. Apparently the integration of IMo121MIiGFis complicated because I & &I 2 dependents on the gluon momentum. Using the fact that (IGoI' - IMo12)IM1:GFis convergent and can be integrated without the gluon mass regulator, we further isolate the IR-divergent part by writing
+ +
-
The same universal soft gluon factor appears in the calculation of the radiative corrections to t 4 Wf b. We can therefore take the result of its phase space integration (with m, # O!) from Eq. (63) in Ref. [21]. The phase space integration can be done w.r.t. the gluon energy ko and the H f boson energy E H + . We can also factorize the threebody phase space into two two-body phase spaces by introducing P = pb k and its P2 scales mass squared z = -z:
+
+
mt
s
The second integration is trivial: dR2 ( p t ;p ~P ), = -2r d cos 8 p . The last integration is done in the P-rest frame with the opposite of the Higgs momentum defining the z-axis, and simplifies to the integration over the polar angle 8 of the gluon in this frame.
Jdn2(p;pbik)
=
J-dd3Pb 2Eb
4k 6 ( k 2 - m z ) 6 4 ( P - p b - k ) =
J 3 d c o s B (16)
wherep: = is the modulus of gluon momentum in P-rest frame. 2+ IMtree12can be written in terms of a linear combination of the invariant scalar products P ' p t , P .k and pt .k with only pt .k dependent on the angle 8. So SdR2(P;pb,k)IMtree12 boils down to Sdcos8 ( ~ ~ . kwhich ) ~ , can be
153
found in Appendix A of the Ref. [22]. Finally the remaining z-integration can be reduced to the following classes of integrals, results of which are given in [21]:
R(m,n):=
TxT, s zm dz
Zmaz
S(n) :=
1
( z - $)*In
Zmin
(
1 - y2 1- y2
+ z + dF + z - dF
dz,
(17c)
where A' = X(1, y2, z ) . The total O(a,) result is obtained by summing up the contribution from the loop and real emissions. The total unpolarized O(cr,) correction agrees with [12] and the total polarized O(a,) correction is given by
289) - 16y2 - 8y3
+ (2 - 9y2 + y4
-
E2(4
-
+8p3n ((1 - y ) 2 - €2
+ 7y4 + ~ ' ( 4+ 8y - 1 4 ~ ' )+ 7c4
3 8PiWO + 3y2) + 2 E4) P-Y + In(€) Y2 Y2
) + (3 - 3y2 + 2c2(4+ y2)
-
2c4)
1-Y
+4pop3(2~i2(1- - '1 - 2 ~ i 2 ( 1 - - y ) - LiZ(w-) PP+ (1 - y2) - E 2 +Liz(w-) 2 In( K P )
+
-(2
€2
+ y2 - ~ ' ( 3+ 2y2) + c4) (2Li2(y) - Li2(w-)
-
Liz(w+))]
}
.
154
T h e separate virtual a n d real emission results as well as t h e detailed discussions on t h e various mass limiting cases of t h e total NLO corrections to t h e rate a n d t h e asymmetry parameter are given in [lo]. Quite similar techniques have been used in [9] a n d in [21] to calculate O(a,) polarization effects in top quark decays.
References G. Mahlon and S. Parke, Phys. Rev. D 55,7249 (1997) . J.H. Kuhn, A. Reiter and P.M. Zerwas, Nucl. Phys. B 272,560 (1986) . J.H. Kuhn, Nucl. Phys. B 237,77 (1984). J.G. Korner, A. Pilaftsis and M.M. Tung, 2. Phys. C 63,575 (1994). S. Groote and J.G. Korner, 2.Phys. C 72,255 (1996). S. Groote, J.G. Korner and M.M. Tung, 2.Phys. C 74,615 (1997). S.J. Parke and Y. Shadmi, Phys. Lett. B 387,199 (1996). A. Brandenburg, M. Flesch and P. Uwer, Phys. Rev. D 59,014001 (1999). S. Groote, W.S. Huo, A. Kadeer, J.G. Korner, hep-ph/0602026 M.C. Mauser, J.G. Korner, hep-ph/0211098 A. Czarnecki and S. Davidson, Phys.Rev. D 47,3063 (1993). A. Czarnecki and S. Davidson, Phys. Rev. D 48,4183 (1993). C.S. Li et al., Phys. Lett. B285,137 (1992). J. Liu and Y.P. Yao, Phys. Rev. D 46,5196 (1992). J. Reid, G. Tupper, G. Li, and M.S. Samuel, 2.Phys. C 51,395 (1991). C.S. Li and T.C. Yuan, Phys. Rev. D 42,3088 (1990); Phys.Rev. D 47,2156(E) (1993). 17. J. Liu and Y.P. Yao, Report No. UM-TH-90-09(1990) (unpublished); Int. J . Mod. Phys. A 6,4925 (1991). 18. J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, “The Higgs Hunter’s Guide”, (Addison-Wesley, Reading, MAA, 1990). 19. A. Denner, Fortschr. Phys. 41,307 (1993). 20. E. Braaten and J.P. Leveille, Phys. Rev. D 22,715 (1980). 21. M. Fischer, S. Groote, J.G. Korner and M.C. Mauser, Phys. Rev. D 65, 054036 (2002). 22. M. Jezabek, J.H. Kuhn Nucl. Phys B 320,20 (1989).
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
SELECTED TOPICS IN TOP QUARK PHYSICS J.G. KORNER
Institut f i r Physik der Johannes-Gutenberg- Universitat, Staudinger Weg 7,0-55099 Mainz, Germany I discuss three different selected topics in top quark physics. The first topic concerns the next-to-next-to-leading order calculation of the hadroproduction of top quark pairs and the role of multiple polylogarithms in this calculation. I report on an ongoing next-to-next-to-leading order calculation of heavy quark pair production in hadron collisions where the loopby-loop part of the calculation is about to be completed. Calculating the loop-by-loop part allows one to take a glimpse at the mathematical structure of the full NNLO calculation. The loopby-loop contributions bring in a new class of functions introduced only eight years ago by the Russian mathematician Goncharov called multiple polylogarithms. The second topic concerns a next-to-leading order calculation of unpolarized top quark decays which are analyzed in cascade fashion t b+W+ followed by Wf + I + q .Finally, I present some next-to-leading order results on polarized top quark decays which are analyzed in the top quark rest system. --$
+
Keywords: multiple polylogarithms, top quark decays, radiative corrections
1. Introduction I begin my talk with a few remarks on present top quark yields at the Tevatron and on expected top quark yields a t the LHC which will start running a t the end of 2007. After a slow start in early 2001 Tevatron I1 started reaching design peak luminosities of 8.5. 1031cm-1s-1 in 2004. The best weekly performance was in early 2006 with a weekly integrated luminosity of 25 pb-'. If Tevatron I1 could perform at this rate it would be able to collect 1.3 ft-' in a year. There has been a three months shutdown in the spring of 2006 with some (electron cooling) improvements on the p beam. The hope is that there will be a factor two or three improvement in luminosity after resumption following the shutdown. At the time of writing this has not been realized so far after a few months of post-shutdown running although the machine is performing quite well with continuous im-
155
156
provements. Such a factor would be dearly needed if one wants to reach the projected total of 8fb-1 when the machine is closed down in 2009. At a c.m. energy of fi = 1.96TeV with a(ti?)M 6.8pb one expects around 7000 ti? pairs at each detector (CDF and DO) for an integrated luminosity of 1fb-l. Single top production occurs at about 33% of the tf pair production rate but has not been detected so far. Much bigger samples of top quarks will be available at the LHC. Due to the higher energy of the LHC the cross section increases by a factor of 100. Also there will be a ten-fold increase in luminosity at the LHC. Thus one will have l o 7 tf-pairs per year, or one tf-pair every four seconds at each detector (ATLAS and CMS). In a later high luminosity run there will be another factor of ten increase in luminosity such that one will have a tf-pair produced every half second. Again single top production occurs at approximately onethird the rate of tf-production. Singly produced top quarks will be highly polarized because they are produced weakly. This opens the way to study angular correlations between the polarization of the top quark and its decay products which forms the third topic of this talk. The yield of top quark pairs at the International Linear Collider (possibly starting in 2015) will be M (1- 4) . 105/y depending on the c.m. energy M 360 - 800 GeV. In e+ - e--interactions a high degree of polarization of the top (or antitop) quark can be achieved through tuning of the beam polarization. 2. NNLO description of heavy top quark production
The full next-to-leading order (NLO) QCD corrections to hadroproduction of heavy flavors were completed as early as 1988l>'. They have raised the leading order (LO) estimates3 but were still below the experimental results on bottom quark pair production (see e.g. [4]). In a recent analysis theory moved closer to experiment4. First experimental results on hadronic tf-pair production5i6 are in agreement with theoretical NLO QCD predictions7?' within the large theoretical and experimental error bars. A large uncertainty in the NLO calculation results from the freedom in the choice of the renormalization and factorization scales. These scale uncertainties amount to a M 10% theoretical error in the NLO cross section predictions7. The dependence on the factorization and renormalization scales is expected to be greatly reduced at next-to-next-to-leading order (NNLO). This will reduce the theoretical uncertainty. In Fig. 1 we show one generic diagram each for the four classes of gluon
157
Fig. 1. Exemplary gluon fusion diagrams for the NNLO calculation of heavy hadron production
induced contributions that need to be calculated for the NNLO corrections to hadroproduction of heavy flavors. They involve the two-loop contribution (Fig. la), the loop-by-loop contribution (Fig. lb), the one-loop gluon emission contribution (Fig. lc) and, finally, the two gluon emission contribution (Fig. Id). In our work we have concentrated on the loop-by-loop contributions exemplified by Fig. lb. Specifically, working in the framework of dimensional regularization, we have calculated O(E’)results for all scalar massive one-loop one-, two-, three- and four-point integrals that are needed in the calculation of hadronic heavy flavour production”. Because the one-loop integrals exhibit infrared (IR)/collinear (M) singularities up to O(E-~)’one needs to know the one-loop integrals up to O(E’) since the one-loop contributions appear in product form in the loop-by-loop contributions. It is exactly the O(e2)terms in the scalar massive three- and four-point integrals that bring in multiple polylogarithmslO~ll. Calculating the loop-by-loop contributions allows one to obtain a glimpse of the complexity that is waiting for us in the full NNLO calculation.
158
This complexity does in fact reveal itself in terms of a very rich polylogarithmic structure of the Laurent series expansion of the scalar one-loop integrals as well as the appearance of multiple polylogarithms of maximal weight and depth four. To underscore the statement that the loop-by-loop contributions reveal the mathematical structure of a full NNLO calculation let us take a look at the paper by Bernreuther et a l l 2 who calculated the O ( $ ) contributions to the oneloop vertex correction of the process V -+ Q&. These are needed for the loop-by-loop part of a NNLO calculation of heavy quark pair production in e+ - e--annihilations. The result can be expressed in terms of onedimensional harmonic polylogarithms of maximal weight three. The same paper also lists results on the corresponding O(co) two-loop vertex corrections which contain onedimensional harmonic polylogarithms of maximal weight four. This shows that the same mathematical complexity appears in the loop-by-loop contributions as in the two-loop contribution. Let me mention that heavy quark pair production in e+ - e--annihilations is a somewhat simpler problem than heavy quark pair production in hadronic collisions because of the appearance of one additional mass scale in the latter case. This explains why one has only one-dimensional harmonic polylogarithms in e+ - e--annihilations case compared to the multiple polylogarithms appearing in the hadronic collision calculation. Even then the two-loop vertex correction to V + QO listed in [12] takes up more than twelve pages. The scalar four-point integrals appearing in the calculation of the loopby-loop evaluation are the most difficult to calculate. They contain a very rich structure in terms of polylogarithmic functions. For example, the c2-coefficients of the Laurent series expansion of the four-point integrals contain logarithms and classical polylogarithms up to order four (ie. Li4) in conjunction with the <-functions ((2), ((3) and <(4) and products thereof, and a new class of functions which are now termed multiple polyl~garithms'~. Since this is a conference on mathematical physics it is appropiate to dwell a little on the subject of multiple polylogarithms. A multiple poly-
159
logarithm is represented by
(
dt x2x3...xk - t dt 23
...X k
-
t
:o)mz-l
o...o
($.)
mk-1
dt
-
1-t’
where the iterated integrals are defined by dt
0
x
... 0 --
tn
J 5J 0
0
dtn-1 an-l
-
tn-l
x
... x
~
a1 dt - tl 0
The indices mk and k are positive integers. The multiple polylogarithms are classified according to their weight w = m l m2 ... mk and their depth k . We mention that a very efficient program for the numerical evaluation of multiple polylogarithms has recently been developed in Mainz which, characteristically, is based on the language GiNaC 14. The classical polylogarithms, Nielsen’s generalized polylogarithms, the one- and two-dimensional harmonic polylogarithms are all special cases of Goncharov’s multiple polylogarithms (see e.g. [15]). For example, the classical polylogarithms
+ + +
are multiple polylogarithms of weight n and depth 1. In our original Feynman parameter calculation our results were written down as onedimensional integral representations given by the integrals
(2) and
where the ~i take values f l and the aj’s are combinations of the kinematical variables of the process. The numerical evaluation of these
160
one-dimensional integral representations are quite stable. The functions FuIuzu3 (a1,a2, a3, a4)and F,, (a1,a2, a3, a4) are related to multiple polylogarithms of maximal weight and depth four as shown in [ll]. We are now in the process of computing the full loop-by-loop contributions including the spin and colour algebra arising from squaring the full oneloop amplitudes as given in [16]. A first result has been obtained for the Abelian case of photon-photon production of heavy quark pairs”. 3. Decays of unpolarized and polarized top quarks
After this brief mathematical detour I return to the physics of top quark decays. In the SM the top quark decays almost 100% to a W+ and a bottom quark. Also, the top quark decays so fast that it retains its initial polarization when it decays. I describe both unpolarized and polarized top quark decays. In the unpolarized case I analyze top quark decays in cascade fashion as a two step process involving the decay t -+b W+ and W+ ---t Z+ vl in the respective rest frames of the top quark and the W+-boson. In the polarized case I perform the decay analysis in the rest system of the top quark. I discuss polar and azimuthal correlations involving the polarization of the top quark and the momenta of the decay products in the decay t ( r )-+ xb ’1 vl. The decay of a polarized top quark into a W+-boson and a jet with &quantum numbers t(t) -+ Z+ vl is desribed by altogether eight invariant structure functions (see e.g. [18, 19]),
+
+
+ +
xb + +
Hp” =
( - gpy H I + pfp,”
H2
- i@YPuPt,p4u H 3 )
+
There are three unpolarized structure functions H1,2,3 and five polarized structure functions from the set G 1 , 2 , 3 , 6 , 8 , 9 a. In general the invariant structure functions are functions of qo and q2. In the narrow resonance approximation for the W+-boson, which we shall adopt in this talk, one has q2 = mk. The aim of the game is to measure the different unpolarized and polarized structure functions (or moments thereof) and to compare them to theoretical predictions. The different structure functions can be separated since they contribute to the rate with different dependencies on the aIn physical expressions the three structure functions G3, G8 and GQcontribute only in two pairs of linear combination^'^
161
electron energy and, in the case of the polarized structure functions, they can be measured through polar and azimuthal correlations involving the polarization direction of the polarized top quark.
Unpolarized top quark decays
+
In the decay t -+ xb W+ the W+ is polarized. The W + is self-analyzing in the sense that the angular decay distribution of its decay products W + -+ I+ ul can be used to reconstruct the polarization of the W + .We shall analyze the unpolarized decay in cascadetype fashion, i.e. we shall analyze the decay W + + Z+ ul in the rest frame of the W + . This brings in the three unpolarized helicity structure functions HT+,HT- , H L (or for short H+, H - , H L ) , which are linearly related to the three unpolarized invariant structure functions H I ,H2, H3 vialg
+
H+ = H i + ltlmt H3, H- = H I - ltlmt H3,
(5)
(6)
+ lf12mm,”H2,
H L = m&Hi
(7)
The polar angle decay distribution is given by
+
1 dr --
3 3 = ?(i+ cos8)2 H+ + -(1 - cos8)2 7-1- sin2 8 7 - 1 ~, (8) rdcose 8 8 4 where the angle 8 is defined in Fig.2. The %+, 7-1- and 7 - 1 ~are the normalized transverse-plus, transverseminus and longitudinal helicity structure functions, resp., such that ‘H+ 7-17 - 1 ~= 1. From the polar angle dependence in Eq. (8) or from matching mquantum numbers in the W + rest frame decay (see Fig. 2) it is clear that
+
+
7-1+ :
favours forward 1+
‘l-t- :
favours backward I+
Translated to the top quark rest frame this implies that F+ (E) produce harder (softer) I f ’ s which can be used to experimentally separate the contributions of the three helicity structure functions. At the Born term level the SM prediction is (mt = 175 GeV, mb = 0) %+(Born) = 0 X-(Born)
1 1 2y2
+ 7-1L(Born) = 2Y2 1 + 2y2 =
~
~
(forbidden) =
0.297
=
0.703,
162
+
Fig. 2. Definition of the polar angle 0 in the rest frame decay of W + -+ I + vl. The two lines "//" indicate a boost to the rest system of the W + . The arrows next to the lepton lines give the helicities of the leptons.
where y = mw/mt. At the Born term level, with m b = 0, H+ is not populated because of angular momentum conservation in the two-body decay process t 4 b+ W + where for m b = 0 the bottom quark has 100% negative helicity and the &quark and the W + are in a back-to-back configuration. The present experimental results on 'H+ are consistent with zero within large error bars. For example, using 230 pb-' DO quotes a value of 'H+ = 0.00 f 0.13 (stat) f 0.07 (syst)20. CDF finds H+ = O . O O ~ ~ : ~ ~ ( s syst) tat or X+ < 0.27 at the 95% confidence level2I. Using the same data sample of 200 pb-', CDF quotes a value of H L = 0.74+:::: for the longitudinal helicity of the W+-boson, also compatible with the SM prediction. The vanishing of H+ is no longer true for additional gluon or photon emission, or when one takes into account bottom mass effects. When all of these are taken into account one has22323
+
X+ = O.O0102(QCD) + O.OOOOS(EW) + 0.00039(mb # 0 ) ,
(9)
where the numbers give the O(a,) QCD corrections, the O ( a )electroweak corrections and m b # 0 corrections (mb = 4.8GeV). Numerically the correction to H+ occurs only at the pro mille level. It is safe to say that, if top quark decays reveal a violation of the SM ( V - A ) current structure that exceeds the 1%level, the violations must have a non-SM origin. The results for the corresponding corrections to 'H- and 'HL are listed in terms of rates normalized to the total Born term rate, i.e. f'i = ri/r'(Born). The normalized partial Born term rates f'i(Born) are factored out. Corrections coming from NLO QCD, from the NLO electro-weak corrections (EW), from the W + finite width correction (BW) and from m b # 0 effects are listed separately. One has
163
F- =0.297 [l- 0.0656(QCD) + 0.0206(EW) - 0.0197(BW) - 0.00172(mb # O ) ] FL =0.703 [l 0.0951(QCD) + O.O132(EW) 0.0138(BW) - 0.00357(mb # O ) ] -
-
Written in terms of the normalized 'Hi this translates into a +2.4% upward shift from 'H-(Born) = 0.297 and a -1.2% downward shift from 'HL(Born) = 0.703. Judging from the fact that 7 - l ~and 'H- will eventually be measured to better than 1%it is quite clear that one has to take radiative corrections into account when comparing experiment with theory.
'
0.68 170
'
I
1
175 rn, [GeVI
180
Fig. 3. Top mass dependences of the ratio 'HL = r'L/I'. Full line : Born term. Dashed line: Corrections including (QCD), electroweak (EW), finite-width (FW) and ( m b # 0) Born term corrections.
In Fig.3 we show the top mass dependence of the ratio 'HL = I'L/l?. The horizontal displacement of the Born term curve and the corrected curve is M 3.5 GeV. One would thus make the corresponding mistake in a top mass determination from the measurement of 7 - i ~if the Born term curve was used instead of the corrected curve. If one takes mt = 175 GeV as central value, a 1%relative error on 'HL would allow one to determine the top quark mass with an error of x 3 GeV.
164
4. Polarized top quark decays Contrary to the analysis of unpolarized top quark decays described in the last subsection, polarized top quark decay will be analyzed altogether in the rest frame of the decaying top quark. This is the natural choice for an experimental analysis. Choosing a particular two-particle rest subsystem is only of advantage if that particular subsystem is resonance dominated as was discussed in the unpolarized decay case. The general angular decay distribution of the rest frame decay of a polarized top quark decaying into a jet xb and a lepton 1+ and a neutrino is given byz4
where the polar and azimuthal angles O p and 4 describe the orientation of the polarization of the top quark relative to the decay plane formed by the decay products of the top quark. The scaled energy and the scaled mass of the W+ are denoted by 40 = qO/mt and y = mw/mt. As usual we define a scaled lepton energy through x1 = 2El/mt. P is the magnitude of the top quark polarization. ??A stands for the unpolarized rate, and rB and rc stand for the polar and azimuthal correlation rates. In [25] we have considered three different helicity systems to analyse the polar and azimuthal correlations in the rest frame decay of a polarized top quark as shown in Fig. 4. It is important to realize that correlation measurements in each of the helicity frames constitute independent measurements of the invariant polarized structure functions. To illustrate this point let us consider the contribution of the invariant polarized structure function GI to the polar and azimuthal correlations in the above three helicity systems. The decay rate is proportional to LP”HPv. One then obtains
where the three contributions in the curly bracket refer to the polar and azimuthal correlations in the three helicity coordinate systems with the z-axes along (1) the lepton I+ , (2) the W+-boson and (3) the neutrino
165
Fig. 4. The definition of the polar angle O p and the azimuthal angle $I in the rest frame decay of a polarized top quark in three different helicity systems. The event plane defines the (z,z)-plane with ( l a ) pi 11 z and (&)% 2 0, (2'a) @ 11 z and (pi)%2 0, and (3a) P; 11 z and (Pi)% 50.
vl. From Eq.(11) it is clear that GI contributes quite differently to the correlation functions in the three reference systems. In [25] we have calculated the Born term and NLO QCD contributions to the polar and azimuthal correlation functions d r g and dI'c in the three different helicity systems. We were able to obtain closed form expressions
166
for the totally integrated angular decay distributions. The results are too long to be listed here but can be found in [25]. We mention that we find agreement with [27, 28, 29, 301 for the unpolarized case drA and the polar correlation function d r g in systems 1 and 3. In numerical form one has z-axis along lepton (system (la)) drNL0 dcosQpd4
47r
(1 - 8.54%)
+ (1- 8.72%)PcosOp -0.24%PsinBpcosq!1
1
,
(12)
z-axis along W+-boson (system (2'a))
"[
drNL0 - rA dcosepd4 47r
(1- 8.54%)
+ (0.406 - 11.62%)PcosBp
1
-(0.760 - 8.20%)PsinBpcosq5 , (13) z-axis along neutrino (system (3a))
-(0.919
-
1
8.61%)Psin13~cosc#1. (14)
In all the three expressions we have factored out the Born term rate (0). The first number in the round brackets stands for the LO Born term rate whereas the second number gives the percentage change due to the NLO QCD corrections. Let me first discuss the LO correlation functions. I shall refer to r g / I ' A and I'C/I'A as the polar and azimuthal analyzing power, respectively. In system (la) (I+ along z ) the polar analyzing power is 100% which necessarily implies that the azimuthal analyzing power is zero in this system. In fact, the vanishing of rc in system (la) can be seen to directly follow from the left-chiral (V-A) structure of the SM quark and lepton currents26.The polar analyzing power in the systems (2'a) and (3a) is less than 100% with +41% and -32%, respectively. As mentioned before the LO azimuthal analyzing power in system (la) is zero. In system (2'a) and (3a) the azimuthal analyzing power is reasonably large with -76% and -92%, respectively. Except for the polar correlation in system (3a) all NLO corrections go in the same direction. They reduce the LO results by approximately 10%.
167
This implies that the polar and azimuthal analyzing powers are not changed very much from their Born term values through radiative correction. An exception is system (3a) where the polar analyzing power is changed from -31.8% to -34.4%. This amounts to a 8.2% change in analyzing power through radiative corrections which is surprisingly large.
5. Summary and conclusions In this talk I have covered three selected topics in top quark physics. The first topic concerned the NNLO calculation of hadronic top quark pair production where the loop-by-loop part is now being completed. The other three missing parts of the NNLO calculation (two-loop, one-loop gluon emission, two-gluon emission) are more difficult and will very likely take another five to ten years to complete. Such a largesize calculation will require a dedicated international effort of the theoretical community which will have to be coordinated by one of the big international centers of particle physics. In the second and third topic I discussed NLO QCD predictions for unpolarized and polarized top quark decays which should be amenable to experimentals tests in the coming few years.
Acknowledgements:I would like to thank my collaborators S. Groote, W. S. Huo, A. Kadeer, D. Kubistin, Z. Merebashvili and M. Rogal for participating in the work I have reported on in this talk. My thanks are also due to Riazuddin and F. Hussain for organizing such a wonderful conference in Islamabad, and to all the Pakistani graduate student helpers who made the meeting such a joyful affair.
References 1. P. Nason, S. Dawson and R. K. Ellis, Nucl. Phys. B303, 607 (1988); ibid B327,49 (1989); ibid B335,260(E) (1990). 2. W. Beenakker, H. Kuijf, W. L. van Neerven and J. Smith, Phys. Rev. D 40, 54 (1989); W. Beenakker, W. L. van Neerven, R. Meng, G.A. Schuler and J. Smith, Nucl. Phys. B351,507 (1991). 3. M. Gluck, J.F. Owens and E. Reya, Phys. Rev. D 17,2324 (1978); B. L. Combridge, Nucl. Phys. B151, 429 (1979); J. Babcock, D. Sivers and S. Wolfram, Phys. Rev. D 18,162 (1978); K. Hagiwara and T. Yoshino, Phys. Lett. 80B, 282 (1979); L. M. Jones and H. Wyld, Phys. Rev. D 17,782 (1978); H.Georgi et al., Ann. Phys. (N.Y.) 114,273 (1978) .
168 4. M. Cacciari, S. Frixione, M. L. Mangano, P. Nason, G. Ridolfi, JHEP 0407, 033 (2004). 5. D. Acosta et al., The CDF Collaboration, Phys. Rev. Lett. 93,142001 (2004). 6. V. M. Abazov et al., The DO Collaboration, Phys. Lett. 626B,55 (2005). 7. M. Cacciari, S. Frixione, M. L. Mangano, P. Nason and G. Ridolfi, JHEP 404, 068 (2004). 8. N. Kidonakis and R. Vogt, Phys. Rev. D 68,114014 (2003). 9. J.G. Korner and Z. Merebashvili, Phys. Rev. D 66,054023 (2002). 10. J.G. Korner, Z. Merebashvili and M. Rogal, Phys. Rev. D 71,054028 (2005). 11. J. G. Korner, Z. Merebashvili and M. Rogal, J. Math. Phys. 47, 072302 (2006), arXiv:hepph/0512159. 12. W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia and E. Remiddi, Nucl. Phys. B706,245 (2005). 13. A.B. Goncharov, Math. Res. Lett. 5 (1998), available at http://www.math.uiuc.edu/K-theory/0297 . 14. J. Vollinga and S. Weinzierl, Comput. Phys. Commun. 167, 177 (2005), arXiv: hep-ph/0410259. 15. M. Rogal, doctoral thesis, Mainz 2005. Available at http://wwwthep.physik.uni-mainz.de. 16. J.G. Korner, Z. Merebashvili and M. Rogal, Phys. Rev. D 73,034030 (2006). 17. J.G. Korner, Z. Merebashvili and M. Rogal, hep-ph/0608287 18. A. V. Manohar and M. B. Wise, Phys. Rev. D 49,1310 (1994). 19. M. Fischer, S. Groote, J. G. Korner and M. C. Mauser, Phys. Rev. D 65, 054036 (2002). 20. V. M. Abazov et al., The DO Collaboration, Phys. Rev. D 72,011104 (2005). 21. A. Abulencia et al. The CDF-Run I1 Collaboration, Phys. Rev. D 73,111103 (2006). 22. M. Fischer, S. Groote, J. G. Korner and M. C. Mauser, Phys. Rev. D 63, 031501 (2001). [arXiv:hep-ph/O011075]. 23. H. S. Do, S. Groote, J. G. Korner and M. C. Mauser, Phys. Rev. D 67,091501 (2003), [arXiv:hep-ph/0209185]. 24. J. G. Korner and D. Pirjol, Phys. Rev. D 60,014021 (1999). 25. S. Groote, W. S. Huo, A. Kadeer J. G. Korner and D. Kubistin, to be published. 26. S. Groote, W. S. Huo, A. Kadeer and J. G. Korner, arXiv:hep-ph/0602026. 27. M. Jezabek and J. H. Kuhn, Nucl. Phys. B314,1 (1989). 28. A. Czarnecki, M. Jezabek and J. H. Kiihn, Nucl. Phys. B351,70 (1991). 29. A. Czarnecki, M. Jezabek, J. G. Korner and J. H. Kuhn, Phys. Rev. Lett. 73,384 (1994). 30. A. Czarnecki and M. Jezabek, Nucl. Phys. B427,3 (1994).
MODULI STABILIZATION USING OPEN STRING FLUXES ALOK KUMAR Institute of Physics, Bhubaneswar 751 005, India E-mail: [email protected] http://www.iopb-res. in In this talk we discuss how by turning on gauge fluxes which couple to the endpoints of open strings one can obtain stabilization of closed string moduli. This is done by analyzing supersymmetry constraints and RR tadpole conditions. Stabilization of complex and Kahler moduli is studied in a T 6 / Z z orientifold.’
1. Introduction In discussions on moduli stabilization in IIB string theory, one generally uses closed string 3-form fluxes along the six compactified directions. The fluxes generate a potential in four dimensions, which is a potential for the geometric moduli, as well as the axion-dilaton fields and lead to their stabilization upon minimization. In the implementation process, there are restrictions: The primitivity condition: J A G = 0 : ( J : Kahler form), G: (imaginary self-dual (2, 1) - form flux), can fix some of the Kahler moduli as well, but never all. In fact, this condition is trivial for CY’s In the present talk, we discuss a different procedure for stabilizing the moduli. This is achieved by turning on fluxes of the worldvolume gauge fields on the brane and demanding that the magnetic field that is turned on preserves N = 1 supersymmetry after compactification to four dimensions. Another reason D-branes with fluxes are generally introduced is to obtain stabilized models with chiral fermions. To elaborate, 0 3 branes normally used in the compactification on T 6 / & orientifold are replaced by D9’s with magnetic fluxes along six compactified directions. We show that this is a consistent compactification. In the process, the moduli can also be fixed. *based on work with I. Antoniadis and T. Maillard, hepth/0505260 and with S. Mukhopadhyay and K. Ray, hep-th/0605083.
169
170
The spectrum with such fluxes is given by the Landau energy levels, which are described by the harmonic oscillator term plus a term proportional to spin. The oscillator frequency is given by the magnetic field. The interplay between the two terms leaves one chirality of fermion massless. The other becomes massive, but pairs up with the opposite chirality from a massive level This process repeats at all levels. In string theory, this aspect is discussed in the following and many other pieces of work: [l,2, 3, 4, etc.]. It is also known that by turning on constant fluxes, one generates non-commutativity. Such magnetized tori are therefore also known as noncommutative tori. 2. Supersymmetry
We now discuss the supersymmetry property of the magnetized branes. The discussion is presented following refs. [ 5 , 61. We start with the supersymmetry of D-branes. For a non-magnetized Dp-brane, the supersymmetry conditions are: E L = rOrl ...r p E R , where E L and E R are two spinors, both of +ve chirality in ten-dimensions, one coming from the left-sector of string theory and the other from the right sector. rM’s in our notation represents flat space Dirac gamma matrices in D = 10. Next we discuss the supersymmetry preserved by a magnetized Dpbrane. First consider fluxes turned on along T 2 a T 2 constructed from the compactification of two coordinates X4 and X 5 . On tthis two dimensional torus, Fij has only a single nonzero component identified with Fij = H ~ i j . To understand the supersymmetry, we study the boundary conditions. For worldsheet fermions $, we recall, before the magnetic field is turned on, that $L = $ R ~ ~ = o . However, now one has $R = % $ L ~ ~ = o , by defining b = .rrqLH. Then by using the relation, b = tan8, one obtains $R = e2ie$LIu=o. In real notation the boundary condition of worldsheet fermions changes to:
$i = cos28$;
- sin28$;I0=o,
T&= C O S ~ ~ $ -; sin2O$~I,=o.
(1)
In other words, there is a rotation in the left-sector in directions $: and $,: with respect to the non-magnetized case. Note that for the rotation of vectors by an angle 28, as above, in X4 - X 5 space, spinors transform 45 as: E L eer E L , with rij denoting the antisymmetric product of the ten dimensional gamma matrices. As a result, in general the D-brane supersymmetry condition now has the form, E L = ...rpp(F)ER,with p ( F ) giving the rotation of the spinors. .--)
171
For magnetic fields, which can be of block diagonal form along three T2's of the compactified six dimensional space, we have p = e01y12+e2734+03756. We will come back to a general form of p little later. At the moment, let us analyze some simple cases. The question relevant to us is: when does a magnetized D-brane, say 0 5 , 0 7 or 0 9 , have the same supersymmetry as that of the 0 3 brane? This is because we are studying T6/& orientifold models, where we necessarily have the 0 3 planes having the same supersymmetry property as the 0 3 ' s . Example-1 (D5-D3): without a magnetic field. Supersymmetry of the 0 3 is given as: E L = I'O...I'3ER. On the other hand, supersymmetry of 0 5 has the form: E L = I ' o . . . ~ ~ Both E R . of these can be consistent only if I'J~ER= E R , which is not possible as I'45 has only imaginary eigenvalues. In this case, the situation does not change much in the presence of a magnetic field. With a magnetic field turned on, the condition translates into l?45.eer4r5having eigen values fl. For this to happen, however, 8 should be equal to *$. On the other hand, from the relation b = tan8, we learn that this corresponds to an infinite magnetic field. In other words, the range of 8 is restricted to - ~ / 2< 8 < ~ / 2 . It is, however, possible for a magnetized 0 7 to preserve the same supersymmetry as that of an ordinary 0 3 . For this to happen, we obtain a condition exactly in the same way as above: f81f82 = 0 where 81 and O2 are the spinor rotations associated with magnetic fields in directions, x4,x5 and x6,x7,the directions that are transverse to the 0 3 , but are the longitudinal directions of 0 7 . Note also the relation: bi = tunei between the magnetic field and the spinor rotation angle 8. This implies for a 0 7 compactification on T 4 that the magnetic fields have to satisfy a relation: bl = f b 2 , or written in a covariant notation: Fij = f e i j k l F k 1 . In other words, the magnetic fields are either self-dual or anti-self dual corresponding t o the instanton configurations in 4d (Euclidean gauge theory), now on T 4 .This equation is now rewritten in complex coordinates, zi = xi iyi, i = 1 , 2 , 3 , in a notation we will use below, by identifying directions x4,x6,x8 with xi's and directions x 5 ,x7,x9 with yi's (for i = 1,2,3). In this complex notation, the self-duality condition becomes,
+
For T 6 , we obtain a similar equation, which we will use for the moduli stabilization.
172
2.1. D9 on T0 This is the case which will be of most interest to us, as already mentioned. Let us review the situation again, starting with the non-magnetized case. . . T 3 € R and the D9-brane We have the 0 3 brane supersymmetry: E L = ...r g E R . For both of these to be consistent, one supersymmetry: E L = will have to have: F ~ . . . F ~ = EE RR which is not possible, as r4...r9has only imaginary eigenvalues. Therefore, no D9 can be put together with 0 3 to produce a supersymmetric system. The situation changes when magnetic fields are turned on along the compactified directions of D9. As a result, the supersymmetry condition now becomes
r4...r9ee,r45+e2r67+e3r89E R = E R
(3)
and leads to the condition, f& f 82 f 03 = where, of course, we have turned on magnetic field components only along three factorized T2's. We now obtain the supersymmetry condition for a general (constant) magnetic flux on T 6 . For this we write down the spinor rotation matrix for a general background metric and gauge flux. First, restricting to the internal six dimensional space with a metric g i j = S i j , we can write E X P . [-!jFijrij], where the notation, 'EXP.' stands
A F ) = JZqiTE
for an exponential expansion with complete antisymmetrization in indices of Fij. As a result, the expansion is always finite. Now we discuss the general situation with the D9 branes. The condition we analyze is I'4..9 EXP.(-iFijrij)ER = E R . Also, for general G,
,m I
we make a change: is then written as
.
.
The supersymmetry condition
v'&Z &&'&qm .
1
-F[. 3! 23.Fkl F7
4
rijklmn) E R = E R .
(4)
Moreover, this eqn. can be written in a covariant form (not keeping track of all the factors) as:
173
Then using the property of spinors that Kahler form) one obtains:
Jdet(G
+F )
rmAc= iJmnE etc.,
[-iJA J A J - J A J A F +iJ A F A F
(with J :
+F A F AF]= &
(6) where v6 is the six dimensional volume element. It comes in the process of changing the equation in components to that in terms of Kahler form. In writing this form of the supersymmetry condition, a number of terms are dropped. For example, only Fij (in complex notation) are kept. This amounts to using (a condition mentioned earlier): F(2,o)= 0. In a compact form one finally writes: (d+ For us, 0 = 0, corresponding to 0 3 brane supersymmetry. On the other hand 0 = if one wants a D9 brane supersymmetry. These two cases will correspond to IIB on T 6 / O ( - ) F LR, with R := ( X 5 ,.., X9) --+ -(X5, ..,X g ) or I I B / R on T 6 . One can similarly analyze other situations. The condition we have
+
derived can also be written as e-i6(iJ F ) 3 = w . 6 ,which further implies, (since rcHS is a real quantity), Irn[ePi6(iJ+ F ) 3 ] = 0. In our case, as 0 = 0, we therefore have: J A J A J - J A F A F = 0 The real part of the
+ q3] = s v 6 , which also implies that + F ) 3 ]= J JG + F ) can be verified.
above condition is Re[e-ie(iJ the condition J Re[e-ie(iJ Now, the BI action is VDBI=E/
MI0
& G + F ) =gs~ /T 6 & G + F ) /
="I gs
+
M4
4
IM4 6
Re[eaie(iJ F ) 3 ]
T6
(7)
This implies a condition: Re[e-ie(iJ+ F ) 3 ]> 0. This condition can be seen from the requirement of the right sign for KE of the 4d gauge field. For 0 = 0 we then obtain: F A F A F - J A J A F > 0. To summarize: the key supersymmetry conditions for us are: F(z,o)= 0, J A J A J - J A F A F = 0 , and F A F A F - J A J A F > 0. Above discussion is presented following6 3. Fixing Complex Structure Moduli
We now show, using [8]and 191, how the condition: = 0, for a set of brane-stacks denoted by index a, fixes the complex structure moduli. For
174
this, first we introduce a set of brane-stacks which have fluxes of various types, meaning having different components of F turned on, with different magnitudes. The brane-stacks are labeled by an index 'u'. The complex structure matrix (in this case a 3 x 3 matrix), appears in this condition through the definition zz = x2 r2.,yJ . F(2,o) = 0 are now used to fix ~ ' 3 ' s .In addition we also use the fact that flux components F3Cqx3,F;y3 , Fg,y3 are rationally quantized: qaFG G
+
27rp2"3 = 2 7 r s . A precise form of nna ,3 will be clear later on, using the 7G mapping from the worldvolume to space-time. Using the definition of pa's given here, the complex structure matrix, r , satisfies the equation: a a F&,o) = 0 --t rTpgX7 - T T ply - pYxr
+ pty = 0.
(8) Then by specifying pgx,p&, etc., for a set of branes, one aims to fix 7's. One can show that the off-diagonal components of 7- can be forced to be zero, by taking appropriate fluxes p x z y 3pxzx3 , etc. along various brane13 - 21 - 23 - 31 = +32 = 0 . stacks: 7-12 -7-7-7-7For diagonal components of 11
P'2
p
1
7-
one can obtain:
P23 2
h = F ~ i K l ,';55=+=K2, , P d V 2 33
=
P~~ P~~ + = K3, and 7-117-22 = -+ = -K4, px1x2
P53&!'
7-227-33
,
PdV3
=
P~~ -+ = -K5, , ~ px2x3
~
P63
1
~= 7-+ - l!~
, -KS,
pz3z1
with solution given as: 7-11 = i d m , r2' = i f i , r33= i d m K 3 . We therefore see that by specifying fluxes, pa's along a set of brane stacks, one can fix the complex structure moduli. To stabilize the Kahler moduli one makes use of the conditions, J A J A J - J A F ~ A F=~0 , withtheconstraint, F a A F a A F a - J A J A F a > 0, where u denotes the brane-stack. Without going into detail, we mention that by specifying fluxes, as mentioned above for the complex-structure stabilization, one can obtain solution for J's: the off-diagonal Kahler moduli are zero: Jij = 0 and diagonal components of Jij are stabilized to the string scale. This is possible for many combination of fluxes and branes. However, one needs to satisfy an additional constraint for building any model. 4. RR Tadpoles
Before giving explicit model we discuss the RR-tadpole cancellations. Constant fluxes generate RR charges, corresponding to lower dimensional
175 branes. Cancellation of all these charges implies that the worldvolume theory is free of anomalies. The amount of charge that is generated can be seen by looking a t the WZ couplings of the brane. The total action is given by
+
I = VDBI Vwz
(9)
We have already looked a t VDBI. VWZ has a general form (using the fact that RR forms are even or odd under the orientifolding O(-)FL):
Vwz
= ~9
C Na a
Lo
(C4 A
Fa A F a A F a
+ Cg A F a ),
(10)
where Na's are the number of branes in a stack with fluxes FG. One therefore obtains new contributions to the 3-brane and 7-brane tadpoles. Using the Jacobi matrix giving the map from the worldvolume to spacetime", W i = the tadpole cancellation conditions are read from CaNaWaFaA Fa A F a N = 16 and CaNaWaFa = 0 , where N is the number of ordinary D3-branes which one can also put to cancel the 3-brane tadpole, W , = detW and Na is the number of branes in the a'th stack. In practice, however, it is difficult to obtain stabilization in T 6 / & orientifolds as generated 7-brane tadpoles are required to cancel among themselves, consistent with supersymmetry and other requirements. However, a no-go theorem does not exist as yet. This problem can be solved by turning on nonzero Fayet-Ilioupoulos (FI) parametersg, and then obtaining the supersymmetric vacua by turning on vacuum expectation values of the charged scalars. The supersymmetry condition is now modified. The models, have limitations, however, as they can not be shown to be exact solutions of string theory in general. In another work", we used non-abelian gauge fluxes where we showed that it may now be possible to solve supersymmetry and tadpole cancellation conditions simultaneously. We, however, do not go into details of these models in this talk.
E,
+
References 1. C. Bachas, arXiv:hep-th/9503030. 2. R. Blumenhagen, L. Goerlich, B. Kors and D. Lust, JHEP 0010, 006 (2000), [arXiv:hep-th/0007024]. 3. R. Blumenhagen, D. Lust and T.R. Taylor, Nucl. Phys. B 663,319 (2003), [arXiv:hep-th/0303016]. 4. J.F.G. Cascales and A.M. Uranga, JHEP 0305, 011 (2003), [arXiv:hepth/0303024].
176 5. M. Berkooz, M. Douglas and R. Leigh, NucZ.Phys. B 480, 265 (1996), [arXiv:hep-th/9606139]; V. Balasubramanian and R. Leigh, Phys. Rev D 55, 6415 (1997), [arxiv: hep-th/9611165]; E. Witten, JHEP 0204, 012 (2002), [arXiv:hep-th/0012054]. 6. M. Marino, R. Minasian, G.W. Moore and A. Strominger, JHEP 0001, 005 (2000), [arXiv:hep-th/9911206]. 7. For a recent review see S.P. Trivedi, talk in strings 2004, and references therein. 8. I. Antoniadis and T. Maillard, Nucl. Phys. B 716, 3 (2005), [arXiv:hept h/0412008]. 9. I. Antoniadis, A. Kumar and T. Maillard, [hep-th/0505260], (revised version: to appear). 10. M. Bianchi and E. Trevigne, arXiv:hep-th/0502147 and arXiv:hept h/0506080: 11. A. Kumar, S. Mukhopadhyay and K. Ray, hepth/0605083.
TOPOLOGICAL STRINGS A N D SPECIAL HOLONOMY MANIFOLDS JAN DE BOER Inatituut voor Theoretische Fysica, Valckenierstraat 65, 1018XE Amsterdam, The Netherlands E-mail: jdeboerOscience.uva.nl
ASAD NAQVI Department of Physics, University of Wales, Swansea, SA2 BPP, UK E-mail: [email protected]
ASSAF SHOMER Santa Cruz Institute for Particle Physics, 1156 High Street, Santa Crut, 95064 CA, USA E-mail: [email protected] We define new topological theories related to sigma models whose target space is a 7 dimensional manifold of Gz holonomy. We show how to define the topological twist and identify the BRST operator and the physical states. Correlations functions at genus zero are computed and related to Hitchin’s topological action for three-forms. We conjecture that one can extend this definition to all genus and construct a seven-dimensional topological string theory. In contrast to the four-dimensional case, it does not seem to compute terms in the low-energy effective action in three dimensions.
1. Introduction Topological strings on Calabi-Yau manifolds describe certain solvable sectors of superstrings and as such provide simplified toy models of string theory. There are two inequivalent ways to twist the Calabi-Yau sigma model. This yields topological theories known as the A-model and the B-model, which a t first sight depend on different degrees of freedom: the A-model apparently only involves the Kahler moduli and the B-model only the complex moduli. However, this changes once branes are included, and it has been conjectured that there is a version of S-duality which maps the A-
177
178
model to the B-model’ . Subsequently, several authors found evidence for the existence of seven and/or eight dimensional theories that unify and extend the A and B-models2-6. This was one of our original motivations to take a closer look a t string theory on seven-dimensional manifolds of Gz holonomy, and to see whether it allows for a topological twist, though we were motivated by other issues as well, such as applications to M-theory compactifications on G2-manifolds, and as a possible tool to improve our understanding of the relation between supersymmetric gauge theories in three and four dimensions. The outline of this note is as follows. We will first review sigma-models on target spaces of G2 holonomy, and the structure of the chiral algebra of these theories. The latter is a non-linear extension of the c = N = 1, superconformal algebra that contains an N = 1 subalgebra with c = This describes a minimal model, the tricritical Ising model, which plays a crucial role in the twisting. We then go on to describe the twisting, the BRST operator, the physical states, and we end with a discussion of topological G2 strings. Here we briefly summarize our findings. A more detailed discussion will appear elsewhere7. There is an extensive literature about string theory and M-theory compactified on G2 manifolds. Many of the results we describe were first obtained in [B]. The world-sheet chiral algebra was studied in some detail in [9, 8, 10, 111. For more about type I1 strings on G2 manifolds and their mirror symmetry, see e.g. [12, 13, 14, 15, 16, 17, 18, 19, 20, 211. A review of M-theory on G2 manifolds with many references can be found in [22].
9,
6.
2. Ga sigma models We start from an
N
= (1,l)sigma model describing d chiral superfields
S
=
s
x’1= p ( z ) + O ? y ( z )
+
d 2 z d20 (GPu B,,)DeX’”DgX’.
(1)
+
The super stress-energy tensor is given by T ( z , O ) = G(z) OT(z) = -1G 2 ’1” DeXfid,X”. This N = (1,1) sigma model can be formulated on an arbitrary target space. However, the target space theory will have some supersymmetry only when the manifold has special holonomy. This condition ensures the existence of covariantly constant spinors which are used to construct supercharges. The existence of a covariantly constant spinor on the manifold also implies the existence of covariantly constant p-forms
179
given by
This formal expression may be identically zero. The details of the target space manifold dictate which p-forms are actually present. If the manifold has special holonomy H c S O ( d ) , the non-vanishing forms (2) are precisely the forms that transform trivially under H . The existence of such covariantly constant p-forms on the target space manifold implies the existence of extra elements in the chiral algebra. For example, given a covariantly constant p form, q5(p) = q5il...i,dxi1 A . . . A dxiP satisfying V+i l...i, = 0, we can construct a holomorphic superfield current given by
J ( p ) ( z0) , = q5il...i,DoXi1 . . . DoXap which satisfies DgJ(,) = 0 on shell. In components, this implies the existence of a dimension and a dimension current. For example, on a Kahler manifold, the existence of a covariantly constant Kahler two form w = gij(d@ A d& - d& A d @ ) implies the existence of a dimension 1 current J = gij$'+j and a dimension current G'(z) = gij(+iaz& - +jaZq9), which add to the (1,l)superconformal currents G ( z ) and T ( z ) to give a ( 2 , 2 ) superconformal algebra. A generic seven dimensional Riemannian manifold has SO(7) holonomy. A Gz manifold has holonomy which sits in a Gz subgroup of SO(7). Under this embedding, the eight dimensional spinor representation 8 of SO(7) decomposes into a 7 and a singlet of Gz, and the latter corresponds to the covariantly constant spinor. The p-form (2) is non-trivial only when p = 3,4. In other words, there is a covariantly constant 3-form 4(3)= 4 $ d x i A d x j A d x k . The hodge dual 4-form is then also automatically covariantly constant. By the above discussion, the 3-form implies the existence of a superfield current J ( 3 ) ( z 0) , = q5iiLDoXiDoXjDoXk = Q, OK. Explicitly, is a dimension current = 4 i(3) jk$a+'+k and K is its dimension 2 superpartner
5
;
;
+
'
'
K = 4$L$i$kb'q5k.Similarly, the 4-from implies the existence of a dimension 2 current X and its dimension superpartner M . The chiral algebra of Ga sigma models thus contains 4 extra currents on top of the two G, T that constitute the N = 1 superconformal algebra. These six generators form a closed quantum algebra which appears explicitly e.g. in [9, 8, 101 (see also [ 111).
180
An important fact, which will be crucial in almost all the remaining analysis, is that the generators @ and X form a closed sub-algebra: if we define the supercurrent GI = x@and stress-energy tensor TI = - i X a we recognize that this is the unique N = 1 super-conformal algebra of the minimal model with central charge c = known as the %-critical Ising Model. This sub-algebra plays a role similar to the U(1) R-symmetry of the N = 2 algebra in compactifications on Calabi-Yau manifolds. In fact, with respect to the conformal symmetry, the full Virasoro algebra decomposes into two commutinga Virasoro algebras: T = TI T, with TI(z)T,(w) = 0. This means we can classify conformal primaries by two quantum numbers, namely its tri-critical Ising model highest weight and its highest weight with respect to T,:\primary) = 1h1,h,). Perhaps it is worth emphasizing the logic here: classically, we find a conformal algebra with six generators in sigma-models on manifolds of G2 holonomy. In the quantum theory we expect, in the absence of anomalies other than the conformal anomaly, to find a quantum version of this classical algebra. In [9] all quantum extensions were analyzed, and a two-parameter family of quantum algebras was found. Requiring that the quantum algebra has the right central charge (necessary to have a critical string theory) and that it contains the tricritical Ising model (necessary for space-time supersymmetry) fixes the two-parameters. This motivates why this is the appropriate definition for string theory on Gz manifolds.
&
+
3. Tri-Critical Ising Model Unitary minimal models are labeled by a positive integer p = 2 , 3 , . . . and occur only on the “discrete series” at central charges c = 1 - P(P+l). -3- The Tri-Critical king Model is the second member ( p = 4) which has central charge c = It is at the same time also a minimal model for the N = 1 superconformal algebra. The conformal primaries of unitary minimal models are labeled by two integers 1 5 n’ 5 p and 1 5 n < p . The weights in this range are arranged into a “Kac table”. The conformal weight of the primary ,@ ,, is h,,, = Ipn’-(p+1)nl2--l . In the Tri-critical Ising model ( p = 4) there are 6 4P(P+ 1) primaries of weights 0, Below we write the Kac table for
&.
&, A,i,&, &.
aThis decomposition only works for the Virasoro part of the corresponding N = 1 algebras. The full N = 1 structures do not commute. Notice, for example, that the superpartner of CJ with respect to the full N = 1 algebra is K whereas its superpartner with respect to the N = 1 of the tri-critical Ising model is X .
181
the tricritical king model. Beside the Identity operator ( h = 0) and the nf = 1 supercurrent ( k = the NS sector (first and the third column) contains a primary of weight h = and its N = 1 superpartner ( h = &). The primaries of weight are in the Ramond sector (middle column).
4)
&, &
&
The Hilbert space of the theory decomposes in a similar way, 'FI = A central theme in this work relies on the fact that since the primaries Q n t n form a closed algebra under the OPE they can be decomposed into conformal blocks which connect two Hilbert spaces. Conformal 1' 1 blocks are denoted by @n:,n,m,m, which describes the restriction of @ n t , n to a map that only acts from to 'FIL~J. An illustrative example, which will prove crucial in what follows, is the block structure of the primary @ Z , J of weight 1/10. General arguments show that the fusion rule of this field with any other primary a, is 42,l) x d(n,,n) = &-l,n) (b(d+l,n). The only non-vanishing conformal blocks in the decomposition of @ 2 , l are those that connect a primary with the primary right above it and the primary right below in the n'+l,n Kac table, namelyb, and (b2,1,n,,n. This can be summarized formally by defining the following decomposition' @n,nt'FInt,n.
?-t,t,,
+
Similarly, the fusion rule of the Ramond field @1,2 with any primary is (6(1,2) x (b(n+,n) = $(n,n-l) (6(n',n+l),showing that it is composed of two blocks, which we denote as follows @1,2 = @ Conformal blocks transform under conformal transformations exactly like the primary fields they reside in but are usually not single-valued functions of ~ ( 2 ) .
+
@c2
@L2.
bNote the confusing notation where down the Kac table means larger n' and vice-versa. cWe stress that this decomposition is special to the field @pz,l and does not necessarily hold for other primaries which may contain other blocks.
182
3.1. Chiral Primary States The chiral-algebra associated with manifolds of G2 holonomyd allows us to draw several conclusions about the possible spectrum of such theories. It is useful to decompose the generators of the chiral algebra in terms of primaries of the tri-critical Ising model and primaries of the remaindere. The commutation relations of the G2 algebra imply that some of the generators of the chiral algebra decompose as’: G(z) = @2,1 €3 $%, K ( z ) = @3,1 @ $% and M ( z ) = ~ @ Z J €3 x% b[X-1,%,1] @ $%, with $, x being primaries of the indicated weights in the T, CFT and a, b are constants. Ramond ground states of the full c = SCFT are of the form I 0 ) and I $). The existence of the I 0) state living just inside the tricritical Ising model plays a crucial role in the topological twist. Coupling left and right movers, the only possible RR ground states compatible with the G2 chiral algebraf are a single 0 ) €3~ ~ & , O ) R ground state and a certain number of states of the form $ ) L €3 I&, $)R. By studying operator product expansions of the RR ground states we get the following “special” NSNS states I O , O ) L@ IO,O)R, I&,$ ) L €3 I&, $ ) R ~In, 6 5 2 ) @~ In, 6 5 2 ) ~ and I % , O ) L €3 I;,O)R corresponding to the 4 NS primaries Qn/,1 with n’ = 1 , 2 , 3 , 4in the tri-critical Ising model. Note that, for these four states, there is a linear relation between the Kac label n / of the tri-critical Ising model part and the total conformal weight htotal = In fact, it can be shown that, similar to the BPS bound in the N = 2 case, primaries of the Gz chiral algebra satisfy a (non-linear) bound of the form
+
&,
&,
&,
9.
which is precisely saturated for the four NS states listed above. We will therefore refer to those states as “chiral primary” states. Just like in the case of Calabi-Yau, the field maps Ramond ground states to NS chiral primaries and is thus an analogue of the “spectral flow” operators in CalabiYau .
&
dWe loosely refer t o it as “the G2 algebra” but it should not be confused with the Lie algebra of the group Gz. eThese are fields that are primary under T, and do not transform under T I . fOtherwise the spectrum will contain a 1-form which will enhance the chiral algebra. Geometrically this is equivalent t o demanding that bl = 0.
183 4. Topological Twist
To construct a topologically twisted CFT we usually proceed in two steps. First we define a new stress-energy tensor, which changes the quantum numbers of the fields and operators of the theory under Lorentz transformations. Secondly, we identify a nilpotent scalar operator, usually constructed out of the supersymmetry generators of the original theory, which we declare to be the BRST operator. Often this BRST operator can be obtained in the usual way by gauge fixing a suitable symmetry. If the new stress tensor is exact with respect to the BRST operator] observables (which are elements of the BRST cohomology) are metric independent and the theory is called topological. In particular, the twisted stress tensor should have a vanishing central charge. In p r a ~ t i c e ~for ~ ithe ~ ~N , = 2 theories, an n-point correlator on the sphere in the twisted theory can conveniently be definedg as a correlator in the untwisted theory of the same n operators plus two insertions of a spin-field] related to the space-time supersymmetry charge, that serves to trivialize the spin bundle. For a Calabi-Yau 3-fold target space there are two SU(3) invariant spin-fields which are the two spectral flow operators U*;. This discrete choice in the left and the right moving sectors is the choice between the +(-) which results in the difference between the topological A / B models. In [8] a similar expression was written down for sigma models on G2 manifolds, this time involving the single Gz invariant spin field which is the unique primary @1,2 of weight It was proposed that this expression could be a suitable definition of the correlation functions of a putative topologically twisted Gz theory. In other words the twisted amplitudes are defined ash
A.
(Vl(~1). . . Vn(Zn))tvist
( C ( w ) K ( z l ) . . vn(Zn)C(O))untvist.
(5)
In [8] further arguments were given, using the Coulomb gas representation of the minimal model, that there exists a twisted stress tensor with vanishing central charge. This argument is however problematic] since the twisted stress tensor proposed there does not commute with Felder’s BRST operators26 and therefore it does not define a bona fide operator in the minimal model. In addition, a precise definition of a BRST operator was lacking. gUp to proper normalization. hUp t o a coordinate dependent factor that we omit here for brevity and can be found in
[71.
184
We will proceed somewhat differently. We will first propose a BRST operator, study its cohomology, and then use a version of ( 5 ) to compute correlation functions of BRST invariant observables. We will then comment on the extension to higher genus and on the existence of a topologically twisted GZ string. 5 . The BRST Operator
Our basic idea is that the topological theory for GZ sigma models should be formulated not in terms of local operators of the untwisted theory but in terms of its (non-local)' conformal blocks. By using the decomposition ( 3 ) into conformal blocks, we can split any field, whose tri-critical Ising model part contains just the conformal family @ z , I , into its up and down parts. For example, the N = 1 supercurrent G ( z ) can be split as
G ( z )= GI(.)
+Gf(z).
(6)
We claim that G1 is the BRST current and G f carries many features of an anti-ghost. As we comment later, it does not appear to be exactly equal to the anti-ghost though. The basic N = 1 relation 2 c / 3 2T(O) G(z)G(O)= (G'(z) G T ( z ) (G'(0) ) Gf(O)) 7 +(7) Z
+
+
N
proves the nilpotency of this BRST current (and of the candidate antighost) because the RHS contains descendants of the identity operator only and has trivial fusion rules with the primary fields of the tri-critical Ising model and so (GL)' = (GI)' = 0. More formally, denote by P,, the projection operator on the sub-space 'Hn,of states whose tri-critical Ising model part lies within the conformal family of one of the four NS primaries an,,l.The 4 projectors add to the identity
Pl
+ Pz + P3 + P4 = 1
(8)
because this exhausts the list of possible highest weights in the NS sector of the tri-critical Ising modelj. We can now define our BRST operator in 'It should be stressed that this splitting into conformal blocks is non-local in the simple sense that conformal blocks may be multi-valued functions of z(Z). jFor simplicity, we will set P,I = 0 for n1 5 0 and n' 2 5 , so that we can simply write C,, P,r = 1 instead of (8).
185
the NS sector more rigorously as
The nilpotency Q2 = 0 is easily proved:
where we could replace the intermediate Pnt+l by the identity because of property 6 and the last equality follows since L-1 maps each to itself. Q does not commute with the local operator Oa(l),o,0 ~ ( ~ , , 5 , and Oa(4),0corresponding to the chiral states (0,O) , 1 , I ,): and ,:I 0) (for brevity we will denote those 4 local operators juts by their minimal model Kac label Oil i = 1,2,3,4). However, one can check that the following blocks,
&, g) &
m
which pick out the maximal "down component" of the corresponding local operator, do commute with Q and are thus in its operator cohomology. Thus the chiral operators of the twisted model are represented in terms of the blocks 11 of the local operators corresponding t o the chiral states. Furthermore, it can be shown easily that those chiral operator form a ring under the OPE. By doing a calculation a t large volume, we see that the BRST cohomology has one operator of type 018 01, b2 b3 of type 0 2 8 0 2 , b4 bs of type 0 3 8 0 3 , and one of type 0 4 8 0 4 . The total BRST cohomology is thus precisely given by H * ( M ) .Also, one finds that the b3 operators of type 0 1 8 0 1 are precisely the geometric moduli of the G2 target space. In the topological G2 theory, genus zero correlation functions of chiral primaries between BRST closed states are position independent. Indeed, the generator of translations on the plane, namely L-1 , is BRST exact
+
+
(12) This is a crucial ingredient of topological theories. Moreover, it can be showns that the upper components G- 4 I , :)L 8 G- 4 I , : ) R correspond to exactly marginal deformations of the CFT preserving the G2 chiral algebra, completely in agreement with
&
186
their identification as the geometric moduli of the theory. Focussing momentarily on the left movers, we can show that [Q, { G - ; , 0 2 } ] = dd2 k, so that the very same deformation is physical, namely Q exact, also in the topological theory. Note that the deformation is given by a conventional operator that does not involve any projectors. Combining with the right-movers, we find that the deformations in the action of the topological string are exactly the same as the deformations of the non-topological string as is expected because both should exist on an arbitrary manifold of G:! holonomy. Most correlation functions at genus zero vanish. The most interesting one is the three-point function of three operators Y = 0 2 &I 0 2 . These correspond to geometric moduli (or B-field moduli that we ignore for now). If we introduce coordinates ti on the moduli space of G2 metrics, then we obtain, from a large volume calculation,
One might expect, based on general arguments, that this is the third derivative of some prepotential if suitable ‘flat’ coordinates are used. We do not know the precise definition of flat coordinates for the moduli space of G2 metrics, but if we take for example M = T7and take coordinates such that $ is linear in them, then we can verify
The prepotential appearing on the right hand side is exactly the same as the action functional introduced by Hitchin in [27, 281. A similar action was also used as a starting point for topological M-theory in [3] (see also [2]). This strongly suggests that our topological G2 field theory is somehow related to topological M-theory.
6. Topological Ga Strings In the case of N = 2 theories, the computation of correlation functions at genus zero outlined above can be generalized to higher genera23i24.An n-point correlator on a genus-g Riemann surface in the twisted theory can be defined as a correlator in the untwisted theory of the same n operators plus (2 - 29) insertions of the spin-field that is related to the space-time supersymmetry charge. For a Calabi-Yau %fold target space on a Riemann kRecall that
A2
was defined in eq. (11).
187
surface with g > 1, the meaning of the above prescription is to insert 29 - 2 of the conjugate spectral flow operator. To generalize this to the G2 situation, we would like to have something similar. However, there is only one G2 invariant spin-field. This is where the decomposition in conformal blocks given in section 3 is useful: the spin-field @2,1 could be decomposed' in a block and in a block At genus zero we needed two insertions of so the natural guess is that at genus g we need 29 - 2 insertions of However, this is not the full story. We also need to insert 39 - 3 copies of the anti-ghost and integrate over the moduli space of Riemann surfaces to properly define a topological string theory. The anti-ghost is very close to G t , and the fusion rules of the tricritical Ising model tell us that there is indeed a non-vanishing contribution to correlation functions of 29 - 2 @el's and 39 - 3 G t . This prescription would therefore work very nicely if we would have found the right anti-ghost. The candidates we tried so far all seem to fail in one way or another. One possible conclusion might be that a twisted stress tensor does not exist and that there is only a sensible notion of topological G2 sigma models but not of topological G2 strings. The fact that so far so many properties of the N = 2 topological theories appeared to hold also in our Gz model leads us to believe that a sensible extension to higher genera indeed exists. Identifying the correct twisted stress tensor remains an open problem. Barring this important omission the coupling to topological gravity goes pretty much along the same lines as for the N = 2 topological string (details can be found in [7]).
@il
@ill
@Fl.
@cl.
7. Conclusions An important
application of topological strings stems from the that its amplitudes agree with certain amplitudes of the physical superstring. Just like N = 2 topological strings compute certain Fterms in four dimensional N = 2 gauge theories, one might wonder whether the G2 topological string similarly computes F-terms in three dimensional N = 2 gauge theories. Since G2 manifolds are Ricci-flat, we can consider compactifying the type I1 superstring on R1)2x N7 where N7 is a 7 dimensional manifold of G2 holonomy. This reduces the supersymmetry down to two real supercharges in 3 dimension from each worldsheet chirality so we end up with a low 'In terms of the Coulomb gas representation, one of these can be represented as an ordinary vertex operator, the other one involves a screening charge.
188 energy field theory in 3 dimensions with N = 2 supergravity. By studying amplitudes in some detail we observe that, except perhaps at genus zero, the amplitudes do involve a sum over conformal blocks, and the topological Gz string therefore seems t o only compute one of many contributions to a n amplitude. In four-dimensions, the possibility t o look at (anti)-self dual gravitons allows one t o essentially restrict attention to a single conformal block, but a similar mechanism is not available in three dimensions. Nevertheless, we believe that topological G2 strings are worthwhile t o study. They may possible provide a good definition of topological M-theory, and a further study may teach us many things about non-topological G2 compatifications as well. We leave this, as well as the generalization t o ~ p i n ( 7 )compactifications ~~ and the study of branes in these theories t o future work. Acknowledgement It is a pleasure t o thank Robbert Dijkgraaf, Wolfgang Lerche, Hiroshi Ooguri and Cumrun Vafa for useful discussions. We would also like t o thank the organizers of the 12th. Regional Conference on Mathematical Physics in Islamabad, Pakistan. This work was partly supported by the stichting FOM.
References 1. A. Neitzke and C. Vafa, “N = 2 strings and the twistorial Calabi-Yau”, arXiv:hep-t h/0402 128; N. Nekrasov, H. Ooguri and C. Vafa, JHEP 0410,009 (2004), [arXiv:hepth/0403167]. 2. A. A. Gerasimov and S. L. Shatashvili, em JHEP 0411, 074 (2004), [arXiv:hep-th/0409238]. 3. R. Dijkgraaf, S. Gukov, A. Neitzke and C. Vafa, “Topological M-theory as unification of form theories of gravity”, arXiv:hep-th/O411073. 4. N. Nekrasov, “A la recherche de la m-theorie perdue. Z theory: Chasing m/f theory” , arXiv:hep-t h/0412021. 5. P. A. Grassi and P. Vanhove, “Topological M theory from pure spinor formalism”, arXiv:hep-th/O411167. 6. L. Anguelova, P. de Medeiros and A. Sinkovics, “On topological F-theory”, arXiv:hep-th/0412120. 7. J. de Boer, A. Naqvi and A. Shomer, “The topological G(2) string”, arXiv:hep-th/O506211. 8. S. L. Shatashvili and C. Vafa, ‘‘Superstrings and manifold of exceptional holonomy”, arXiv:hep-th/9407025. 9. R. Blumenhagen, Nucl. Phys. B 381,641 (1992). 10. J. M. Figueroa-O’Farrill, Phys. Lett. B 392, 77 (1997), [arXiv:hept h/9609113].
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B 610, 545 (200l), [arXiv:hepth/0101116]; B. Noyvert, JHEP 0203,030 (2002). [arXiv:hep-th/0201198]. B. S. Acharya, “N=l M-theory-Heterotic Duality in Three Dimensions and Joyce Manifolds”, arXiv:hep-th/9604133. B. S. Acharya, Nucl. Phys. B 524,269 (1998), [arXiv:hep-th/9707186]. T. Eguchi and Y. Sugawara, Phys. Lett. B 519, 149 (2001), [arXiv:hepth/0108091]. R. Roiban and J. Walcher, JHEP 0112,008 (2001), [arXiv:hep-th/Ol10302]. T. Eguchi and Y. Sugawara, Nucl. Phys. B 630, 132 (2002), [arXiv:hepth/0111012]. R. Blumenhagen and V. Braun, JHEP 0112, 006 (2001), [arXiv:hepth/0110232]; JHEP 0112,013 (2001), [arXiv:hep-th/0111048]. R. Roiban, C. Romelsberger and J. Walcher, Adw. Theor. Math. Phys. 6,207 (2003), [arXiv:hep-th/0203272]. K. Sugiyama and S. Yamaguchi, Phys. Lett. B 538, 173 (2002), [arXiv:hept h/0204213]. T. Eguchi, Y. Sugawara and S. Yamaguchi, Nucl. Phys. B 657,3 (2003), [arXiv:hep-th/0301164]. M. R. Gaberdiel and P. Kaste, JHEP 0408, 001 (2004), [arXiv:hepth/0401125]. B. S. Acharya and S. Gukov, Phys. Rept. 392, 121 (2004), [arXiv:hept h/0409191]. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Commun. Math. Phys. 165,311 (1994), [arXiv:hep-th/9309140]. I. Antoniadis, E. Gava, K . S. Narain and T. R. Taylor, Nucl. Phys. B 413, 162 (1994), [arXiv:hep-th/9307158]. E. Witten, “Mirror manifolds and topological field theory”, arXiv:hepth/9112056. G. Felder, Nucl. Phys. B 317,215 (1989) [Erratum-ibid. B. 324,548 (1989)]. N. Hitchin, “The geometry of three-forms in six and seven dimensions”, arXiv:math.dg/0010054. N. Hitchin, “Stable forms and special metrics”, arXiv:math.dg/0107101. J. de Boer, A. Naqvi and A. Shomer, “Topological Strings on Exceptional Holonomy Manifolds”, to appear.
11. D. Gepner and B. Noyvert, Nucl. Phys.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
VACUA I N N=4 GAUGED SUPERGRAVITY SUDHAKAR PANDA Harish-Chandra Research Institute, Chhatnag Road, Allahabad 211019, India E-mail: pandaomri. ernet.in In this talk, we summarise the results of our investigation on the vacua in four dimensional N = 4 gauged supergravity coupled to six vector multiplets. The gauging involves both compact and non-compact gauge groups.
The de Sitter (dS) space-time is a maximally symmetric solution of Einstein's equations with positive cosmological constant (A). In D-dimensions, dS space can be represented as a hyperboloid in R1iD and is defined by
In D = 4,this geometry is a solution of Einstein's equation in vacuum with a cosmological constant A = 3 H 2 , (> 0) in the units of 87rG = 1. This can be interpreted as the geometry generated by the interaction with gravity of a system with uniform matter density, p = 3 H 2 and pressure p = - p . Locally this can be described by the FRW metric in D = 4 with the scale factor a ( t ) > 0 and = A/3 = H 2 . This metric describes the evolution of our universe with accelerated expansion which is favoured by experimental evidence in cosmology. In fact it is believed that the universe is in a dS regime with A 10-12'M4P' In inflationary scenarios of cosmology, the accelerated expansion is triggered by slow evolution of a uniform scalar field @ with energy dominated by a positive potential V ( @ with ) V ( @ )>> b2. Such a model can have critical points @po in the field space defined by N
a,pV(@)l~,, = 0 with V(Qp0) = A > 0.
(2)
String theory, being a natural theory invoving gravitational interactions, is
190
191
expected to invoke the inflationary models and in particular should admit dS space as a solution. However, in 2000, Maldacena and Nunezl studied a supergravity description of field theories on curved manifolds. They discovered a no-go theorem for having a de Sitter solution in a four dimensional gravity theory if such a theory is dimensionally reduced from a theory in higher dimension, provided the parent theory did not admit a de Sitter vacuum. Though this analysis is carried out in a perturbative framework, its implication in the context of Super String threoy as a viable theory for quantum gravity is very crucial. This is because of the fact that string theory is consistent only in ten dimensions and the theory governing various dynamics in our four dimensional world is believed to be a compactified version of this. Since string theory in ten dimensions does not admit a de Sitter vacuum, the above no-go theorem implies that any four dimensional theory obtained from string theory cannot admit a de Sitter vacuum. On the other hand, as already mentioned, the experimental observations favour a de Sitter Universe. This led many researchers to study supergravity theories with different local supersymmetries and with special emphasis on de Sitter v a c ~ u a ,~since - ~ these supergravity theories are low energy limits of string theories and are field theoretic frameworks close to string theory. This scheme relies on the expectation that some non-perturbative physics could play a crucial role in evading the no-go theorem6. Since the supergravity theories, by construction, are equipped with high degree of supersymmetries, the only way to deform the action for allowing a non-trivial scalar potential is to promote some of the global symmetries of the theory to local symmetries. This procedure is known as gauging. We studied a N = 4 theory? which is constructed in four dimensions and based upon a superconformal formulation of gravity. This has a larger symmetry group compared to the usual super-Poincare group. The higher symmetry group allows us to find, in a simpler way, the invariant action with simple transformation rules for the fields in the supergravity theory, including its coupling with matter fields. However, for keeping the physical degrees of freedom of the theory to be the same, one needs to add compensating fields. The additional symmetries are to be broken explicitly by imposing various constraints on the fields. For example, recall that in the general theory of relativity, when formulated in terms of the gauge fields, e;, wEb, associated with translation and local Lorentz rotations respectively, one imposes the constraint that the curvature (RE,) vanishes. This constraint is solved to express wEb in terms of e: and more importantly the
192
local translation gauge transformations give rise to the general coordinate transformations. Thus the resulting theory has the required symmetries of general coordinate transformations and the local Lorentz transformations. For conformal gravity, one assigns a gauge field to each of the generators of the conformal group i.e. translations ( P a ) local , Lorentz transformations (Mab), dilatation ( 0 ) and the conformal boosts (K,). These generators satify the usual conformal algebra. Let the respective gauge fields be e;, wEb, b, and f:. Note that the degrees of freedom associated with these gauge fields are far larger than that of the gravitational field. The constraints
REv (P) = Oand RE! ( M ) Eby = 0,
(3)
where Eby is the inverse of eb,, reduce the degrees of freedom and leave only e: and b, fields as independent gauge fields. These constraints automatically reduce the P-gauge transformation to general coordinate transformations. However, we are still left with twenty degrees of freedom associated with the above independent gauge fields. Out of this fifteen degrees of freedom can be fixed with the use of conformal gauge symmetries. The remaining five degrees of freedom coincide with the dimension of the massive spin-2 representation of the Poincare algebra. But we are one short of the number of degrees of freedom associated with the gravitational field. We need to find a description, in the conformal framework, for the spin-0 degree of freedom i.e. the overall scale of the metric field. This conformal mode can suitably be described in terms of a real scalar field (having conformal/Weyl weight 1) with the Lagrangian density given as
L
=
- 1/2 e $ DaD,$ with D,$
=
E,” (8, - b,) $,
(4)
where e is the determinant of the gauge field e;. This Lagrangian density is invariant under local M , D, K gauge and the general coordinate transformations. Choosing K-gauge i.e. b, = 0 and D-gauge i.e. when $ = $0, which is a fixed constant taking the value 4; = 6 / ~ ’( K being the gravitational coupling constant) , the above Lagrangian density reproduces the conventional Einstein-Hilbert Lagrangian density:
L =
- -I 2K2
e R with R = Rl,”b(M)E,”Eby,
(5)
where the prime denotes the absence of f; term in the curvature term. This demonstrates how ordinary gravity emerges from gauge fields of the conformal group. One can consider the coupling of extra n matter scalar fields to gravity by adding their conformally invariant actions to the first one, described by
193
the Lagrangian density:
C
=
1/2 e ~ R 4R S DaDa#’
+
where R, S take the values 1 , 2 , 3 , .....,n 1 and ~ R is S a diagonal matrix with entries f l . Proceeding as before by fixing various gauges and demanding that the kinetic energy of the physical fields must be positive (which actually fkes the signs of the entries of q) , the above Lagrangian density can be brought to the following form
with I
(-,
=
2,3, ....., n
+ 1, provided
the entries in the q-matrix are
+, +, .......,+) and the following constraint on the fields are satisfied:
This is clearly the known expression for matter scalar fields coupled with gravity but here it arises in a conformal formulation of gravity. Note that the above constraint on the scalar fields play a crucial role since it says that the physical scalar degrees of freedom are described by a n-dimensional submanifold of (n+ 1)-dimensional space and the scale of the manifold is set by the Newton’s constant hidden in K . In fact, the a-model can be identified with the coset space SO(l,n)/SO(n) by noting that the solutions of the above constraint can be viewed as the first row of the SO(1,n) matrix. It is interesting to observe the emergence of the non-trivial a-models in the Poincare formulation starting from the conformal approach. The above conformal approach can be generalized to the superconforma1 symmetry group for constructing supergravity theories. To obtain a N = 4 supergravity theory in this formulation, we need to include the generators Qi,SZ (i = 1,2,3,4), which generate supersymmetry, super conformal boost respectively, as well as the SU(4) generator for phase rotations of chiral parts of & a and Si.We can perform the same analysis as before by introducing gauge fields for each generator belonging to the superconformal group and imposing the required constraints. It turns out that the N = 4 Weyl multiplet contains two complex scalar fields (Pa ( a = 1 , 2 ) which transform under a global S U ( 1 , l ) and a local U(1) symmetry and are subjected to the constraint that q!P q5a = 1. In our basis we have d1 = &, q52 = - 4;. Thus these fields actually provide two real scalar degrees of freedom and can be parametrized by the coset space SU(1,1)/UU).
194
The N = 4 supergravity multiplet, in this formulation, can be constructed by introducing six compensating vector multiplets. This is achieved by coupling six N = 4 vector multiplets to the scalars q5a in the Weyl multiplet such that the above mentioned global S U ( 1 , l ) symmetry is respected. Note that starting with the global vector multiplet and acting on a vector field A, with four different supersymmetry generators, one gets four fermionic fields and six complex scalar fields, which we denote by @j and they obey @j = (&)* = - 1/2 & j k l & l . Thus we are left with six real scalar degrees of freedom from each vector multiplet. Besides the compensating vector multiplets one can also couple an arbitrary numer ( n )of matter vector multiplets to the Weyl multiplet. There are two important ingredients which play a crucial role for having a nontrivial scalar potential from the resulting supergravity theory in our formalism. The first one is the gauging. Note that the gauge symmetry for the above n 6 vector multiplets is an Abelian symmetry i.e. 6 A; = a p e R where R = 1 , 2 , ....,n + 6 . This Abelian symmetry can be extended to a non-Abelian symmetry acting on subsets of the vector fields. Equivalently, the gauge group is a product of several factors like SU(2) x SU(2), SO(3) x SO(3) etc if there are only six vector fields. Such gaugings produce a scalar potential V(q5a,qhij) which is a function of the above mentioned scalar fields. The second crucial ingredient is called the SU(1, 1)-angle. In coupling the vector multiplets to the S U ( 1 , l ) scalars &, it turns out that all the parameters specifying a matrix representative of the group S U ( 1 , l ) are not relevant. For example, the matrix C belonging to this group can taken to be
+
*: ( i) ,
C=
with
+ +
a = 1/2 ePia(s 1/s - it) b = 1/2 eia(-s l/s it),
+
where the parameters s and t correspond to to the non-compact directions and a is the angular variable corresponding to the compact direction. It turns out that t can be set to zero (since it corresponds to a total derivative term in the Lagrangian density of the coupled theory) and s can be absorbed into the vector field by a scaling. Thus, a is the only non-trivial parameter for gauging purposes and one can associate an angle ai for the vector fields A, in the i-th subgroup of the product gauge group. In effect, these angles rotate one factor group with respect t o another in the product of gauge groups. We also define the real scalar fields 2, with a = 1 , 2 , ...,6 which are
195
the scalars coming from each of the vector multiplets (q5ij) but in a basis such that 2, transforms as a vector under SO(6).Taking into account all these scalars coming from n 6 vector multiplets, we can denote them as 2," where R = 1 , 2 , ...,n 6. One finds that 2," transforms globally under SO(6,n) and locally under SO(6) x SO(n). Performing an analysis analogous to what we did for the conformal formulation of gravity, one can arrive at the following Lagrangian density for N = 4 supergravity coupled to n matter vector multiplets:
+ +
e-l L: = - 1/2 R
+ L : $ +~ L:$, + L : T ~-
~ ( 4Z ),
(9)
where K E denotes the kinetic energy term for indicated fields, the details of which are not a concern for this talk but the reader can consult the original paper. The object of interest for this talk is the scalar potential V(q5,Z) given by
+ 2 ZTW)
V(q5,Z) = [-1 ZRU zsv (,,w 4 _ _
zRSTUVW
]
3
(10)
@:R) ~ R S T@(u)~ U R W ,
36 where ~ R is S a (n+ 6 ) x (n+ 6) diagonal matrix with the first six elements being -1 and the next n elements being +1 and ZRS=ZR a zs a zRSTUVW = Eabcdef Z R Z S Z T a b c @ ( R )=
eiaR
41+ e-iaR
22 2,"
+2
27
$tR)
(11)
+4fRl.
The last line above reflects the freedom in coupling the vector multiplet, labelled by R, to the Weyl multiplet via the all important S U ( 1 , l ) angle (YR which can be reinterpreted as a modification of the SU(1, 1)-scalars coupling to the multiplet. f ~ =sf z s~~ V are T the totally antisymmetric structure constants of the gauge group. The scalar potential being proportional to the structure constants reassures the importance of gauging for the theory to have a non-zero scalar potential. We have already alluded to the constraint on the SU(1, 1)-scalars. The real scalar fields, from the vector multiplets, are also subjected to the constraint that 2,"
VRS
=
-
dab,
(12)
where 7 ~ are s the components of the invariant metric in the vector representation of SO(6,n) in a basis where it takes the matrix form as mentioned above. This constraint restricts these scalars to the coset
196
SO(6, n ) / ( S 0 ( 6 ) x S o ( n ) ) Hence . the scalars 2: can be viewed as the upper six rows of a S0(6,n)-matrix. Note that there are 6n physical scalars in this coset. Thus, the number of physical scalars from the two cosets in the theory add up to 6n 2 when n number of matter multiplets are coupled to the supergravity multiplet. From now on we will restrict ourselves to n = 6, which possibly makes contact with string theory. For this case we have thirtyeight scalars. However, the scalar potential can further be analysed only when we can solve the constraints on these scalars. A convenient parametrization of the S U ( 1,1)-scalars, which solves the constraint $a 4a = 1 is given by
+
To solve the constraint on the matter scalars, it is convenient to split the indices R , S , ... of ~ R Sin A , B ,... = 1,...,6, ( ~ A B = - ~ A B and ) I , J ,... = 7 , ..., 12, ( ~ I J= 6 1 ~ )Defining . 2," = X," and 2,' = Y'-s a 1 the constraint is solved if we parametrize the 6 x 6 matrices X and Y as below:
+
X =
y
=
1 (G G-' 2 1 - (G - G - 1 2
+
-
+ BG-l-
G-lB
-
BG-lB)
(14)
B G - ~- G - ~ B- B G - ~ B ) .
Here G is an invertible symmetric 6 x 6-matrix and B is an antisymmetric 6 x 6-matrix whose respective elements, namely 21 and 15 together, account for the 36 matter scalars. In the scalar potential, we have, Z A B = ( X t X ) A B ZA' , = (XtY)A' and Z I J = ( Y t Y ) l JHere . t denotes transpose. Thus it is a function V ( Tp, , G, B ) of the thirtyeight scalar fields. For simplicity, we consider a truncated sytem of scalar fields for the matter sector, namely a (positive) and b and take the other thirtyfour scalars to be zero. This is achieved by choosing:
G
=
a
):(
and B
=
b
( ")' -13
0
(15)
where I3 is the 3 x 3 unit matrix. Thus the scalar potential becomes a function of the four scalars T , p, a and b. Its explicit function can be determined only when the gauge group for gauging is fixed. For an example, we consider here the gauge group S O ( 2 , l ) x S O ( 2 , l ) x (U(1))' by assigning ( A t ,A;, A;) and (A;, A:, Aho) to the two S0(2,1)-groups with (91,~ 1 and (92,~ 2 being ) the respective coupling constants and the associated
)
197
SU(1,l) angles. Using the known structure constants for this group, the scalar potential is determined and found to be
V ( r , ' p , a , b ) = V(r,'p)V ( a , b ) with
+
+ + +
92 cos(2a2 cp)] 7 1 V ( a ,b) = -(a2 b2 1)2 [b2 (a2 b2 1)2 2a2 (a2 - b2) 16a6 This potential can be extremised with respect to the four scalar fields and we find that the extremum corresponds to an absolute minimum where the value of the potential, VO,is found to be
VO
+ +
= 191 572 sin(w - m)l.
+
3.
(17)
The right hand side of this equation, being an absolute value, is a positive quantity for a1 # a2 and non-zero values of the two coupling constants. Hence this vacuum corresponds to a de Sitter vacuum. This brings out the essential feature of our model i.e. we need product groups for gauging as well as different angles for coupling the vector fields with the Weyl multiplet. However, if the other thirtyfour scalar fields, which were put to zero in the above analysis, are turned on, this minimum turns out to be unstable since the mass matrix involving all the matter scalar fields develop negative eigenvalues and hence tachyonic modes appear in the spectrum. This aspect of stability and the gauging with various product groups are studied in detail in [8]. An interesting observation for gauging with the group (S0(3))4 is that there are no negative eigenvalues in the mass matrix but when the matter scalars are at the minimum of the potential, the two SU(1, 1)-scalars are at the the maximum of the potential giving rise to the possibility of unstable/metastable vacuum. This conclusion remains unchanged even for gauging the theory with CSO groupsg where a cosmological scaling solution is found. The possibility of a connection of this supergravity model to String theory has been studied by considering a group manifold (SU(2) x SU(2)) reduction of N = 1 supergravity in ten dimensions, coupled to Y-M multiplets, to four dimensionslO. This analysis reveals that the field contents of these two four dimensional theories are the same and the two SU(l.1)scalars can be identified with the dilaton and the axion of the String theory. Besides, the scalar potentials of the two theories can be brought to the same form only when the S U ( 1 , l ) - angles are put to zero. This observation raises
198 the possibility of the angles having origin in the non-perturbative physics in String theory. Certainly, more has to be done for a complete understanding of the vacua of this supergravity theory.
Acknowledgments
I am grateful t o M. de Roo, D. B. Westra, M. Trigiante and M.G.C. Eenink for collaboration and sharing their insight on the subject. The results discussed here are the output of these collaborations. I thank the organisers of this workshop for the kind invitation and making my stay at Islamabad a very enjoyable and a memorable one too.
References 1. 2. 3. 4. 5.
6. 7. 8. 9. 10.
J. Maldacena and C. Nunez, Int. J . Mod. Phys. A 16,822 (2001). P. Fre, M. Trigiante, and A. Van Proeyen, Class. Qu. Grav. 19, 4167 (2002). M. de Roo, S. Panda and D. B. Westra, JHEP 02,003 (2003). K. Behrndt and S. Mahapatra, JHEP 0401,068 (2004). R. Kallosh, A. Linde, S. Prokushkin and M. Shmakova, Phys. Rev. D 65, 105016 (2002). S. Kachru, R. Kallosh, A. Linde and S. Trivedi, Phys. Rev. D 68,046005 (2003). M. de Roo, Nucl. Phys. B 255, 515 (1985); M. de Roo and P. Wegmans, Nucl. Phys. B 262,644 (1985). M. de Roo, S. Panda, M. Trigiante and D. B. Westra, JHEP 0311, 0022 (2003). M. de Roo, D. B. Westra and S. Panda, aeXiv:hep-th/0606262. M. de Roo, M. G. C. Eenink, S. Panda and D. B. Westra, JHEP 0506, 077 (2006).
NEUTRINO PHYSICS FUAZUDDIN National Centre for Physics, Quaid-i-Azam University Campus, Islamabad, Pakistan Recent progress in neutrino physics, which is certainly one of the most exciting areas of research at present, is reviewed. The implications of neutrino oscillations and mass squared splitting between neutrinos of different flavor on the pattern of the neutrino mass matrix is discussed. Neutrinos are providing evidence for new physics but the scale of the new physics is not yet pinned down. The heavy right handed Majorana neutrinos, needed to understand the tiny masses of light neutrinos at new physics scale, may provide an explanation for baryogenesis through leptogenesis.
1. Introduction The neutrino was the first particle postulated by a theoretician, W. Pauli, in 1930, to save conservation of energy and angular momentum in p decays. Ever since its direct observation in the 1950’s by Reines and Cowan, this elusive particle which has almost no interaction with matter, has contributed to some of the most important discoveries in Physics. 7 Nobel Prizes since 1950 involve neutrinos in one way or other. Certainly one of the most exciting areas or research at present is neutrino physics. Neutrinos are fantastically numerous in the universe and so to understand the universe we must understand neutrinos. What is known about neutrinos: Neutrinos are elementary particles with spin 1/2, are electrically neutral and obey Fermi-Dirac Statistics. Neutrinos play an important part in stellar dynamics: The energy of the sun is generated through the fusion process: 4p +4 H e
+ 2e+ + 2ve + 26.7MeV.
There are about 7 x l O ’ ~ m - ~ sec-l neutrinos from the sun reaching the earth.
199
200
If neutrinos were not there, the stars would not shine and we would not be here. Neutrinos occur in three flavors, each flavor associated with the corresponding charged lepton:
All neutrinos detected are left-handed i.e. their spins point in the opposite direction to their momenta. All anti-neutrinos are right-handed. Neutrinos “oscillate” from one species to another with a high probability. This means that a neutrino produced in a well-defined weak eigenstate ua, can be detected later in a distinct weak eigenstate up. Such a flavor change is observed and the simplest way to explain this phenomenon is to postulate that neutrinos have distinct non-zero mass, and the neutrino mass eigenstates are different from the neutrino weak eigenstates, the latter being generally coherent superpositions of the former
i
The neutrinos undergo oscillations as they propagate and the oscillating probability Pap is a function of the propagation distance L, neutrino mass difference Amzj = m; - m:, neutrino energy E, and the elements of the lepton mixing matrix Uai. For example for two flavor conversion:
Pap = sin2 28 sin2 [1.27Am2/E,] L , where L is measured in meters and Am2 = mi - rnf in units of eV2 while the neutrino energy E , is measured in MeV and 8 is the mixing angle in a 2 x 2 (two flavor) scenario, with
u=
(
cos8 sin8 - sin 8 cos 8
1.
The above result illustrates the quantum mechanical phenomenon of interference which provides a sensitive method to probe extremely small effects. For interferometery to work, one needs’ i . a coherent source: luckily there are many coherent sources of neutrinos: the sun, cosmic rays, reactors etc. ii. interference: luckily there are large mixing angles to make interference possible. iii. large baseline to enhance the tiny effects: again luckily many baselines are available: size of the sun, that of the earth etc.
201
Thus nature is very kind to provide all the ingredients for neutrino oscillations (interferometry) to work, thereby providing us a unique tool to study physics at very high energy scales. For 3 flavors, it is customary to parameterize the mixing matrix elements U,i (not all independent) by three mixing angles 1912, 013, 1323 and one complex phase 6:
The mass eigenstate Ivi) (i = 1, 2, 3) has a well defined mass mi and it is customary to order the mass eigenvalues such that mf < m i , Am:, < IAmbI: Am:, Am:,
> 0 +mi > rn: < 0 +mi < m:
normal mass hierarchy inverted mass hierarchy.
Detailed combined analysis of all neutrino data are consistent at 3a level with’
5 4.8 x 10-3eV2, 0.70 5 sin22812 5 0.95, 5.2 x 10-5eV2 5 IArn?,I _< 9.5 x lOP5eV2, sin21313 5 0.0047.
sin226323 2 0.92, 1.2 x 10-3eV2 5
Currently, there is no constraint on the CP-odd phase S or on the sign of Am13. Since the oscillation data are only sensitive to mass squared differences, they allow for 3 possible arrangements of the different mass levels two of which are shown in Fig. 1 while the third has degenerate neutrinos i.e. ml M m2 M m3. The most important conclusion one draws is that “NEUTRINOS HAVE MASS”. However oscillation experiments do not tell us about the overall scale of masses, but they are exceedingly tiny. The most straightforward limit on the absolute value of neutrino masses is obtained by looking for structure near the end point of the electron energy spectra in tritium P-decay. These searches reveal1
v i
202
rm?
Fig. 1.
Neutrino mass level arrangments: Normal and Inverted mass hierarchy
2. The Spectrum Scale of v-mass from Present to Projected Goal2 The following are the current bounds on neutrino masses: Tritium /3 decay x m i 5 2eV
N
x
0.2eV,
i
Cosmology
mi 5 0.69eV
i PPOV
mpp
=
x
N
(0.05 - 0.1) eV,
IU,imil- 0 . 0 2 e ~
i
50.3eV By combining all these bounds it is safe to say that all neutrinos have masses less then l e v . Fig.2 depicts the value of all known fundamental fermions: One sees that the gap between neutrino masses and the lightest charged fermion is deserted in contrast to that between me and mt which is populated3. Further
which needs to be understood.
203
fermion masses
Mass spectrum of quarks and leptons.
Fig. 2.
3. Origin of Neutrino Mass The neutrino occurs in just one helicity state (left handed). This together with lepton number conservation implies m, = 0. However, there is no deep reason that it should be so. There is no local gauge symmetry and no massless gauge boson coupled to lepton number L , which therefore is expected to be violated. Thus one may expect a finite mass for the neutrino. However the Standard Model (SM) conserves L , nor does it contain any chirally right-handed neutral fields, but only left-handed ones VL. If one allows right-handed neutrinos Ni which are S U ( 2 ) x U(1) singlets, then one can write Yukawa interactions
C,,,
= LSM- h,iE,HNi
+ h.c.,
where L is the left handed lepton doublet, H is the Higgs doublet. After electroweak symmetry breaking,
+
Cmass= -h ( H )DLNR h.c. This does not mix neutrinos and antineutrinos, so it conserves L. The neutrino mass matrix is
Since ( H )
-
m, = h ( H ) .
175 GeV, the magnitude of the neutrino mass requires h 5 at least 6 orders of magnitude smaller than electron Yukawa couplings. A natural explanation of the smallness of m, is not contained in the above equation.
204
4. Majorana Neutrinos
An interesting question about the intrinsic nature of neutrinos] raised by the discovery of neutrino mass is: Are neutrinos their own antiparticles? i.e. for given helicity h Vi(h) = vi(h). Now NR being an electroweak isospin singlet, all the SM principles] including electroweak isospin conservation allow a “Majorana mass term”
CM
= -MN~$NR-th.c.,
where NE is the charge conjugate of NR. This converts N to and as such does not conserve L. The most economical way to add neutrino mass to the SM is to allow neutrinos to have Majorana mass arising from AL = 2 nonrenormalizable interactions of the form [L is the lepton doublet]
G
L,ff = -LHLH
M
After electroweak symmetry breaking i.e. replacing the Higgs field by its expectation value (H) = v,
= m,vvl
which is nothing but the neutrino mass
G M
my = -v2. Such an effective interaction can be generated by 1L y = LiHhijeRj EHhijNRj - - N ~ M N R h.c., 2 i, j = 1,2,3 for 3 leptons families. The lepton number violating term is introduced by the third term. M is the Majorana mass matrix while hijare Yukawa couplings. After spontaneous symmetry breaking] a Dirac mass term is generated] (mD)ij = hijv, assumed to be small compared to M . Light neutrino mass matrix Mu arising from diagonalizing the 6 x 6 neutrino mass matrix is
+
and takes the seesaw form m u = -rn;M-’mD mNi = Mi.
+
205
This matrix has an eigenvalue
by requiring the existence of a scale M , associated with new physics. With v = 175GeV, the above number is of the order needed to explain neutrino m, M 0.045 eV anomaly for M N 1015GeV or so, not much different from Grand Unification (GUT) scale and other scales which have been proposed for new physics. If neutrinos are Majorana particles, the 3 x 3 leptonic matrix U may contain 3 CP violating Majorana phases q5i associated with the neutrino (self conjugate) mass eigenstate ui. In their presence,
Can one test the Majorana character of the neutrino? SM conserves lepton number L , so that the L non-conservation we seek can come only from Majorana mass terms and as such will be challenged by the smallness of m,. Search of neutrinoless double P-decay (pictured in Fig. 3) is the only approach that shows considerable promise of meeting this challenge4.
Fig. 3. Neutrinoless double beta decay.
The above process does not exist if Fi # ui, helicity of Vi cannot be exactly +1 but contains a small piece, of order mi/E,i having helicity -1.
206
Thus contribution of ui exchange is proportional t o mi Amp[Ov/3P]a IU:imiI ImppI =
.
= mpp’,
lcOsel3(jmll e-2i61 cos2 812 + lmzl e-zidz
sin2el2)
+ sin2 eI3 im31 epzi6I
Current experiments rule out m p p L l e v and the present upper bound is I 0.3eV. A controversial and yet-to-be confirmed analysis of 76Ge decay data by the Heidelberg-Moscow group claims5 that m p p lies between O.1leV t o 0.58eV. If confirmed this result is of fundamental importance giving the first indication of lepton number violation and that Majorana neutrinos can exist in nature. 5 . Neutrino Mass Models
Phenomenological models derived by the data have to involve some peculiar features of neutrino mixing such as6 (i) Some or (all) neutrino masses could be quasi-degenerate in absolute value. (ii) IUe31 << all other entries in the neutrino mixing matrix U . (iii) I UcL3 I N IU731, maximal atmospheric mixing. Such models in turn give predictions for parameters not measured so for. The good thing about such models is that they are falsifiable. Broadly speaking neutrino mass matrices which are consistent with the mass-squared mass difference are exemplified as f 0 1 l o w s ~ ~ ~ :
I Texture
Hierarchy
I
Iue3I
Normal (N) E l l
100 Inverted (I) 011
I
.-O
Quasi-Degenerate (QD)
001 Measurement of whetherIUegI2>> 0.01 and/or lcosO231 >> 0.01 will allow us to determine the best path to follow as far as understanding of neutrino masses and lepton mixing is concerned. Predictions for Maximal and Minimal values of (rn)eff in units of meV for neutrinoless double p decay, using IArnf31 = 2.6 x 10-3eV for NH and
207
sin2013 0.0 0.02 0.04
(m):; 2.6 3.6 4.6
(m)yf min 19.9 19.5 19.1
max
(m)$f max
50.5 49.5 48.5
(m):: 79.9 74.2 68.5
min
Except for NH, the next generation experiments are supposed to be able to probe the above values. 6 . Leptogenesis
Understanding the origin of matter i.e.
v=
nB
-
nB
=
(6 f 3) x
727
is one of the fundamental questions of Cosmology. The answer may come from Particle Physics. Three ingredients are necessary to generate v, the observed Baryon asymmetry of the Universe: i. Baryon number B violation ii. C P violation iii. Departure from thermal equilibrium. In SM, B and L symmetries hold at the classical level. However non perturbative quantum effects imply B L violation although B - L is preserved. However the phase in the quark mixing matrix could not have produced near enough C P violation to explain 7. As a result there is considerable interest in that the excess resulted from Leptogene~is~. In the see-saw mechanism each light neutrino is accompanied by a heavy neutrino N . Both are Majorana particles. Thus there is a CP violation coming from Majorana phases. The C P violation leads to unequal rates for the leptonic decays
+
N
4
If
+ Higgs-
and N
1-
4
+ Higgs+.
We shall restrict our discussion to the case of hierarchical Majorana neutrino masses, M1 << M2, M3, so that if the interactions of NI = N are in thermal equilibrium when N2 and N3 decay, the asymmetry produced by N2 and N3 can be erased before Nl decays. The asymmetry is then generated by the out of equilibrium C P violating decays of N -+ l H versus N + i H a t the temperature T M = MI << M2, Ms where T is the temperature of the thermal bath after inflation. A CP asymmetry N
El
=
r ( N -+
& H ) - r (N 4 & H * ) ( N 4 & H ) r (N 4 & H * )
+
208 in the decay produces net asymmetry of SM leptons. This asymmetry is partially transformed into a baryon asymmetry by non-perturbative B L violationg
+
I Y
YB = CYL = CIE-EI, S
where IE I 1 is an eficiency factor which can be obtained through solving the Boltzman equation. n / s is the ratio of the Nl equilibrium density to the entropy density. C 1/3 tells us what fraction of lepton asymmetry is converted into baryon asymmetry, due to B L violation processes.
- -
+
1
z - ( 6 f 3) x 10-l'
7
can be explained if ~1 2
7. Conclusions To conclude, various neutrino mass patterns and corresponding neutrino mass matrix types are possible. Further the absolute values of the neutrino masses are not yet determined. However, one thing is certain that neutrinos are providing evidence for new physics but the scale of the new physics is not yet pinned down. The heavy right handed neutrinos at new physics scale may provide an explanation for baryogenesis through leptogenesis. If the past is any guide, neutrinos will enrich physics still further.
References 1. R. N. Mohapatra et al., hep-ph/0412099 2. Paul Langacker, Int. J. Mod. Phys A20,5254 (2005) 3. H. Murayami, in Proceedings of the 21st International Symposium on Lepton and Photon Interactions at High Energies, Editors H. W. K. Cheung and T. S. Pratt, (World Scientific Singapore, 2004), p.484. 4. Boris Kayser, hep-ph/0504052 5 . H. V. Klapdor-Kleingrothaus et al., Mod. Phys. Lett. 16,2409 (2001) 6. A. de Gouvea, hep-ph/0503086 7. A. de Gouvea, hepph/0401220 8. F. Vissani, JHEP 9906,022 (1999); F. Feruglio, A. Strumia and F. Vissani, Nucl. Phys. B637,345 (2002);S.Pascoli and S. T. Petcov, Phys. Lett. B580, 280 (2004) 9. M. Fukugita and T. Yanagida, Phys. Lett. B174,45 (1986) ; G. F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, hep-ph/0310123; W. Buchmuller,
209 P. Di Bari and M. Plumacher, hep-ph/0401240; A. Pilaftsis, Phys. Rev. D56, 5431 (1997); Nucl. Phys. B504 , 6 1 (1997); W. Buchmuller and M. Plumacher, Int. J. Mod. Phys. A15,5047 (2000)
WHAT STRING THEORY HAS TAUGHT US ABOUT THE QUANTUM STRUCTURE OF SPACE-TIME M.M. SHEIKH-JABBARI Institute for Studies in Theoretical Physics and Mathematics (IPM), P. 0.Box 19355-5531, Tehran, I R A N jabbariotheory. i p m . ac. ir String theory is the most widely accepted model for quantum gravity and as such it should give an answer t o the fundamental question: what is the nature of quantum space-time? A handle on this question was provided through two of the recent conceptual advances in string theory, the (BFSS-type) matrix models and the AdS/CFT duality. In my talk I’ll focus on the second and discuss two places where the AdS/CFT duality has shed light on the question posed in the title. In particular I’ll focus on a matrix theory formulation I proposed to describe the DLCQ of strings on the Ads geometry, the tiny graviton matrix theory, according to which the quantum structure of the AdS5 x S5 geometry contains two noncommutative fuzzy three spheres as well as a noncommutative Moyal plane. We’ll argue that the appearance of noncommutative geometries is a generic feature of these matrix theories. Keywords: AdS/CFT, Matrix Models, DLCQ, Tiny Graviton Matrix Theory
1. Introduction and Motivation Gravity is described by General Relativity (GR) and is a classical field theory whose dynamical field is the space-time metric. In other words, the “geometry” of space-time are the dynamical degrees of freedom which are involved in gravity and is described by the Einstein-Hilbert action
where g is the determinant of the space-time metric gPv and R is its Ricci scalar curvature. On the other hand Quantum Mechanics (QM) starts with the noncommutative phase space
[Xi&] = i mij , [Xi, Xj] = 0 , [Pi,Pj] = 0 . 210
(2)
211
When Special Relativity is added to a quantum mechanical system we need to use quantum field theory (QFT), which is obtained from a classical field theory upon (for example path integral) quantization. The question which then arises: is whether the same procedure can be applied to GR (as a classical field theory) to obtain the theory of quantum gravity (Q.Gr)? And of course the well-known answer is no, because of the non-renormalizability of the Einstein-Hilbert action and the problem with infinities appearing a t loop level. So far, String Theory has appeared as the leading candidate for quantum gravity and starts with a simple,.elegant idea that everything including the fabric of space-time itself, is made out of (configurations) of “fundamental strings” or everything including the geometry itself is a state in the Hilbert space of string theory. As a model for quantum gravity string theory should then tell us What is the Quantum Space-Time emerging from string theory? or How does the space-time probed by strings (or other objects available to a string theorist) look like? In order to answer the above question I’ll very briefly review the developments of string theory in the recent ten years and in particular focus on the BFSS matrix model ideas’ and the AdS/CFT duality2. As the conjunction of the above two ideas we present the tiny graviton matrix theory (TGMT) conjecture3 which is a 0 1 dimensional supersymmetric U ( J )gauge theory proposed to describe the DLCQ of type IIB strings on the AdSs x S5 or the corresponding plane-wave geometry. We elaborate on the structure of the quantum space time emerging from the TGMT.
+
2. D-branes, the key objects One of the main developments in string theory happened when we learned how to include the D,-branes in perturbative string theory via open strings with Dirichlet boundary conditions4. D,-branes are extended objects with p+ 1dimensional worldvolume which carry one unit of the Ramond-Ramond p l-form charge. They are 1/2 BPS objects in the string theory (on flat space background). The low energy effective field theory which resides on a D,-brane is a p+ 1 dimensional supersymmetric U(1) gauge theory (with 16 supercharges) and is described by the Dirac-Born-Infeld action, in the lowest order in the string tension a’, that is a p + l dimensional supersymmetric Yang-Mills (SYM)’. Geometrically the 9 - p scalars of the supersymmetric
+
212
+
p 1 dimensional gauge theory with 16 SUSY can be interpreted as the directions transverse to the D,-brane in the ten dimensional bulk space-time. D-branes have the remarkable property that when we put N of them on top of each other we end up with a U ( N ) gauge theory as the worldvolume theory6. Therefore, the coordinates transverse to the stack of N D-branes turn into N x N hermitian matrices of U ( N ) .This is remarkable because this means that coordinates of the space-time viewed by D-brane probes are matrices, rather than c-numbers, and are inherently noncommuting. In particular, if we take N DO-branes, we are dealing with a 0 1 dimensional U ( N ) SYM, which is conjectured to describe M-theory on the eleven dimensional flat space, though in the Discrete Light-Cone Quantization (DLCQ) in the sector with N units of the light-cone momentum1. That is, each DO-brane is carrying one unit of the light-cone momentum. According to the BFSS matrix model, the space-time, the whole supergravity dynamics is described by 9 N x N X imatrices (together with their fermionic counterparts). The positions of the N DO-branes then appear as the eigenvalues of these matrices. The other important line of development has been the AdS/CFT duality according to which quantum gravity (type IIB string theory) on the Ads5 x S5 background is dual to/described by an N = 4 four dimensional SYM. The parameters of the two sides are related as:
+
#five - form fluxes on the S5 = rank of the gauge group
,~,
The radius of the Ads5 (or the S 5 ) ,R, is given as R4 = l i N , where I, is the ten dimensional Planck length or in the string units, l,, the 't Hooft coupling X is the Ads radius X = ( R ~ d s / l , Moreover, )~.
-
Physical String states c--) gauge inv. opt's in SYM String scattering Amplitudes
n-point functions in SYM.
(4)
For more details see [7]. The AdS/CFT is a very nice framework to address the question of our interest, as now geometry (and gravity) is going t o be described by a nongravitating gauge theory, which we know how to deal with systematically. There is, however, a subtlety that the AdS/CFT is a kind of strong/weak duality, in the sense that wherever the gauge theory is perturbative the gravity picture is not accessible (recall that the Ads radius is equal to the 't Hooft coupling) and vice-versa. Nonetheless, as we are dealing with a supersymmetric gauge theory we can study the BPS states/operators,
213
which are protected by supersymmetry and hence in the BPS sector the perturbative results are also valid at strong coupling where the gravity picture is applicable. Among the BPS states the simplest one is the 1/2 BPS sector. To give a classification of these operators let us review the field content of the N = 4 SYM:
1 gauge field A,, 6 real scalar fields q&, i = 1 , 2 , . . . 6. 4 Weyl fermions I = 1,2,3,4,a = 1,2.
$A,
+
4 s $6 = 2.The 1/2 BPS Opt's (technically called Chiral Primary operators) are those which are only made out of gauge invariant combinations of 2.For example T r Z J , : Tr2J"TrZJ-J1 : or in the most general Let
form: K
K
TrZJi:
: i=l
(C
Ji
(5)
= J).
i=I
All 1/2 BPS Opt's with # Z-fields= J (# 2-fields is a conserved charge, the R-charge) have the same scaling dimension A, which is exact (protected by supersymmetry) and equal to their R-charge J . In the sector with a given R-charge J, one can classify all the chiral primaries. It is a solution to the simple well-posed mathematical problem of partition of J into any number of non-negative integers, i.e. Finding set of { Ji} E Z,EL1Ji = J . Solutions to the above can be given by Young Tableaux of J boxes, which in turn are equivalent to all irreps of U ( N ) with J boxes. There are two key questions: (i) What are the dynamics of the chiral primaries in the N = 4 SYM? (ii) What are the corresponding gravity solutions? 2.1. The Gauge Theory Picture
The action of U ( N )N = 4 gauge theory on R x S3 is 1 6
+ fermions
1
.
214
Assuming no dependence on S3 and that only
+ i @ e = 2 (and not
Z t ) is turned on:
which are nothing but simple N x N harmonic oscillators. We, however, still have the gauge symmetry and not all the N 2 harmonic oscillators are independent. One can use the gauge symmetry to diagonalize 2, Z = diag( 2 1 , z2, . . . , ZN) but then &/2
BPS =
/
DZtDZ esl/2BPS VoWJ(N))
/n
(7)
N
=
D Z ~ e:iag
i=l
x n ( z i- zj). i>j
ni,j(~i
The - z j ) factor is the Van der Mond determinant (Jacobian of transformations)8. The above system is in fact a system of N free fermions with the z-plane as their phase space. The 1 / 2 BPS condition is then translated into
rIz = iZ+,n,,
=
42,
(8)
which implies that [ Z , Z t ]= 1.
(9)
One can show that the above system is also equivalent to a system of N 2d fermions in the external magnetic field. The BPS condition tells us that the fermions are sitting in the Lowest Landau Level (LLL)'. 2.2. The Gravity Picture
Lin-Lunin-Maldacena (LLM)lo constructed all the solutions t o supergravity equations with exactly the symmetry of the 1/2 BPS chiral primary operators. The LLM solutions are geometries with SO(4) x SO(4) isometries and have globally defined time-like (or light-like) Killing directions: ds2 = -hP2(dt
+
+ h2(dy2 + d z f ) + yeGdRi + ye-Gdf?;,
1 h-2 = 2ycoshG , z = - tanhG, y E [O,cm), 2 1 23 3 = - a y z ; E i j d j Z = yayV,.
€..a.v. Y
(10)
(11)
215
The above two equations imply that $ z equation: +ay(y3ay@) = o or
+
8,".
= @ satisfies a six dim. Laplace
1 + yay(-ayz) Y
= 0.
(12)
Smoothness condition then forces z ( z i ;y) to take values" 1 2
z ( q y = 0) = f-
(13)
Some comments are in order: 0 All the LLM solutions are then given by
As zo(zi;O) only takes values f 1 / 2 , one may use a Black/White colorcoding on z i plane to distinguish regions with zo = +1/2 from the regions with zo = -1/2. Quantization of the fiveform flux implies:
Area of the Black region (in 10 dim. Planck units) should be quantized. This in turn implies that [XI,z2] = iz;
(15)
+ [z,z+]= i$ .
(16)
or if we introduce 2 = L ( x ~ izz),
Jz
That is, both (semi-classical) gravity and gauge theory lead us to the same result: The
plane in the LLM geometry is a NonCommutative Moyal plane.
(51,z2)
But this is not a satisfactory uniform picture. What about the rest of the directions?!. ... To answer this question we need a non-perturbative formulation of string theory, that is the tiny grawiton matrix theory, TGMT. 3. The Tiny Graviton Matrix Theory
The TGMT conjecture3 is:
216
DLCQ of type IIB string theory on the Ads5 x S5 or the ten dim. max. supersymmetric plane-wave background in the sector with J units of the light-cone momentum is described by a U (J ) 0 1 dim. supersymmetric gauge theory; i.e. a U ( J ) SUSY QM, with the Hamiltonian
+
(17) where I , J , K = 1 , 2 . .. , 8, I = {i, u } and i , j , Ic = 1,2,3,4 and a , b, c = 5,6,7,8. L Cis~ a J x J unitary matrix where
L ~ = ~ J ,, JTr&=O,
(18)
[ F I F2, , F3, F4] = 8'"'FiFjFkfi
(19)
and
is the quantized Nambu 4-bracket. Nambu brackets are a direct generalization of Poisson brackets. The algebra defined by these brackets is closely related to the Quantum version of the Area Preserving Difleomorphisms of a four (or three) dimensional surface. The TGMT Hamiltonian is obtained from the Discretized or Quantized version of the action of a D3-brane in the ten dimensional plane-wave background3. The TGMT enjoys the following U (J ) gauge symmetry:
X',II' L5
+ UX',II'U-l --f
uc5u-?
, U
E
U(J), (20)
Although a gauge theory, TGMT is not a Yang-Mills theory. The U ( J ) gauge theory of the TGMT is the Quantized (or discretized) version of the Area Preserving Diffeomorphisms of the three-brane. TGMT is the theory of J tiny gravitons, the cousins of the DO-branes. The tiny gravitons we are dealing with here are in fact small size spherical three-branes which are 1/2 BPS objects in the AdSs x S5 or the plane-wave background. In this sense the TGMT parallels the BFSS. Conceptually what TGMT shares with BFSS is that in both cases it is the gravitons or gravity waves or metric fluctuations, carrying one unit of the light-cone momentum, which are used to formulate quantum gravity.
217
3.1. Half BPS configurations of TGMT
1/2 BPS configurations of the TGMT are all the matrices which render the Hamiltonian zero". That is,
IIr = 0, X" = 0 [Xi, xj,Xk,L5]= -12Gjklxl,
(21)
where
is the tiny graviton scale or the "fuzziness". The TGMT can be either used to describe DLCQ of strings on the plane-wave background for which case R- and p are independent parameters, or to describe strings on the A d s 5 x S5 in which case3
leading to 14 = - -1x 1 4 N
p'
The solutions to (21) are Noncommutative, Fuzzy three sphere 5';. To completely define S$, besides the above four-bracket equation, we need the radius: A
R&
C X: = l 2 J. i=l
In the 1 4 0, Rs$= f i x e d , limit we recover the round three sphere. It is worth noting that the radius squared of the fuzzy three sphere is quantized in units of L 2 . The above mentioned single sphere solution is only one of the solutions to the four-bracket equation. In general we can have fuzzy sphere solutions which are of the form of concentric fuzzy three spheres, the sum of their radii squared (in units of 1 2 ) is J . For more details see [ll]. 3 . 2 . Relation to LLM geometries
What we learn from the TGMT about the LLM geometries is then: The two three spheres in the LLM geometries are classical versions of a fuzzy three sphere, whose radii are quantized.
218
Quantization of the three sphere radii implies that the ( 5 1 , 5 2 ) plane in the LLM geometries is a NC Moyal plane. Note that in the class of the LLM geometries corresponding to the TGMT solutions the q appears as the light-like circle and ( 5 1 , 5 2 ) plane is indeed a cylinder, a NC, fuzzy cylinder, i.e. l 2
The radius quantization also implies the discreteness of the spectrum of y coordinate. Explicitly, Ay2 2
14. As we see, within the TGMT setup, all the eight transverse directions
in the quantized LLM geometries are quantized as explained above. 4. Summary and Outlook 0 A picture of Quantum Space-Time should emerge from theory of Quantum Gravity and in particular String theory. 0 One may even start with a specific Quantum Space-Time and try to build a Q.Gr. based on that. This may turn out to be string theory or otherwise?! 0 Within string theory we have some examples in which we have encountered NC (Quantum) structure in space-time. Here I reviewed some cases which involve gravity. 0 The NC structure uncovered depends on the probe and also the background. However, the point is that in each case it is possible to find an appropriate probe. Besides the conceptual point mentioned above, is there anything fundamentally in common in different cases?! This is yet to be explored. 0 In the string theory configurations, it turns out that the NC structure is strongly correlated with the fluxes in the background. Things to be studied in further detail: 0 F’urther analysis of the TGMT is needed to reveal more details of the quantum gravitylspace-time. 0 Generically the NC space-times show IR/UV mixing. This may have implications for the Cosmological Constant problem.. ...... 0 The above ideas could be used to find a “HOLOGRAPHIC” formulation for gravity on various backgrounds and may also be used to resolve issues in counting micro-states of a black-hole ......... I would like to thank the organizers of the 12th Regional conference in Mathematical Physics, and especially Faheem Hussain for the nice organization and warm hospitality during the conference.
219
References 1. T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D 55, 5112 (1997), [arXiv:hep-th/9610043]. 2. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. 3. M. M. Sheikh-Jabbari, JHEP 0409, 017 (2004), [arXiv:hep-th/0406214]. 4. J. Polchinski, Phys. Rev. Lett. 75, 4724 (1995), [arXiv:hep-th/9510017]. 5. J. Polchinski, String Theory, Vol. lt3 8, (Cambridge University Press, 1998). 6. E. Witten, Nucl. Phys. B 460, 335 (1996), [arXiv:hep-th/9510135]. 7. 0. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. 0 2 , Phys. Rept. 323,183 (2000), [arXiv:hep-th/9905111]. 8. S. Corley, A. Jevicki and S. Ramgoolam, Adv. Theor. Math. Phys. 5 , 809 (2002), [arXiv:hep-th/Oll1222]. D. Berenstein, JHEP 0407, 018 (2004), [arXiv:hep-th/0403110]. 9. D. Berenstein, Phys. Rev. D 71, 085001 (2005), [arXiv:hep-th/0409115]. A. Ghodsi, A. E. MosaiTa, 0. Saremi and M. M. Sheikh-Jabbari, Nucl. Phys. B 729, 467 (2005), [arXiv:hep-th/0505129]. 10. H. Lin, 0. Lunin and J. M. Maldacena, JHEP 0410, 025 (2004), (arXiv:hepth/0409174]. 11. M. M. Sheikh-Jabbari and M. Torabian, JHEP 0504,001 (2005), [arXiv:hept h/0501001]. 12. M. Ali-Akbari, M. M. Sheikh-Jabbari and M. Torabian, “Tiny graviton matrix theory / SYM correspondence: Analysis of BPS states,” arXiv:hepth/0606117.
GAUGE THEORY DESCRIPTION OF THE FATE OF THE SMALL SCHWARZSCHILD BLACKHOLE SPENTA R. WADIA
Tata Institute of Fundamental Research, Homi Bhaba Road, Mumbai-400005, India E-mail: wadia0theor-y. tifi. res. in In this talk we discuss the fate of the small Schwarzschild blackhole of Ads5 x S5 using the AdS/CFT correspondence at finite temperature. The third order N = 00 phase transition in the gauge theory corresponds t o the blackhole string transition. This singularity is resolved using a double scaling limit in the transition region. The phase transition becomes a smooth crossover where multiply wound Polyakov lines condense. In particular the density of states is also smooth at the crossover. We discuss the implications of our results for the singularity of the Lorentzian section of the small Schwarzschild blackhole.
(Talk given at the 12th Regional conference in Islamabad, Pakistan, based on hep-th/060504 1 )
1. Blackholes and string theory Over the past several years the subject of blackhole physics has had a symbiotic relationship with string theory. Blackholes provide a perfect theoretical laboratory to develop and test string theory. In turn string theory provides a consistent framework to deduce blackhole thermodynamics from quantum statistical mechanics. For a review see [l]. This relationship has been most fruitful in the case of the 5-dimensional supersymmetric blackhole solution of Strominger and Vafa whose microscopic degrees of freedom are D1 and D5 branes. Here it was shown that the Bekenstein-Hawking formula for blackhole entropy gave the same result as Boltzmann’s formula. The Strominger-Vafa blackhole is extremal and its microscopic degeneracy is that of the D1/D5 system in its ground state. Small excitations around the ground state correspond to small deviations from extremality and a small non-zero temperature. Such blackholes emit Hawking radiation which can be calculated as an emission/absorption pro-
220
221
cess in the microscopic theory. All these calculations of the microscopic theory match with semi-classical general relativity because of the high degree of supersymmetry in the system. The effective field theory of the D1/D5 system is a symmetric product 2-dim. superconformal field theory with (4,4)supersymmetry. Its central charge determines the thermodynamics of the blackhole and its 2-point functions compute the Hawking radiation formulas. The key reason why the microscopic calculations remain applicable in the gravity domain is because the high degree of supersymmetry circumvents the strong coupling problem. Our aim in this talk is to present some progress in trying to understand blackholes which are far from being supersymmetric, using the AdS/CFT correspondence’. Here unlike the supersymmetric case we have to face the strong coupling and intermediate coupling problems of the non-abelian gauge theory. There has been a lot of work in connection with blackholes in AdS5. Almost all this work is qualitative and deals in arriving a t a phase diagram for the gauge and bulk theory in the large N limit using both weak coupling expansions in the gauge theory and the supergravity correspondence at strong c ~ u p l i n g ~ - ~ . 2. Small 10-dim. Schwarzschild blackhole in Ads5 x S5 The problem of the fate of small Schwarzchild blackholes is important to understand in a quantum theory of gravity. In a unitary theory this problem is the same as the formation of a small blackhole. An understanding of this phenomenon has bearing on the problem of spacelike singularities in quantum gravity and also (to some extent) on the information puzzle in blackhole physics. It would also teach us something about non-perturbative string physics. We focus attention on the fate of the small 10-dim. Schwarzschild blackhole in Ads5 x S 5 , which (for reasons we will explain) is amenable to a precise quantitative treatment under reasonable assumptions’. In order to avoid confusion we mention that the blackhole we are referring to, is the small Schwarzchild blackhole of 10-dim. spacetime rather than the small blackhole in Ads:,.The latter, unlike our case, is uniformly spread over S 5 , and hence has the Gregory-Laflamme instability18. In the past Susskindg, Horowitz and Polchinski (SHP)1° and 0 t h e r ~ l l - l ~ have discussed this, in the framework of string theory, as a blackhole-string transition or more appropriately a crossover. Their proposal was that this crossover is parametrically smooth and simply amounts t o a change of description of the same quantum state in terms of degrees of freedom ap-
propriate to the strength of the string coupling. The entropy and mass of the state change at most by O(1). By matching the entropy formulas for blackholes and perturbative string states, they arrived at a crude estimate of the small but non-zero string coupling at the crossover. The SHP description is difficult to make more precise because the blackhole-string crossover occurs in a regime where the curvature of the blackhole is O(1) in string units, so as to render the supergravity description invalid. It is also clear that besides 1, related effects, the string coupling is non-zero and its effects have to be taken into account. Presently our understanding of string theory is not good enough for us to make a precise and quantitative discussion of the crossover. Hence we will discuss the problem using the AdS/CFT correspondence. When the horizon of this blackhole approaches the string scale l,, we expect the supergravity (geometric) description to break down and be replaced by a description in terms of degrees of freedom more appropriate at this scale. Presently we have no idea how to discuss this crossover in the bulk IIB string theory. Hence we will discuss this transition and its smoothening in the framework of a general finite temperature effective action of the dual S U ( N ) gauge theory on S3 x 5''. In fact it is fair to say that in the crossover region we are really using the gauge theory as a definition of the non-perturbative string theory. The use of the AdS/CFT correspondence for studying the blackholestring crossover requires that there is a description of small Schwarzschild blackholes as solutions of type IIB string theory in AdSs x S5. Fortunately, Horowitz and Hubenylg have studied this problem with a positive conclusion. This result enables us to use the boundary gauge theory to address the crossover of the small Schwarzschild blackhole into a state described in terms of 'stringy' degrees of freedom. 3. Gauge theory and effective action
At finite temperature we are dealing with the N=4,S U ( N ) SYM theory on S3 x S1. At large but finite N , since S3 is compact, the partition function and all correlation functions are smooth functions of the temperature and other chemical potentials. There is no phase transition. However in order to make a connection with a dual theory of gravity, which has infinite number of degrees of freedom, we have to take the N -+ 00 limit and study the saddle point expansion in powers of It is this procedure that leads to non-analytic behavior. It turns out that by taking into account exact results in the expansion it is possible to resolve this singularity and recover a
&.
&
223
smooth crossover in a suitable double scaling limit. The gauge theory is very hard to deal with as we have to solve it in the expansion for large but finite values of the 'tHooft coupling A. Inspite of this there is a window of opportunity to do some precise calculations because it can be shown that the effective action of the gauge theory at finite temperature can be expressed entirely in terms of the Polyakov loop which does not depend on points on S3: U = P e x p i Aodr) , where
&
f ( Ao(r) is the zero mode of the time component of the gauge field on S3.
This is a single N x N unitary matrix, albeit with a complicated interaction among the winding modes TrU". Two comments are in order: i) Since the 10-dimensional blackhole sits a t a point in S 5 , one may be concerned about the spontaneous breaking of SO(6)R-symmetry and corresponding Nambu-Goldstone modes. However using a supergravity analysis2, we have concluded that the symmetry is not spontaneously broken. Instead we have to introduce collective coordinates for treating the zero modes associated with this symmetry. ii) The circumstance, that the order parameter U in the gauge theory is a constant on S3,matches well on the supergravity side with the fact that all the zero angular momentum blackhole solutions are also invariant under the SO(4) symmetry of S3. The blackhole may be localized in S 5 , but it does not depend on the co-ordinates of S3 on which it is uniformly spread. The coefficients of the effective action depend upon the temperature, the 't Hooft coupling A and the vevs of the scalar fields. 4. Effective action and multi-trace unitary matrix model
The partition function of the gauge theory can now be written as a general unitary matrix model, Z(A, T)=
J dU eS(').
Gauge invariance requires that the effective action of U be expressed in terms of products of Tr U", with n an integer, since these are the only gauge invariant quantities that can be constructed from A0 alone. S ( U ) also has a ZN symmetry under U -+ e% U . These requirements fix the effective action to be of the form P
+
S ( U , U + )= C a i n U i T r U + i C"S,&(U)T,-,(U+), i=l S,P
(2)
224
where
z,2 are arbitrary vectors of nonnegative entries, and
ai and a f , ~are , parameters that depend on the temperature p-’ and the coupling A. Reality of the action (2) requires a f f , = a5k‘f‘ In fact, using the explicit perturbative rules to compute S(U,U t ) in (2), one can show that the o f f , are real, therefore
5. A lemma in matrix theory The general unitary matrix model can be analyzed due a lemma that enables us to express a multi-trace unitary matrix model in terms of a single trace matrix model2. Since the effective action (2) is a polynomial in n u i , TrUtZ, we can use the standard Gaussian trick twice to write the partition function (1)as
where
In the above formula we have introduced the definition
and the free energy F(gk,&) is defined by
225
6. Critical behavior in the matrix model
The eigenvalues of a unitary matrixa U are the complex numbers eioi. In the large N limit, we can consider an eigenvalue density, p(8), defined on the unit circle by,
The density function is non-negative and normalized,
It is well known that in the limit of N -+ m, p(8) can develop gaps, i.e. it can be non-zero only in bounded intervals. For example, in the case of a single gap, when p(8) is non-zero only in the interval (-$, $), it is given by the classical formula
A well known example of a p(8) which does not have a gap is
At a = 1, p ( n ) = 0, and a gap will begin to open. For a > 1 the functional form of p(8) is as given by (12). In general the condition p(n) = 0 defines a critical surface in the space of couplings of the effective action. The saddle point distribution of the eigenvalues of the matrix U may or may not have a gap, depending on the values of parameters g k in (8). The opening/closing of the gap in the eigenvalue distribution signals the GrossWitten-Wadia (GWW) third order phase transition in the matrix model 15-17
In the large N expansion, the functional dependence of F ( g k , & ) on "Phase structure of a generic unitary matrix model has been discussed in [24]
226 gk,gk
depends on the phase, and we quote from the known results
20-22,
f ( g k ,&),
FA1),FA2) and Gn(gk,g k ) are calculable functions using standard techniques of orthogonal polynomials. Two comments are in order: i) In the above, we have assumed for simplicity that the eigenvalue distribution has only one gap. (In principle we cannot exclude the possibility of a multi-gap solution. But here, since we are interested in the critical phenomena that results when the gap opens (or closes) we will concentrate on the single gap solution.) Near the boundary of phases, the functions Fn(g) and Gn(g) diverge. It is well known that in the leading order N , F(gk,g k ) has a third order discontinuity at the phase boundary. This nonanalytic behavior is responsible for the large N GWW type transition. In the O ( N - 3 ) scaling region near the phase boundary (the middle expansion in (14)) this non-analytic behavior can be smoothened by the method of double scaling. This smoothening is important for our calculation of the double scaled partition function near the critical surface. 2) In the gapped phase of the matrix model, F(gk,g k ) has a standard expansion in integer powers of &,which becomes divergent as one approaches the critical surface. In the double scaling region (14), ( g - g c ) O ( N - S ) , and the the perturbation series (14) is organized in an expansion in powers of N - 3 . The reason for the origin of such an expansion is not clear from the viewpoint of the bulk string theory. However, it is indeed possible to organize the perturbation series, in the scaling region, in terms of integral powers of a renormalized coupling constant. We will come back to this point later. In the ungapped phase the occurrence of O ( e P N )terms is also interesting. Here too we lack a clear bulk understanding of the non-perturbative terms which naturally remind us of the D-branes.
-
6.1. Critical surface of the large N phase transition
We now describe the critical surface in the space of couplings across which there is a GWW phase transition.
227
From ( 1 4 ) we can easily find the density of eigenvalues in the ungapped phase,
and
pk = k g k .
For a set of real g k , the lagrangian (8) is invariant under u + Ut. We will assume that the gap opens at 8 = 7r according to p ( -~ 8) (T - 8)2, which characterizes the first critical pointb. At the boundary of the gappedungapped phase (critical surface) we have p(7r) = 0. In terms of the critical fourier components pg, it is the equation of a plane with normal vector
-
B k =
c
(-1)"pg
+&) = - 1
(15)
k=-oo
Now since p i = kgi (up to non-perturbative corrections), we get the equation of a plane
x M ~~
(-l)"(g&
+ &)
=
-1,
k=-w
where g i are the values of g k at the critical plane. Since the metric induced in the space of g k from the space of P k is G k , p = k 2 b k , k ' , the vector that defines this plane is
We mention that the exact values of g i , where the thermal history of the small blackhole intersects the critical surface, are not known to us as we do not know the coefficients of the effective lagrangian. However this information, which depends on the details of dynamics, does not influence the critical behavior. The information where the small blackhole crosses the critical surface is given by the saddle point equations, which are in turn determined by the O ( N 2 )part of the action ( 6 ) .
7. Saddle point equations at large N The saddle points of ( 6 ) corresponding to the N=4 SYM theory are in correspondence with the bulk supergravity (more precisely IIB string theory) saddle points. For example, the Ads5 x S5 geometry corresponds to a general the mth critical point is characterized by p ( x - 8)
N
(x - €')zm.
228
saddle point such that (TrU”) = 0 V n # 0. Hence the eigenvalue density function is a uniform function on the circle. Now, depending on the coefficients in ( 6 ) , the saddle point (TrU”) can have a non-uniform gapped or ungapped eigenvalue density profile. Changing the values of the coefficients, by varying the temperature, may open or close the gap and lead to non-analytic behavior in the temperature dependence of the free energy at N = 00. We will interpret this phenomenon, the GWW transition, as the string-blackhole transition. As we shall see this non-analytic behavior can be smoothened out by a double scaling technique in the vicinity of the phase transition. The O ( N 2 )formula for the free energy leads to the large N saddle point equations for the multi-trace matrix model (5). By the AdS/CFT correspondence, the solutions to the saddle point equations are dual to supergravity/string theory solutions, like A d s 5 x S5 and various A d s 5 x S5 blackholes. The number and types of saddle points and their thermal histories depends on the dynamics of the gauge theory (i.e. on the numerical values of the parameter a j and C X ; , ~ , which in turn are complicated functions of X and p). These issues have been discussed in the frame work of simpler models in [7], where the first order confinement/deconfinement transition and its relation with the HawkingPage type transition in the bulk has also been discussed. Here we will not address these issues, but focus on the phenomenon when an unstable saddle point crosses the critical surface (16). (see Fig(7)) In a later section we will use the AdS/CFT correspondence to argue that in the strongly coupled gauge theory, a 10 -dimensional “small blackhole” saddle point reaches the critical surface p ( r ) = 0. The interpretation of this phenomenon in the bulk string theory, as a blackhole to excited string transition will also be discussed. 8. Double scaled partition function at crossover
We will assume that the matrix model (8) has a saddle point which makes a gapped to ungapped transition as we change the parameters of the theory(&k,k” a j ) by tuning the temperature p-’. We will also assume that, this saddle point has one unstable direction which corresponds to opening the gap as we lower the temperature. These assumptions are motivated by the fact that the small (euclidean) Schwarzchild blackhole crosses the critical surface and merges with the A d s 5 x S5and that it is an unstable saddle point of the bulk theory. To calculate the doubled scaled partition function near this transition point, we basically follow the method used in [7]. We
-
229
Fig. 1. Critical plane in the p space and thermal history of the saddle point
expand the effective action (8) around the 1st critical point, and we simultaneously expand the original couplings a j , g j , i j j and a z , around ~ their critical values a;, /3$, g$ = 0, and a5k , e ' For clarity we define ~ ( pp,, a ) =
-.. C az,g,(-i)lkl+lk'lT-(T G(p)' k
(18)
z,P We also introduce the column vectors,
"),
A = ( %,P
p=(i:>.
g=($)
and expand the above mentioned vector variables g
2 -
-
gc = N - s t ,
p - p" = N - Z n ,
A - A" = i j N - i ( ~ , where ij = N 3 (/3 - pc)and (Y = $$Ip=pc. The expansion of the co-efficients a j and a5 are proportional to the deviation of the tuning parameter /3 k,k'
-
from its critical value, i.e. = N +(pC- /3). Putting the above expression in the partition function we get the final result, 2
-
i(det(H))-i e x p F ( C . l).
(21)
230
We have assumed that the Hessian H does not have a zero mode, but the one negative eigenvalue accounts for the i in front of (21). We have no independent derivation of the existence of the negative eigenvalue except that the dual blackhole has exactly one unstable direction in the euclidean signature. Note that C .t = t is a parameter along the vector C which is normal to the critical surface. It can be proved from the discrete recursion relations of the matrix model that the function F ( t ) in (21) is given in terms of the Painleve I1 function f (t) ,
where f (t) satisfies the Painleve I1 equation, 182f -= tf
+ f3.
2 at2 The exact form of F ( t ) is not known but it is known that it is a smooth function with the following asymptotic expansion.
t3 1 3 63 F ( t ) = - - - log(-t) - -f - + . . . , -t >> 1, 6 8 128t3 1024t6 1 4at3 1 35 3745 (-t>>1. F(t) = - e - 3 2l.r Sat; 18432dtq The 0 ( 1 ) part of the partition function, (21), is universal in the sense that the appearance of the function F ( t ) , does not depend on the exact values of the parameters of the theory. Exact values of the couplings and the O ( N 2 )part of the partition function determine where the thermal history crosses the critical surface (16). However the form of the function F and the double scaling limit of (21) are independent of the exact values of g i . They only depend on the fact that one is moving away perpendicular to the critical surface. This is the reason why in [7] we obtained exactly the same equation when gt # 0 but all other g i = 0.
+--
+ * a * ) ,
8.1. Condensation of winding modes at the crossover
We will now discuss the condensation of the winding Polyakov lines in the crossover region. In the leading order in large N it is not difficult to see that p i = k g i , where p k =< >. In order to calculate subleading corrections it can be easily seen that all the p k ' s condenses in the scaling region,
231
where pig = k g k . This smoothness of the expectation value of the P k ’ S follows from the smooth nature of F ( t ) . The derivative of F ( t ) d’iverges as t -+ -ca and goes to zero as t -+ 00. This behavior tallies with the condensation of winding modes in one phase (the gapped phase) and the non-condensation of winding modes in the ungapped phase. The condensation of the winding modes also indicates that the U ( 1 ) symmetry (which is the ZN symmetry of the S U ( N ) gauge theory in the large N limit) is broken at the crossover, but restored in the limit t -+ 00. 9. Applications to the small blackhole-string transition
We now apply what we have learned about the matrix model (gauge theory) GWW transition and its smoothening in the critical region to the blackholestring transition in the bulk theory. The first step is to identify the matrix model phase in which the blackhole or for that matter the supergravity saddle points occur. We will argue that they belong to the gapped phase of the matrix model. This inference is related to the way perturbation theory in is organized in the gapped, and ungapped phase, as discussed in (14). Note that it is only in the gapped phase, that the expansion is organized in powers of exactly in the way perturbation theory is organized around classical supergravity solutions in closed string theory. Hence at the strong gauge theory coupling(/\ >> l),it is natural to identify the small 10 dimensional blackhole with a saddle point of the equations of motion obtained by using F ( g k , i j k ) corresponding to the gapped phasec. One can associate a temperature with this saddle point which would satisfy l r l >> T >> R-I. As the temperature increases towards Zrl, one traces out a curve (thermal history) in the space of the parameters a i , a k , k ’ of the effective theory. One can also say that a thermal history is traced in the space of pi = ( + T t U i ) , which depends on the parameters of the effective theory. We will now make the reasonable assumption that the thermal history, at a temperature T, ,’;Z intersects the critical surface (15) (equivalently the plane (16)) and then as the temperature increases further it reaches the point pi = ( $ T r U i ) = 0, which corresponds to A d s , x S5. Once the
k
&,
N
‘A saddle point of the weakly coupled gauge theory may also exist in the gapped phase. With a change in the temperature the saddle point can transit through the critical surface. Using the results of [7], it is easy to see that this is precisely what happens for the perturbative gauge theory discussed in [6]. We note that in the corresponding bulk picture, since I , >> R ~ d sthe , supergravity approximation is not valid. It would be interesting to understand the bulk interpretation in this case.
thermal history crosses the critical surface, the gauge theory saddle points are controlled by the free energy of the ungapped phase in (14). The saddle points of eqns. which were obtained using this free energy do not correspond to supergravity backgrounds, because the temperature, on crossing the critical surface is very high T 2 Zrl. Besides this the free energy in the gapped phase has unconventional exponential factors (except at gk = 0 which corresponds to Ads5 x S5).It is likely that these saddle points define, in the correspondence, exact conformal field theories/non-critical string theories in the bulk. Neglecting the exponential corrections, exp( - N ) , it seems reasonable, by inspecting the saddle point equations, that in this phase the spectrum would be qualitatively similar to that around pi = 0. Since this corresponds to Ads5 x S 5 ,we expect the fluctuations to resemble a string spectrum. As we saw in the previous section, our techniques are good enough only to compute a universal O(1) part of the partition function in the vicinity of the critical surface. The exact solution of the free energy in the transition region in (14) enabled us to define a double scaling limit in which the non-analyticity of the partition function could be smoothened out, by a redefinition of the string coupling constant according to lj = N ? (pc - p). This smooth crossover corresponds to the blackhole crossing over to a state of strings corresponding to the ungapped phase. We have also computed the vev of the scaling operator and hence at the crossover the winding modes, pi = (&TYV ' ) ,condense (25). They also have a smooth parametric dependence across the transition. This phenomenon in the bulk theory may have the interpretation of smooth topology change of a blackhole spacetime to a spacetime without any blackhole and only with a gas of excited string states. However in the crossover region a geometric spacetime interpretation is unlikely. We may be dealing with the exact description of a non-critical string in 5-dimensions in which only the zero mode along the S3 directions is taken into account. This interpretation is inspired by the fact that the free energy F ( t ) also describes the non-critical type OB theory as was already discussed in [7]. 10. Density of states and the singularity of the Lorentzian
blackhole The resolution of the singularity of a blackhole, that occurs in its general relativity description, is a fundamental question in string theory. We interpret our result in favor of this resolution in the gauge theory. Since the partition function, in an appropriate scaling limit, is a smooth function of
233
the renormalized coupling constant 3, at the crossover between the gapped and ungapped phase, it is clear that the density of states p ( E ) also inherits the same property. Since p ( E ) is as well a quantity that has meaning when the signature of time is Lorentzian, it would imply that the blackhole-string crossover in the Lorentzian signature is also smooth. This is an interesting conclusion especially because we do not know the AdS/CFT correspondence for the small Lorentzian blackhole. The Lorentzian section of the blackhole has a singularity behind the horizon. Since the gauge theory should also describe this configuration, a smooth density of states in the crossover may imply that the blackhole singularity is resolved in the gauge theory. A more direct Lorentzian calculation is will strengthen this conclusion. 11. Acknowledgment
I would like to acknowledge and thank the organizers of the 12th Regional conference in Islamabad, especially Prof. Faheem Hussain, for efforts that made the conference and the visit to Pakistan a memorable and wonderful experience. This research is supported in part by the J. C. Bose Fellowship of the Dept. of Science and Technology, Govt. of India.
References 1. J. R. David, G. Mandal and S. R. Wadia, Phys. Rept. 369, 549 (2002) [arxiv:hep-th/0203048]. 2. L. Alvarez-Gaume, P. Basu, M. Marino and S. R. Wadia, “Blackhole / string transition for the small Schwarzschild blackhole of AdS(5) x S**5 and critical unitary matrix models,” arXiv:hep-th/0605041 3. E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) [arXiv:hep-th/9803131]. 4. B. Sundborg, Nucl. Phys. B 573,349 (2000) [arXiv:hep-th/9908001]. 5. A. M. Polyakov, Int. J. Mod. Phys. A 17S1, 119 (2002) [arXiv:hepth/0110196]. 6. 0. Aharony, J. Marsano, S. Minwalla, K . Papadodimas and M. Van Raamsdonk, Phys. Rev. D 71,125018 (2005) [arXiv:hep-th/0502149]. 7. L. Alvarez-Gaume, C. Gomez, H. Liu and S. Wadia, Phys. Rev. D 71,124023 (2005) [arXiv:hep-th/0502227]. 8. P. Basu and S. R. Wadia, Phys. Rev. D 73, 045022 (2006) [arXiv:hepth/0506203]. 9. L. Susskind, “Some speculations about black hole entropy in string theory,” arXiv:hep-th/9309145. 10. G. T. Horowitz and J. Polchinski, Phys. Rev. D 55,6189 (1997) [arXiv:hepth/9612146]. 11. A. Sen, Mod. Phys. Lett. A 10,2081 (1995) [arXiv:hep-th/9504147]. 12. M. J. Bowick, L. Smolin and L. C. R. Wijewardhana, Gen. Rel. Grav. 19, 113 (1987).
234 13. M. J. Bowick, L. Smolin and L. C. R. Wijewardhana, Phys. Rev. Lett. 56, 424 (1986). 14. S. W. Hawking and D. N. Page, Commun. Math. Phys. 87,577 (1983). 15. D. J. Gross and E. Witten, Phys. Rev. D 21,446 (1980). 16. S. Wadia, “A Study Of U(N) Lattice Gauge Theory In Two-Dimensions,” EFI-79/44CHICAGO 17. S. R. Wadia, Phys. Lett. B 93, 403 (1980). 18. V. E. Hubeny and M. Rangamani, JHEP 0205, 027 (2002) [arXiv:hepth/0202189]. 19. G. T. Horowitz and V. E. Hubeny, JHEP 0006, 031 (2000) [arXiv:hepth/0005288]. 20. Y. Y. Goldschmidt, J . Math. Phys. 21, 1842 (1980). 21. V. Periwal and D. Shevitz, Phys. Rev. Lett. 64, 1326 (1990). 22. V. Periwal and D. Shevitz, Nucl. Phys. B 344, 731 (1990). 23. L. Gervais and B. Sakita, Phys. Rev. D 11,2943 (1975). 24. G. Mandal, Mod. Phys. Lett. A 5 , 1147 (1990).
Plasma Physics
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ZONAL FLOW GENERATION BY MAGNETIZED ROSSBY WAVES IN THE IONOPHERIC E-LAYER T . D. KALADZE I. Vekua Institute of Applied Mathematics, Tbilisi State Uniuersity, 2 University Str., 0143 Tbilisi, Georgia
D. J. WU Purple Mountain Obseruatory, Chinese Academy of Sciences, Nanjing, 21 0008, China
0. A. POKHOTELOV Institute of Physics of the Earth, 123995 Moscow, 10 B. Gmzinskaya Str., Russia R. Z. SAGDEEV
Department of Physics, University of Maryland, College Park, MD 20740, U.S.A.
L. STENFLO Department of Physics, Umed University, SE-90 187 Umed, Sweden
P. K. SHUKLA Institut f i r Thwretische Physik IV, Ruhr-Uniuersitat Bochum, 0-44780 Bochum, Germany
1. Introduction
Development of anisotropic large scale structures, such as convective cells, zonal flows and jets, is a problem which has attracted a great deal of interest both in plasmas [Hasegawa, Maclennan, and Kodama, [l]]and in geophysical fluid dynamics [Busse, [2]; Rhines, [3]]. Recently it has been realized that zonal flows play a crucial role in the regulation of the anomalous transport in a tokamak [Diamond, Itoh, Aoh and Hahm, [4]]. It is believed that
237
238
the nonlinear energy transfer from small to large length scale component (inverse cascade) is a cause of spontaneous generation and sustainment of coherent large structures, e.g., zonal flows in atmospheres, ocean and plasmas. Both ground-based and satellite observations clearly show that, a t different layers of the ionosphere, there are large scale flow band structures (zonal flows) with nonuniform velocities along the meridians [ Gershman, [5]; Gossard, [6]; Kamide, [7]].It is known [e.g., Petwiashwili and Pokhotelow, [S]] that in the presence of velocity shear in the zonal flow, the nonlinear effects start to play a role in their dynamics. It is thus of interest to take into account the interaction of the planetary waves propagating in the ionosphere with the shear flows. It this way, the ionospheric medium builds up conditions which are favorable to the formation of nonlinear stationary solitary wave structures [Pokhotelowet al., [9, lo]]. In reality, several planetary ionospheres can support both propagating waves and zonal flows and they thus constitute dynamic systems which exhibit complex nonlinear interactions. It should be noted that zonal flows vary on time scales slower than those of the finite-frequency waves. The generation of zonal flows is still not fully clarified. Recently, there has been renewed interest in examining the nonlinear coupling between coherent and incoherent drift waves and zonal flows (or convective cells) in nonuniform magnetoplasmas [e.g., Smolyakov et al., [ll];Manfredi et al., [12]; Shukla and Stenflo, [13]]. It has been found that pseudo-threedimensional drift waves strongly couple with zonal flows whose dynamics is governed by Navier-Stokes equation in the presence of drift wave stresses. The latter is nonlinearly coupled with the Hasegawa-Mima equation in the drift wave-zonal flow theory. As there is a well-known analogy between drift waves and Rossby waves [e.g., Nezlin and Chernikow, [14]], the idea of generation of zonal flows by Rossby waves was put forward by Shukla and Stenflo [15]. Their theory was further developed by Onishchenko et al. [16]. In these papers it was shown that zonal flows in a nonuniform rotating neutral atmosphere can be excited by finite amplitude Rossby waves. The driving mechanism of this instability is due to the Reynolds stresses which are inevitably inherent for finite amplitude small scale Rossby waves. Hence, these investigations provided an essential nonlinear mechanism for the transfer of spectral energy from small scale Rossby waves to large-scale enhanced zonal flows in the Earth’s neutral atmosphere. In addition, the zonal flow generation was considered within a simple model for Rossby wave turbulence, using the classical nonlinear two-dimensional Charney equation
239
to describe the dynamics of solitary vortex structures of the dipole type, i.e. a cyclone-anticyclone pair. This means that the wavelengths of the considered Rossby waves were small as compared with the Rossby radius T R and the nonlinearity is therefore only due to the so-called vector, or Poisson bracket, nonlinearity. Hence, it corresponds to the quasi-geostrophic approximation in geophysical hydrodynamics for which structures are considered as purely two-dimensional, and the perturbations of the free surface of the liquid motion are considered as either absent or negligibly small. In the present paper we will focus our attention at the Earth’s ionosphere. A large amount of observational data has been stored up till now. These data verify the permanent existence of ULF (ultra-low frequency) planetary-scale perturbations in the E- and F-regions of the ionosphere [e.g., Lawrence and Jarvis, [17]]. Among them, special attention must be paid to large-scale Rossby type perturbations propagating at a fixed latitude along the parallels around the Earth. Unlike the neutral atmosphere the ionospheric E- and F-layers consist of neutrals and charge particles whose existence makes the ionosphere conductive. Therefore, the interaction of inductive currents with the inhomogeneous geomagnetic field (varying along the meridians) should be taken into account. Recently Kaladze and Tsamalashwili [18],Kaladze [19, 201 and Kaladze et al. [21] showed that the so-called magnetized Rossby waves can propagate in the E-layer of the ionosphere without perturbing the geomagnetic field. They have typically wavelengths larger than the Rossby radius rR. For the ionosphere we have rR M 1000 - 3000 km. As shown by Kaladze et al. [22], magnetized Rossby wave turbulence should then be described by a more complex equation, namely the so-called generalized Charney equation which includes an additional scalar, Korteweg-de Vries (KDV) type, nonlinearity. This equation corresponds to the intermediate geostrophic approximation in geophysical hydrodynamics, for which the perturbation of the free surface of the atmosphere is taken into account. 2. Linear magnetized Rossby waves in the ionospheric E-layer Let us consider a weakly ionized E-layer that consists of electrons, ions and neutral particles. Due to the strong collisional coupling between the ionized particles and the neutrals the behavior of such a gas is mainly determined by its massive neutral component. The E-layer satisfies the condition n / N << 1, where n and N are the equilibrium number densities of the charged particles and the neutrals, respectively. The presence of charged
240
particles makes the medium electrically conducting. For a typical ionization fraction in the E-layer, the Lorentz force is comparable to the Coriolis force. Hence we must take into account the effects of the spatially inhomogeneous geomagnetic field B and the vertical component of the Earth's rotation S2. Magnetized Rossby waves [e.g., Kaladze and Tsamalashvili, [18]; Kaladze, [19, 2011 represent the ionospheric generalization of tropospheric Rossby waves in the rotating atmosphere with a spatially inhomogeneous geomagnetic field. The theory of magnetized Rossby waves was developed by Kaladze et al. [22]. We introduce a local Cartesian coordinate system with the x-axis directed to the east, the y-axis to the north, and the z-axis in the local vertical direction. Magnetized Rossby waves propagate in the middle-latitude E-layer of the ionosphere and their frequency is [Kaladze et aZ., [22]]
+
Here wk is the wave frequency, k the wave vector, and k l = (kp ?c:)'/~ where k x and k , are the x and y components of the wave vector. The Rossby velocity V R and the Rossby radius r R are defined as V R=
-(a
2 8 + P ) r i = -rR-(f + Y), dY
(2)
and
where HO stands for the atmospheric reduced height, g is the gravitational acceleration, and f is the Coriolis parameter which depends on latitude A, i.e.
f =2RoZ=2RosinX= fo+py,
(4)
with
df 1 df p== --
2RocosXo = > 0. (5) dy RdX R Analogously, considering the inhomogeneous dipole geomagnetic field we write the geomagnetic field parameter as en 2en 7 = -BOz = -- Be, sin X = 70 a y , (6)
fo = 2RosinXo > 0
and
P
+
P
with 2en
70 = ---Be,sinXo P
<0
and
241
(7)
In (4)-(7) the quantities a , P, fo and yo are related to the latitude XO, e is the magnitude of the electron charge, Be, is the equatorial value of the geomagnetic field at a distance R from the Earth’s center, and p = N m is the neutral gas mass density. The factor ( a P ) in Eq. ( 2 ) represents the generalized Rossby parameter, where a corresponds to the contribution from the Lorentz force. The parameters a and P are comparable in m-ls-l) in the ionospheric E-layer, and a magnitude ( p N -a ”/ depends on the ratio n / N . This ionization fraction is distinctly different at the nightside and dayside of the Earth. Thus, unlike the Rossby waves in a neutral atmosphere, the magnetized Rossby waves in the ionospheric E-layer can propagate both westwards and eastwards at a fixed latitude along the parallels. The magnetized Rossby waves do not significantly perturb the geomagnetic field. They are induced by the latitudinal inhomogeneity of the Earth’s angular velocity as well as of the geomagnetic field, determined by p and a , respectively. They are excited by the ionospheric dynamo electric field when the Hall effect due to the interaction with the ionized ionospheric component in the E-layer is included. The dynamics of propagation depends on the generalized Rossby parameter ( a P) and the modified Rossby radius. The Lorentz force counteracts the Coriolis force vorticity and partial or full compensation of the Coriolis deviation by that of the magnetic is therefore possible. Correspondingly, the phase velocities of the linear waves also decrease. The period of these waves is of the order of several days. A magnetized Rossby wave belongs to the ultra-low frequency range ( s-l, its wavelength is 1000 km or larger, and its phase velocity is of the order of the velocity of the local winds, i.e. (1 - 100) m/s. Such waves correspond to longitudinal mode numbers less than 8 - 10. It has been established that the inhomogeneity (latitude variation) of the geomagnetic field and the Earth’s rotation generates magnetized Rossby waves, which propagate along the parallels to the east as well as to the west. These large-scale waves are weakly damped. The features and the parameters of the theoretically investigated electromagnetic wave structures agree with those of the large-scale ULF mid-latitude long-period oscillations and the ionospheric wave perturbations observed in the ionosphere.
+
+
N
242
3. Nonlinear interactions of magnetized Rossby waves and zonal flows in the E-layer
Considering large-scale structures with sizes a 2 T R , it was shown [e.g., Kaladze et al., [22]] that the magnetized Rossby waves turbulence in the ionospheric E-layer could be described by the generalized Charney equation
d z ( h - &Vo",h)
dh + VR-dh + VRh- - (f + y ) r i J ( h , V ; h ) = 0. dX dX
(8)
Within this model, the ionosphere is treated as an incompressible shallow water fluid of depth H = Ho(l+ h ) , where HOis the unperturbed constant depth and h stands for a dimensionless wave amplitude. The Poisson bracket operator J ( a , b) = &adyb - dyad,b represents the vector nonlinearity. The new term in the generalized Eq. (8) is the scalar nonlinearity of the KdV type hd,h. We estimate from Eq. (8) that the vector nonlinearity can compensate the scalar nonlinearity only when the size a of the structure is less than r i g ,or
-
a
< Tig = ( T i R ) 1 ' 3 ,
(9)
where we have introduced the so-called intermediate geostrophic radius rig. Here T R is the Rossby radius defined by (3), and R is the Earth's radius which is the scale of inhomogeneity of both the Earth's angular velocity and the geomagnetic field. In the problems considered by Shukla and Stenjlo [15] and Onishchenko et al. [16] only the vector nonlinearity was kept. This means that only small size dipole structures (see Fig.l), satisfying the inequality (9), were considered in these papers. Below we will consider the generalized Eq. (8), i.e. we will study structures of arbitrary size. The generalized Charney equation (8) for magnetized Rossby waves contains thus both scalar and vector nonlinearities and it can describe solitary vortex structures of arbitrary size. This equation corresponds to the so-called intermediate geostrophic approximation in geophysical hydrodynamics, where a perturbation of the free surface of the liquid plays a role (accordingly the two-dimensional divergence of the velocity differs from zero). The mechanism for self-organization of solitary structures is associated with the compensation of wave dispersion by both the scalar and vector nonlinearities. As a result, a solitary structure is in general intrinsically anisotropic and contains a circular vortex superimposed on a dipole perturbation. The degree of anisotropy increases sharply as the size of the vortex approaches the intermediate geostrophic size (9). The generalized
243
Fig. 1. Dipole Structure due t o the vector nonlinearity
Charney equation (8) for Rossby waves with a = y = 0 was first derived [23]. by P~t~2as~vilz Owing to the presence of the scalar nonlinearity, Eq. (8) breaks the cyclone-anticyclone symmetry and predicts the existence of solitary waves (solitons) with monopole structures, and defined signs; i.e. either cyclones or anticyclones. Such solitary structures were first found in laboratory modeling of solitary Rossby vortices by Antipov et al. 1241. A large-scale dipole vortex splits into two monopoles ( a cyclone and an anticyclone), where a vortex of one polarity i s long-lived whereas the vortex of the opposite polarity disperses (see Fig.2). In case of magnetized Rossby waves, only those anticyclones survive that propagate faster then the maximum velocity of the corresponding linear waves. Thus, we emphasize that the presence of the scalar nonlinearity plays an additional role, similar to an instability, in forming new structures from former dipole structures. Since the zonal flow varies on a much larger time scale than the comparatively small-scale magnetized Rossby waves, one can use a multi-scale expansion, assuming that there is a sufficient spectral gap separating the large- and small-scale motions. Following the standard procedure to describe the evolution of the coupled system (magnetized Rossby waves plus
244
Fig. 2. earity
Formation of a monopole structure owing to the presence of KdV type nonlin-
zonal flows), we decompose the perturbation of the dimensionless ionospheric depth h into its low- and high-frequency parts, that is
h=%+X,
(10)
where x(y, t ) refers to the large-scale zonal flow and X(r, tj to the small-scale magnetized Rossby wave. Averaging Eq. (8) over the small spatial scales, we obtain the evolution equation for the mean flow
where the angular bracket denotes the averaging process. Bn Eq. ( l a ) the term on the right-hand side describes the Reynolds stresses induced by the small-scale magnetized Rossby waves. The nonlinear coupling of the magnetized Rossby waves with the zonal flow is governed by
245
The magnetized Rossby waves are considered as a superposition of a pump wave and two sidebands, that is
h
=
ho
- + h+ + h-,
(13)
where for the pump wave we have
ho = hk exp(i(k. r - Wkt))
+ h i exp(-i(k. r - Wkt)),
(14)
with the frequency W k given by Eq. (1). The change in the zonal flow amplitude is given by
h = hq exp(i(q. r - 0 t ) )+ h: exp(-i(q. r - R t ) ) ,
A
(15)
with q = q? where ? is the unit vector along the latitude. For the magnetized Rossby side-bands we have
-
h+ = hk* exp(i(k+ . r - Wk+t))
+ h i k exp(-i(k+. r
-
Wk.+t)),
(16)
are the frequencies and wave vectors of the magnetized Rossby sidebands. into Eq. ( l l ) , we obtain Substituting ho, and
x+
k2)hk+hi - (k! where the expressions for the Fourier amplitudes hk+ and h i - , found from Eq. (12), are
and
where
246
Substituting (20) and (21) into Eq. (19) we obtain
The dispersion relation (23) is in general too cumbersome for analysis, and it can thus only be solved numerically. In order to simplify it we consider the most interesting case, namely q << Ic, when the typical scales of the zonal flows are much larger than the scales of the magnetized Rossby waves. In this limiting case we can adopt the expansions
where
and
is the latitudinal (9-component) of the Rossby group velocity. We note that both vg and v$ can change sign when
k,
=f(3kiri
-
l)/~i)’/~.
(28)
This occurs on the Rossby wave caustics. Substituting (24) and (25) into Eq. (23), one finds
0,
21 qv,
f
Let us now investigate two special cases.
247
In the case of small-scale turbulence, when a < rig,we obtain from ( 2 9 )
It is thus obvious that a necessary condition for instability is v$/w, < 0. This condition is similar to the Lighthill criterion for modulation instability in nonlinear optics [Lighthill, [ 2 5 ] ] According . to ( 2 6 )
which in the small wavelength limit
>> 1) reduces to
( ~ T R
i.e.
Instability occurs when k; - 3 k i > 0, and the instability condition thus applies to magnetized Rossby pump waves with wave vectors in the cone
The maximum growth is attained at the axis of the cone when lc, this case the mode is purely growing with the growth rate
=
0. In
Expression ( 3 5 ) describes the initial (linear) stage of zonal flow growth due to the parametric instability of small-scale magnetized Rossby waves. For the Rossby regime f y >> wk the last term in the parenthesis of ( 3 5 ) is small. For qrR 1 and k r >> ~ 1 we can then estimate the growth rate to be
-
+
MI f f I (krR)lhOl.
(36)
This estimation shows that y increases as k in the small wavelength limit ( k r > ~> 1). Physically, this instability is the manifestation of an inverse cascade. It shows that the spectral energy of the small-scale magnetized Rossby wave turbulence is transferred into the large scales of the zonal flows, i.e. the magnetized Rossby wave energy is converted into the energy
248
of slow zonal motions. For the typical parameters of the Earth's atmosphere ( f y) FZ lov4 s-l, krR 2 10 and ho 2 lop2, we obtain y 2 s-'. This estimate is consistent with existing observations. Thus, it is possible that the parametric instability of magnetized Rossby waves is responsible for the generation of mean flows in the ionosphere of our rotating Earth. Onishchenko et al. [2004] investigated the special case 4I-R >> 1 to obtain the expression for the maximum growth rate and to define the optimal parameters of the zonal flow. Here we will however consider the opposite case when qrR << 1. We then obtain
+
q2 J
y=-
k2
In the Rossby regime f
2(f+ y)2(krR)61h012- W E .
+ y >>
Wk
and for the case
~ I - >> R
(37) 1 we thus have
q2 g I f + 7 I (krR)31hOl.
(38)
In spite of the fact that the expression (38) contains a small factor q 2 / k 2 , owing to the high value of ( k r R ) 3 >> 1, the growth rate is however significant. As we have mentioned above, the presence of the geomagnetic field causes reduction of the value of (f y). Thus, the obtained growth rate (38) increases as (f T ) - ~ . Let us now investigate the new term in Eq. (29) coming from the contribution of the scalar (KdV) nonlinearity. In the case of large-scale turbulence with krR 1, we estimate that when q / k lop1,
+
+
-
VR
-(f +
-
y)riqk2
(f
+ y ) T R4k
(39)
i(f
v; + y)qVR-.wk
(40)
Thus using the main term we get
n*
qvg f qkzrRlh0l
J
This instability exists for any sign of ui/wk unlike the instability considered above. Substituting the expressions (2) and (31) into (40) one finds
Maximum growth rate is here attained again when k , = 0, i.e.
249
An estimate of this growth rate when krR
+
N
1 gives
+
-
s-’, a p m-ls-l, For the typical parameters f y N and q/lc 10-1 of the Earth’s ionosphere, we rR lo6 m, lhol N s-’. Thus, this growth rate is ten times less than that obtain y obtained in the small-scale turbulence case (see (36)). N
-
-
4. Discussion and Conclusions A novel mechanism for the generation of large-scale zonal flows by smallscale Rossby waves in the Earth’s ionospheric E-layer is considered. The generation mechanism is based on the parametric excitation of convective cells by finite amplitude magnetized Rossby waves. To describe this process a generalized Charney equation containing both vector and scalar (Korteweg-de Vries type) nonlinearities is used. The magnetized Rossby waves are supposed to have arbitrary wavelengths (as compared with the Rossby radius). A set of coupled equations describing the nonlinear interaction of magnetized Rossby waves and zonal flows is obtained. The generation of zonal flows is due to the Reynolds stresses produced by finite amplitude magnetized Rossby waves. It is found that the wave vector of the fastest growing mode is perpendicular to that of the magnetized Rossby pump wave. Explicit expressions for the maximum growth rate as well as for the optimal spatial dimensions of the zonal flows are obtained. A comparison with existing results is carried out. The present theory can be used for the interpretation of the observations of Rossby type waves in the Earth’s ionosphere. In the present study we have demonstrated how zonal flows in the shallow rotating ionospheric E-layer can be excited by finite amplitude magnetized Rossby waves. The driving mechanism of this instability is due to the Reynolds stresses which are inevitably inherent for finite amplitude smallscale magnetized Rossby waves. Hence, our investigation provides an essential nonlinear mechanism for the transfer of spectral energy from small-scale magnetized Rossby waves to large-scale enhanced zonal flows in the Earth’s ionosphere. We have used the generalized Charney equation describing the turbulence of magnetized Rossby waves of arbitrary size. In the case of small-scale turbulence, when krR >> 1, only the vector nonlinearity is responsible for the parametric instability giving the most important growth rate. The peculiar feature of this instability is that it appears solely for
250
magnetized Rossby waves that are localized in a cone bounded by the caustics for which v$ = 0. This can lead to the formation of a so-called caustic shadow in the spectrum of the magnetized Rossby waves. A typical value of the obtained growth rate is loF5 - low4 s-l. But the presence of the geomagnetic field (causing a contribution opposite to that of the Earth’s angular velocity rotation) can increase the growth rate by one order. In the case of large-scale turbulence, when lcrR I 1, the KdV type scalar nonlinearity gives the main contribution in forming the turbulence structures. The corresponding growth rate is positive for the arbitrary sign of Z I $ / W ~but , is one order less than in the case of the small-scale turbulence.
Acknowledgments This research was supported in part by the Natural Science Foundation of China (grants no. 10425312,10373026, and 10333030, by PPARC (grant no. PPA/G/S/2002/00094), by ISTC (grant no. 2990), by the Russian Foundation for Basic Research (grant no. 04-05-64657), the Russian Academy of Sciences through the grants ”Physics of the Atmosphere: Electrical processes and radiophysical methods” and ”Low-frequency wave structures as a response of the magnetosphere to the solar wind perturbations”.
References 1. A. Hasegawa, C.G. Maclennan, and Y. Kodama, Phys. Fluids 22, 2122 (1979). 2. F.H. Busse, Chaos 4,123 (1994). 3. P.B. Rhines, Chaos 4,313 (1994). 4. P.H. Diamond, S.-I.Itoh, K.Itoh, and T.S. Hahm, Plasma Phys. Controlled Fusion 47,R35 (2005). 5. B. I. Gershman, Dynamics of the Ionospheric Plasma, (Nauka,Moscow, 1974)
(In Russian). 6. E. Gossard and W. Hooke, Waves in the Atmosphere, (Elsevier, Amster-
dam,1975). 7. Y. Kamide, Electrodynamic processes in the Earth’s ionosphere and magnetosphere, (Kyoto Sangyo University Press, Kyoto, 1988). 8. V. I. Petviashvili and 0. A. Pokhotelov, Solitary waves in plasmas and in the atmosphere, (Gordon and Breach, Reading - Philadelphia, 1992). 9. 0. A. Pokhotelov , L. Stenflo and P. K. Shukla, Plasma Phys. Reports 22, 852 (1996). 10. 0. A. Pokhotelov, T. D. Kaladze, P. K. Shukla, and L. Stenflo, Phys. Scripta 64, 245 (2001). 11. A. I. Smolyakov, P. H. Diamond and V. I. Shevchenko, Phys. Plasmas 7, 1349 (2000).
251 12. G. Manfredi, C. M. Roch and R. 0. Dendy, Plasma Phys. Control. Fusion 43,825 (2001). 13. P. K. Shukla and L. Stenflo, Eur. Phys. J. D 20, 103 (2002). 14. M. V. Nezlin and G. P. Chernikov, Plasma Phys. Rep. 21,922 (1995). 15. P. K. Shukla and L. Stenflo, Physics Letters A . 307,154 (2003). 16. 0. G. Onishchenko, 0. A. Pokhotelov, R. Z. Sagdeev, P. K. Shukla and L. Stenflo, Nonlin. Proc. Geophys. 11,241 (2004). 17. A. R. Lawrence and M. J. Jarvis,J. Atm. Solar-Terr. Phys. 65,765 (2003). 18. T. D. Kaladze and L. V. Tsamalashvili, Physics Letters A 232,269 (1997). 19. T. D. Kaladze, Physica Scripta T75,153 (1998). 20. T. D. Kaladze, Plasma Phys. Reports 25,284 (1999). 21. T. D. Kaladze, 0. A. Pokhotelov, R. Z. Sagdeev, L. Stenflo and P. K. Shukla, J. Atmosp. Solar-Terr. Phys. 65 757 (2003). 22. T. D. Kaladze, G. D. Aburjania, 0. A. Kharshiladze, W. Horton and Y . H. Kim, J. Geophys. Res. 109,A05302, doi: 10.1029 / 2003JA010049, (2004). 23. V. I. Petviashvili, JETP Lett, Engl. Transl. 32,619 (1980). 24. S. V. Antipov, M. V. Snezhkin and A. S. Trubnikov, Sou. Phys. JETP, Eng. Transl., 55,85 (1982). 25. M. J. Lighthill, J. Inst. Maths. Applics 1, 1 (1965).
QUIESCENT AND CATASTROPHIC EVENTS IN STELLAR ATMOSPHERES SWADESH M. MAHAJAN Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, U.S.A E-mail:mahajan@mail. utexas. edu NANA L. SHATASHVILI Plasma Physics Department, Andronikashvili Institute of Physics, Tbilisi 0177,Georgia Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy E-mai1:shatashOictp. it We show that two fluid theory in which the velocity field is treated at par with the magnetic field has the potential of creating an excellent theory for the observed structures and phenomena in stellar atmospheres. Quasi-steady, fast, and even catastrophic phenomena have an underlying unified description. Simple analysis can capture essential and qualitative aspects of both the quiescent and the violent processes - a violent fate of a given structure is underwritten right at its moment of birth. Simulations are needed t o capture what actually happens near the catastrophe.
1. Introduction Stellar atmospheres are hot and charged - much of the observed phenomena are caused by the motion of these charged hot particles (electrons and protons mostly) in magnetic fields. Magnetic fields play a key role in the formation of stars (planetary systems); they control the atmospheric dynamics, the stellar coronae and the stellar winds, the space weather etc. The stellar magnetic field is generated in the interior of a star like the Sun by a “dynamo mechanism”; rotation and convection are the most important ingredients. The atmospheric magnetic field continually adjusts to largescale flows on the surface, to flux emergence and subduction, and to forces opening up the field lines into space. Stellar magnetic activity is revealed in the wealth of phenomena:
252
253
starspots, nonradiatively heated outer atmospheres, activity cycles, deceleration of rotation rates, and even, in close binaries, stellar cannibalism (for the review, see e.g. [l]). Main topics of interest are: radiative transfer, convective simulations, dynamo theory, outer-atmospheric heating, stellar winds and angular momentum loss. Magnetically active stars shed angular momentum - lose mass through their asterospheric magnetic fields. This process involves the interaction of a topologically complex, evolving coronal magnetic field with an embedded plasma, that is heated throughout the corona and accelerated on its way to interstellar space. Stellar observations suggest that the Sun was magnetically active even before it became a hydrogen-burning star; this activity has been smoothly declining over billions of years - angular momentum is lost through a magnetized solar wind (e.g., Schrijver et al. 2003 [l]). The processes observed in stellar atmospheres are mainly of two types Quiescent and Violent. Among quiescent processes one could list the following: formation of long lived coronal structures, heating, maintenance and slow dissipation of these structures, the solar winds (slow and fast), acceleration, flow generation, waves, surface turbulence, granulation etc. Explosive events like blinkers, sudden cell and network brightenings, flares, coronal mass ejection (CME’s) etc. fall in the ”violent category”. Slow or fast, peaceful or violent - these processes represent conversion of one form of energy into another. And there are really only three forms of energy which play a fundamental role - magnetic, thermal and kinetic; gravity also plays some role but not a determining one. For example the flares convert magnetic energy to heat and motion; CME’s destroy magnetic energy and heat for mass motion - both these processes are catastrophic. The latter takes plasma from the low corona into the SW and can disturb the near-Earth space. Each flux emergence into the atmosphere brings helicity to accumulate additively in a coronal structure while excess magnetic energy is flared away.
~
2. Solar Corona - Observations - Inferences Recent observations reveal that the solar corona is a highly dynamic arena replete with multi-species, multiple-scale spatiotemporal structures ’. The magnetic field was always known to be a controlling player. However a major new element has entered current deliberations: the discovery that strong flows are found everywhere in the low atmosphere, in the subcoronal (chromosphere) as well as in coronal regions, has forced kinetic energy to find its rightful place in the dynamics of the stellar atmosphere. The
254
plasma flows may, in fact, complement the abilities of the magnetic field in the creation of the amazing richness observed in the coronal structures3. There, thus, exists a challenge to incorporate flows into a comprehensive theory of energy transformations for understanding the quiescent and eruptive/explosive events (see e.g. [4, 5, 61).
2.1. H o w does the Solar Corona get to be so hot?
The temperature of the solar corona is a million K (100 eV) (Grotrian 1939; Edlen 1942 [7]):it is much hotter than its photosphere (less than an eV for the Sun and other cool stars). How does it get to be so hot and remain so hot is still an unsolved problem! There has never been a shortage of models or a lack of suggestions. The problem is how to prove or disprove a model with the help of observations. Is it the ohmic or the viscous dissipation? or is it the shocks or the waves that impart energy to the particles? And do the observations support the “other consequences” of a given model? The recently developed notions, that formation and heating of coronal structures may be simultaneous and that the directed flows may be the carriers of energy, open a new channel for exploration3.
2 . 2 . The Solar Wind ( S W )
-
History
In addition to the heating of the solar corona, the solar wind constitutes the other seminal problem in solar physics. Historically the hydrodynamical expansion of the corona was invoked to be the fundamental mechanism that makes the solar wind. But here is this serious difficulty: the particles in the fast component of the solar wind (FSW) have velocities considerably higher than the coronal proton thermal velocity (> 300 kms-’). Naturally such fast particles could not come from a simple pressure driven expansion. The proposed rescue was in postulating acceleration (by a variety of possible processes) of the wind to observed velocities. The search for accelerating mechanisms triggered the birth of an active acceleration industry. The standard suggestion that the expansion/acceleration takes place in the regions called the coronal holes where the magnetic field influence can be neglected, is also out of vogue; recent observations show that the wind comes from everywhere’. How so? How do the charged particles beat the closed line magnetic fields?
255
2.3. Present State of Art
There is sufficient observational evidence to conclude that the energy transport and particle channeling mechanisms in the stellar atmosphere are deeply connected to the challenging problems of coronal heating and stellar wind origing. Neither the Stellar Wind "acceleration" nor the numerous eruptive events in the stellar atmosphere can be treated as isolated and independent problems; they must be solved simultaneously along with other phenomena like plasma heating (which may take place in different stages). Any particular mechanism may be dominant in a specific region of parameter space. For the solar wind there is another serious problem - what is the source of matter and energy? The corona or the sub coronal regions (chromosphere, photosphere, more directly from the sun)? Or both? There is a growing consensus at present: the same mechanism that transports mechanical energy from the convection zone to the chromosphere to sustain its heating rate also supplies the energy to heat the corona and accelerate the SW.
2.4. Towards a General Unifying Model
A theory for general global dynamics in a given region of the solar atmosphere is thus sorely needed. Presentation of such a theory, that accords, in addition to the magnetic field, a place of honor to plasma flow, is the principal goal of this paper. Both the magnetic and the flow fields have origins in the sub-atmospheric region and will jointly participate in the creation of a rich variety of coronal structures. The magneto-fluid equations should cover both the open and the closed field regions. The difference between various sub-units of the atmosphere will come from the initial and the boundary conditions. One then boldly conjectures that the formation and primary heating of the coronal structures as well as the more violent events (flares, erupting prominences and CMEs) are expressions of different aspects of the same general global dynamics that operates in a given coronal region. The plasma flows, the source of both the particles and energy (part of which is converted to heat), interacting with the magnetic field, become dynamic determinants of a wide variety of plasma states reflecting the immense diversity of the observed coronal structures.
256
3. Magneto-fluid Coupling
+
Let V be the flow velocity field of the plasma. Total current j = j o j,; where j , is the self-current (that generates Bs). Total (observed) magnetic field is given as B = BO B, (we follow here [3]). The stellar atmosphere is finely structured with multi-species and multi-scales. Then the simplest approach to study the fine-structure creation in the corona is the two-fluid approach invoking the standard normalizations: the density n to no , the magnetic field to some measure of the ambient field BOand velocities to the Alfvkn velocity VAO.We assign equal temperatures to the electrons and the protons so that the kinetic pressure p is given by: p = pi + p , = 2nT, T = Ti N T,. At the same time, the electron and the proton flow velocities are different: Vi = V , V , = ( V - j / e n ) . Note, that in the nondissipative limit the electrons are frozen in the electron fluid; while the ion fluid (with finite inertia) moves distinctly. The parameters TAO = GM/V;,Ro = 2/30/r,oI a0 = &/&, PO = C ~ ~ / Vare , " , defined with no, To, Bo; c,o = is the sound speed. We note that the Hall current contributions become significant when the dimensionless Hall coefficient a0 = Xio/Ro (& - the characteristic scale length of a system and Xi0 = c / q o is the collisionless skin depth) satisfies the condition: (UO > q , where q is the inverse Lundquist number for the plasma. This contribution is important to be taken into account in: interstellar medium, turbulence in the early universe, white dwarfs, neutron stars, stellar atmosphere. For instance, for a typical solar plasma (in the corona, the chromosphere and the transition region (TR)), this condition - l o p 7 for densities within is easily satisfied (a0 is in the range (1014 - 1 0 ~ ) c m -and ~ q = C2/(47rVAo&)O) 10-14, where Ra is solar radius, and c is the plasma conductivity). In such circumstances, the Hall currents modifying the dynamics of the microscopic flows and fields could have a profound impact on the generation of macroscopic magnetic fields" and fast f l o ~ s ~ > ~ . We remind the reader, that the heating due to the viscous dissipation of the flow vorticity can be essential for finely structured atmospheres3:
+
d-
N
3.1. Construction of a Typical Coronal structure The solar Corona temperature T, varies within (1+ 4) . lo6 K , while densities are found to be 5 1010cm-3. The standard approach is based on
257
the idea, that the Corona is first formed and then heated. Three principal heating mechanisms were proposed: (i) by Alfven Waves, (ii) by Magnetic reconnection in current sheets, (iii) by MHD Turbulence. All of these attempts fall short of providing a continuous energy supply that is required to support the observed coronal structures. In this paper we will develop a totally different concept: the formation and primary heating of a structure are contemporaneous, rather than sequential3. During the very trapping of the primary flows (with Tof N l e v << T c ) ,a part of their kinetic energy dissipates into heat and when the coronal structure appears, it is already hot and shining; the initial and boundary conditions define the characteristics of a given structure. Observations suggest that there are strongly separated scales both in time and space in the solar atmosphere. And that is good. Two distinct eras mark the life-span of a coronal structure: 1) A hectic dynamic period when it acquires particles and energy (accumulation and primary heating). To capture the essence of this era we will use the entire set of time dependent dissipative two-fluid equations. Heating takes place while particles accumulate (get trapped) in a curved magnetic field. Simulations show that the kinetic energy contained in the primary flows can be dissipated by viscosity, and that this dissipation can be large enough to provide the continuous heating up to observed temperatures3; 2) A quasi-stationary period when the structure ”shines” as a bright, high temperature object - a reduced equilibrium description suffices for this period; collisional effects and time dependence are ignored. Note, that in equilibrium, each coronal structure has a nearly constant T, but different structures have different characteristic T’s, i.e., the bright corona ”viewed” as a single entity will have considerable T-variation. - Fast, dynamic: Energy losses from the corona are F N (5 . lo5 + 5 . lo6)erg/cm2 s. If the conversion of the kinetic energy in the primary flows were to compensate for these losses, we would require a radial energy flux
1st Era
-+
For the primary flow parameters, VON (100 900) km/s and n N 9 . lo5 + l o 7 ~ m - the ~ , viscous dissipation of the flow takes place on a time: T2
258
For a flow with TO = 3eV = 3.5 . 104K, no = 4 . 1O8cmW3creating a quiet coronal structure of size L = (2 . lo8 + lolo) cm, tvisc N (3.5 . lo8 + lo1') s. Note, that Eq.(2) is an overestimate - treal << tvisc- for the following reasons: 1)vi = vi(t,r) will vary along the structure, 2) the spatial gradients of the V- field can be on a scale much shorter than L (defined by the smooth part of B-field). Simulations show, that when the primary flows interact with the ambient closed magnetic fields, the magneto-fluid coupling can assure a sufficient material and energy supply for the simultaneous formation and heating of a hot coronal structure (see figures 1-4). 2nd Era - Quasi Equilibrium: We contend that the standard magnetohydrodynamics (MHD) theory (single fluid) is inadequate to model this period. Since MHD does not adequately treat the fundamental contributions of the velocity field, equilibrium states (relaxed minimum energy states) encountered in MHD do not have enough structural richness. In a two-fluid description, the velocity field interacting with the magnetic field provides the new pressure confining states and the possibility of heating these equilibrium states by dissipation of short scale kinetic energy. We now summarize a simple equilibrium theory (magnetofluid theory) first proposed and analyzed in [ll,31. The simplest two-fluid equilibria (T = const n-l Vp 4 TV Inn; generalization to homentropic fluid: p = const. nY is straightforward) are contained in the dimensionless equations see (Mahajan et al. [3] for details):
-
1 -Vxbxb+V n
+Vx(VxV)=O,
V . (nV) = 0, V.b=0. The system allows the following relaxed state solution b
+ aoV x V = d n V,
augmented by the Bernoulli Condition
259
where a and d are dimensionless constants related to ideal invariants. The magnetic and the generalized helicities are: hl = J ( A . b) d32,
h2 =
J
(A
(9)
+ V) . (b + V x V)d32.
The system is obtained by minimizing the energy E while keeping hl and h2 invariant. Equations (7) yield
=
(10)
J ( b . b + n V . V ) d32,
ff2
~ V ~ V x V + c u o V x n
which must be solved with (8) for n and V . Equation (8) is solved to obtain, (dT)
= Tco/T),
The variation in density can be quite large for a low /30 plasma if gravity and the flow kinetic energy vary on length scales comparable to the extent of the structure. A very instructive analytical model calculation can be done for varying temperature but constant density (n = 1);Equation (ll),then, simplifies to (where Q is either V or b): a;VxVxQ+ao 3.2. Curl Curl Equation
With p
=
(l/a
-
c
--d
-
1
(
3
V X Q + 1--
Q=O.
(13)
Double-Beltrami states
d) and q = (1- d/a), Eq. (13) is cast as (aoV x -X)(aoV x -p)b
= 0,
(14)
where X(X+) and p ( L ) are the solutions of the quadratic equation
If Gx is the solution of the Beltrami Equation (ax and a, are constants) V x G(X) = XG(X),
then
(16)
260
is the general solution of the double curl equation. The velocity field, corresponding to the magnetic field of Eq.(17) is:
V
b
=-
a
+ aoV x b = (: + a o A ) axG(A) +
(i+
n o p ) a,G(p).
(18)
The Double curl equations, representing states arising out of the magnetofluid coupling, are fully given in terms of the solutions of Eq. (16). 3.3. Double Beltrami States
A desirable and interesting property of the new magneto-fluid states is that there are two scales in equilibrium (unlike the standard case)" - a possible clue for answering the extremely important question: why do the coronal structures have a variety of length scales, and what are the determinants of these scales? The scales could be vastly separated. They are determined by the constants of the motion - the original preparation of the system (these constants also determine the relative kinetic and magnetic energy in the state in quasi-equilibrium); the scales could be a complex conjugate pair (the fields will be obtained by an appropriate real combination - change in the topological character of the flow and magnetic fields); the short scale is a result of a singular perturbation on the standard MHD system that introduces classes of states inaccessible to MHD. It is all a consequence of treating flows and magnetic field co-equally. These vastly richer structures can and do model the quiescent solar phenomena rather well - construction of coronal arcade fields, slow acceleration, spatial rearrangement of energy etc. become possible. 3.4. A n Example of structural richness
In a coronal structure, the magnetic field is relatively smooth but the velocity field must have a considerable short-scale component if its dissipation were to heat the plasma. Can a Double Beltrami state provide that? SubAlfvhnic Flow is characterized by a d >> 1 ==+ X (d - .)/a0 d a, N
-
p = d/ao,
V
1 a
+ da,G(p), b = axGx + a,G(p). = - axGx
(19) (20)
While the slowly varying component of velocity is smaller by a factor (ap1 N d - ' ) compared to the similar part of the b-field, the fast varying component
261
is a factor of d larger than the fast varying component of the b-field! As a result one gets for an extreme sub-Alfvknic flow (e.g. JVI d-' O.l)] N
N
The velocity field is equally divided between slow and fast scales while the magnetic field is mostly on the slow scale3. 4. Acceleration of Plasma Flows
The most obvious processes for acceleration (if rotation is ignored) are the conversion of magnetic and/or the thermal energy to plasma kinetic energy. The well known magnetically driven transient but sudden flow-generation models are: Catastrophic models; Magnetic reconnection models; Models based on instabilities. Quiescent pathway can be in action when Bernoulli mechanism, converting thermal energy into kinetic, is operative and the general magneto-fluid rearrangement of a relatively constant kinetic energy is possible (for instance, going from an initial high density-low velocity to a low density-high velocity state). 4.1. Eruptive and Explosive events, Flaring
Let us try to construct a theory of catastrophic changes by examining the double Beltrami (DB) equilibrium states (the analysis presented below follows [4]).We investigate the fate of the DB states when there characteristic parameters are subjected to change. Let's assume that: 1) the parameter change is sufficiently slow / adiabatic] 2) at each stage, the system can find its local DB equilibrium, 3) in slow evolution the dynamical invariants: hl I hz, and the total (magnetic plus the fluid) energy E are conserved. Can such a slowly evolving structure suffer a catastrophic loss of equilibrium? The general equilibrium solution was shown to be
b = CpGfi(P)+ CxGx(A)1
The transition may occur in one of the following two ways: The roots (A - large-scale] p - short-scale) of the quadratic equation] determining the length scales for the field variation] go from being real to complex or the amplitude of either of the two states (Cfi,v)ceases to be real. The
262
three invariants hl, hz and E (quantum numbers) provide three relations connecting 4 parameters A, p, CA,C, that characterize the DB field. In order to determine if a catastrophe is possible, we will take the large scale X as a control parameter (an observationally motivated choice). We will investigate if we run into a loss of equilibrium as X is slowly varied. The cause of this slow change could be the structurestructure interaction. Working with a simple 2D Beltrami ABC field with periodic boundary conditions (see Fig.5) , we deal with quasi-quilibrium structures that have real Alp initially. width=Either of the following constitutes a loss of equilibrium : (1) either of (C,/,)2 becomes zero (starting from positive values) for real X f i / v (see Fig.6); (2) the roots Xfi/,, coalesce (A, H A), for real A,/ and C,,, (see Fig.9). In the Solar atmosphere the observed equilibria are found to be with vastly separated scales (for a variety of sub-Alfvknic flows) rendering the second possibility not of much relevance. The analysis shows: 0
0
0
0
that the conditions for catastrophic changes in slowly evolving solar structures (sequence of DB magneto-fluid states), leading to a fundamental transformation of the initial state, can and are derived. for E > E, = 2(hl f the DB equilibrium suddenly relaxes to a single Beltrami state corresponding to the large macroscopic size (Fig. 7, 8). all of the short-scale magnetic energy is catastrophically transformed to the flow kinetic energy. Seeds of destruction lie in the conditions of birth (Fig. 7, 8). the proposed mechanism for the energy transformation works in all regions of the Solar atmosphere with different dynamical evolution depending on the initial and boundary conditions for a given region.
a),
4.2. Non-uniform density case
Analytical methods, that can be successfully used for constant density models, have to yield to numerical methods for the varying density cases. A 1D numerical simulation for sub-Alfvknic flows5 carried out on the closed system, eqs. (3), (7), (8), with g ( r ) = T , O / T , gives:
-v n
x
v x v + a0 v x
[(A
-d)
nV]
+ (1 -
t)
V = 0,
(24)
263
n = exp
(-
I)
v,2
[2go - - - 29 + 2Po v2 2PO
’
where the height “2” from the stellar center is the nontrivial dimension. The boundary conditions were taken at 20> (1 2.8. R,, where the influence of ionization can be neglected. Initial, boundary and DB parameters were chosen to satisfy : lbol = 1, fi = 0.01VAO (with = = V,o); a 100, ( a - d ) / a 2 The chosen physical parameters are: d (no; Bo; TO;VAO):10l1 ~ r n - ~100G; ; 5 e V ; 6 0 0 k r n / ~ (YO ; 0.007 << 1. Code limitations dictate the range of (YO; they are much larger than their actual values ( 5 in corona/sub-corona). It is found that lbI2 = const over a distance for all runs and, for small (YO, there exists some height where the density begins to drop precipitously with a corresponding sharp rise in the flow speed. The effect is stronger for low ,B plasmas. There is a catastrophe in the system and the local acceleration is determined by local Po(T0) (see figures 10, l l ) . The Bernoulli condition (26) yields an indirect estimate for the height a t which the observed shock-formation may take place (for all (YO):
+
N
-
-
v,,
-
It is also shown, that when Po(r, t ) goes up the density fall (V-amplification) gets smoother. Final velocities go up with Vo[krn/s] d-lVAo. For instance, the primary flow with an initial speed of 3.3 km/s ends up acquiring 100 km/s at (2 - 20) 0.09 Ro. N
-
-
4.3. Where do the flows come from? Reverse Dynamo -
Flow generation If flows are to be assigned such an important role in determining the coronal dynamics, we have to make sure that the physical processes that we invoke allow the possibility of generating these flows. In the history of astrophysical (as well as laboratory) plasmas, researchers have always actively sought the sources of the magnetic field. One of the chief quests has been the so called “Dynamo mechanism” - a generic process of generating macroscopic magnetic fields from an initially turbulent system. The standard Dynamo is defined by the generation of macroscopic fields from (primarily microscopic) velocity fields. The relaxations observed in reverse field pinches, for example, are vivid illustrations of the Dynamo in action. The myriad
264
phenomena in stellar atmospheres (heating, field opening, the wind) are impossible to explain without knowing the origin and nature of the magnetic field structures. In the so called kinematic dynamo, the velocity field is externally specified and is not a dynamical variable. In ”higher” theories - MHD, Hall MHD, two fluid etc, the velocity field evolves just as the magnetic field does - the fields are in mutual interaction. The question is that if the short-scale turbulence generates the macroscopic magnetic fields, then will the turbulence, under appropriate conditions, generate macroscopic plasma flows? If the process of conversion of short-scale kinetic energy to long-scale magnetic energy is termed the “dynamo” (D) then the mirror image process of the conversion of short-scale magnetic energy to long-scale kinetic energy could be called the “Reverse dynamo” (RD)6. We could safely extend the definitions in the following manner : Dynamo(D) - Generation of macroscopic magnetic field from any mix of short-scale energy (magnetic and kinetic). Reverse Dynamo(RD) - Generation of macroscopic flow from any mix of short-scale energy (magnetic and kinetic). Theory and simulation show6 that: (1) D and RD processes operate simultaneously - whenever a large scale magnetic field is generated there is a concomitant generation of a long scale plasma flow. (2) The composition of the turbulent energy (how much kinetic, how much magnetic) determines the ratio of the macroscopic flow/macroscopic magnetic field. 4.4. Reverse Dynamo relationship
-
Theory
We present here a unified D-RD theory in a minimal two fluid model (incompressible, constant density HMHD ignoring gravity)6. We begin with the standard dimensionless equations,
( V - V X B ) X B, dV
-=
at
V,=V-VxB,
Y)
v x (V x V )+ (V x B)x B v (P + -
(28) (29)
and break the total fields into ambient fields and perturbations
B
= bo
+ H + b,
v = wo + U + v ,
where bo and w o are equilibrium fields. The fluctuations are split into macroscopic ( H , U ) and microscopic ( b, w ) components. The principal departure of this purely analytic calculation lies in the choice of the equilibrium fields: we take them to be the DB pair obeying
265
+ vo2/2) = 0, bO - + v x bo = vo, bo + v x Vo = dwo, U
the Bernoulli conditions, V ( p 0
(30)
which, as shown earlier, may be solved in terms of the single Beltrami (SB) field (V x G ( p ) = p G ( p ) ) .Inverse scale lengths A, p are fully determined in terms of a, d (hence, initial helicities). In what follows, X ( p ) will denote the micro (macro) scale. We shall also assume that the microscopic fluctuations are smaller than their equilibrium values: Ibl << bo, IvI << vo. No ordering will be made with respect to the macroscopic ( H , U ) and the ambient fields. A summary of the calculation is given below (see Ref. [6]). The evolution equations of the macrofields are written as:
&U
=
[
U x (V x U )
+ (V x (V x -
210)
x H
1+
(VOx
(V x v ) )
+ (V x bo) x b + (V x b ) x bo)
(V(V0
[I
+V x H
. v))- v
where the terms in .. are nonlinear and
(”:”’> ~
W,O
,
= vo - V
x bo. The ensemble
averages of the rest of the terms on the r.h.s. of above equations have to be found after solving for v o and bo. The short scale fields (microfields) v and b obey: dV
-=
at
+
- ( U . V)VO ( H . V ) b o ,
(33)
Since one can, in principle, solve the above set for v and b in terms of U and H ( and the given ambient fields), one can go back to eqs. (31,32) and have a closed set of equations for the macroscopic fields. This closure model of the Hall MHD equations is rather general - two essential features are: 1) a closure of the full set of equations - feedback of the micro-scale is consistently included in the evolution of H IU ; 2) role of the Hall current (especially in the dynamics of the micro-scale) is properly accounted for”.
266
4.5. Choice of Initial Fields
The model calculation is best done by assuming that the original equilibrium is predominantly short-scale. Thus from the DB fields we keep only the X part. Notice that our initial state has short scale energy of both kinds - magnetic and kinetic without any restriction on their relative magnitudes. On the short scale, we have the following: Do
= bo ( X + a - l ) ,
(35)
yielding 2)eo
= Do -
v x bo = bo u-l
(36)
and leading to:
b = (u-~H - U ).Vbo, W =
( H - (A
+ u-')
(37)
U ).Vbo.
(38)
Carrying out appropriate averages over the short scale ambient fields (all expressed in terms bo) will give us the time behavior of the macro fields U and H . Spatial averages with isotropic ABC flow yield (see for details [S]):
U
= bNi+
--V 2 3
x [((hli)l)U-hH],
(39)
where N1 and N2 are the time derivatives of the nonlinear terms displayed earlier - they will not change on short-scale averaging. The factor b i measures the ambient micro-scale energy. Notice that H evolves independently of U but evolution of U does require knowledge of H . Neglecting the nonlinear terms and writing eqs. (39) and (40) formally as
H
=
-r(X)(V x H ) ,
0=V
+
x [s(X)U q(X)H],
(41)
and Fourier analyzing, we find the growth rate of H , U , w4 = r2k2
w2 = -/rl(k),
and the increasing macro-fields are related as6
(42)
267 4.6.
A Nonlinear Solution in Linear Clothing
Since a choice of a , d (and hence of A) fixes relative amounts of ambient microscopic energy, it also fixes the relative amount of energy in the nascent macroscopic fields U and H . The linear solution, presented above, has a remarkable feature: when satisfied, it makes nonlinear terms strictly zero it is an exact (a special class of) solution of the nonlinear system and thus remains valid even as U and H grow to larger amplitudes (this behavior appears again and again in Alfvknic systems: MHD - nonlinear Alfvh wave: Walen [12]; in HMHD - Mahajan & Krishan, [12]). Let us now work out some explicit cases to demonstrate both the dynamo(D)and reverse dynamo(RD) in action. It will depend on the initial mix of the kinetic and magnetic energies. 4.6.1. Analytical Results - An Almost straight dynamo
-
-
-
For a d >> 1, inverse micro scale A a >> 1implying 0 0 a bo >> bo i.e, the ambient micro-scale fields are primarily kinetic. The generated macrofields have exactly complementary ordering: U a-l H << H . This is an example of the straight dynamo mechanism: super-Alfvknic “turbulent flows” generate magnetic energy far in excess of kinetic energy - super-Alfv6nic “turbulent flows” lead to steady flows that are equally sub-Alfvhic. It is important to note that the dynamo effect must always be accompanied by the generation of macro-scale plasma flows. This realization can have serious consequences for defining the initial setup for the later dynamics in the stellar atmosphere. The presence of an initial macro-scale velocity field during the flux emergence processes is, for instance, always guaranteed by the mechanism exposed above. 4.6.2. Analytical Results
-
-
-
An Almost Reverse dynamo
-
When a d << 1, the inverse micro scale is A a - a-l >> 1, leading to 00 abo << bo. The ambient energy, now, is mostly magnetic (photospheres/chromospheres). This magnetically dominant initial micro-scale system creates macro-scale fields that obey U a - l H >> H , and are kinetically abundant. From a strongly sub-Alfvknic turbulent flow, the system generates a strongly super- Alfvhic macro-scale flow [reverse dynamo]. Initial dominance of fluctuating/turbulent magnetic field plus magneto-fluid coupling leads to efficient/significant acceleration. Part of the magnetic energy will be transferred to steady plasma flows (a steady super-Alfvknic
-
268
flow); a macro flow will be accompanied by a weak magnetic field. Hence, RD mechanism can explain the results of recent observations: fast flows are found in weak field regions (Woo et al, [8]). The main results of the analysis may be summarized as: 0
0
0
Dynamo and "Reverse Dynamo" mechanisms have the same origin - are manifestations of the magneto-fluid coupling; U and H are generated simultaneously and proportionately. The greater the macro-scale magnetic field (generated locally), the greater the macro-scale velocity field (generated locally); Growth rates of macro-fields are defined by DB parameters (by the ambient magnetic and generalized helicities) and scales directly with the ambient turbulent energy b i (v,").The larger the initial turbulent magnetic energy, the stronger the acceleration of the flow.
-
These findings can have significant impact on the evolution of largescale magnetic fields, their opening up with respect to fast particle escape from stellar coronae, and on the dynamical and continuous kinetic energy supply of plasma flows observed in astrophysical systems. Notice that initial and final states have finite helicities (magnetic and kinetic). The helicity densities are dynamical parameters that evolve selfconsistently during the flow acceleration. 4.7. A Simulation of Dynamical Acceleration
A 2.5D numerical simulation of the general two-fluid equations in Cartesian geometry was carried out in [3, 131. The simulation system contains an ambient macroscopic field, and several other effects (not included in the analysis) : 1) dissipation and heat flux; 2) the plasma is compressible and embedded in a gravitational field (extra possibility for micro-scale structure creation). We follow the fate of an initially gaussian up-flow (the peak is located in the central region of a single closed magnetic field structure) passing through an arcade like magnetic field (see figures 12-14). The initial characteristics of the magnetic field and flow are: B = V x A B,2; A(0;A,; 0); b = B/Bo; b,(t; 2 ; z # 0) # 0; Bo, = 100 G (uniform in time); primary flow is weak and symmetric with: lV0,l << C,O (C,Ois a sound speed) and time duration t o = 100 sec; Vomaz(xo; z = 0) = Vo, = 2.18. 105cm/sec; nomaa: = 1012cm-3; T ( z ; z = 0) = const = lOeV (see
+
~31).
269
We find that the acceleration is significant in the vicinity of the magnetic field maximum with strong deformation of field lines; there is an energy redistribution due to magneto-fluid coupling and dissipation effects. A part of the flow is trapped in the maximum field localization area, accumulated, cooled and accelerated; the accelerated flow reaches a speed greater than 100Icm/s in less than 100sec. Then accelerated flow follows to the maximum magnetic field localization areas, perhaps an expression of the RD mechanism as the flow passes through a series of quasi-equilibria. In this relatively expanded era 1000 sec of stochastic/oscillating acceleration, the intermittent flows continuously acquire energy. After this stage the bifurcation of all parameters follows; the flow starts to accelerate again. The acceleration is highest in the strong field regions (newly generated!). Note, that at the initial stage of acceleration the macroscopic magnetic energy converts to macroscopic flow energy; while at the second stage of acceleration (after the quasi-equilibrium) the microscopic magnetic energy is converted to macroscopic flow energy (RD). N
Simulation Summary
0
0
0
0
The dissipation and the Hall term ( w L Y O ) (through the mediation of micro-scale physics) play a crucial role in the acceleration/heating processes. Initial fast acceleration in the region of maximum original magnetic field plus the creation of new areas of macro-scale magnetic field localization, with simultaneous transfer of the micro-scale magnetic energy to flow kinetic energy, are signatures of the manifestations of the combined effects of the D and RD phenomena. Continuous energy supply from fluctuations (dissipative, Hall, vorticity) implies the maintenance of quasi-steady flows for significant period. In the simulations the actual h l , hz are dynamical. Even if they are not in the required range initially, their evolution could bring them in the range where they could satisfy conditions needed to efficiently generate flows. Thus there are likely to be several phases of acceleration. The dissipation effects play a fundamental role in setting up these distinct stages (modifying generalized vorticity and hence modifying field-lines, creation of micro scales (shocks, fast fluctuations,etc.)).
270
5. Conclusions
A two fluid theory in which t h e velocity field is treated at par with t h e magnetic field has t h e possibility of becoming an excellent theory for t h e observed structures and phenomena in the stellar
0
0
atmosphere. Quasi-steady, fast, and even catastrophic phenomena have a n underlying unified description. Simple analysis can capture essential and qualitative aspects of both quiescent a n d violent processes. A violent fate of a given struct u r e is underwritten right at its moment of birth. Simulations are needed to capture what actually happens near t h e catastrophe.
References 1. Schrijver, C.J., DeRosa, M.L. and Title, A.M. Astrophys. J . 590,493 (2003); Schrijver, C.J. and. Title, A.M. Astrophys. J . 551, 1099 (2001); Schrijver, C.J., Mem. S.A.A. 76,766 (2005). 2. Schrijver, C.J., et al., Solar Phys. 187,261 (1999); Aschwanden, M.J., Poland A.I. and Rabin D.M., Annu. Rev. A&A 39, 175 (2001); Winebarger, A.M., DeLuca, E.E. and Golub, L., Astrophys. J. 553, L81 (2001); Wilhelm, K. Annu. Rev. A&A 360,351 (2001); Choudhary, D.P., Shrivastava, N. and Gosain, S. Annu. Rev. A&A. 395,257 (2002); Christopoulou, E.B., Georgakilas, A.A., and Koutchmy, S., Solar Phys. 199, 61 (2001). 3. Mahajan,S.M., Miklaszewski, R., Nikol’skaya, K.I. and Shatashvili, N.L., Phys. Plasmas 8 , 1340 (2001) 4. Ohsaki, S., Shatashvili, N.L., Yoshida, Z. and Mahajan, S.M., Astrophys. J. 559,L61 (2001); Ohsaki, S., Shatashvili, N.L., Yoshida, Z. and Mahajan, S.M., Astrophys. J . 570 (2002). 5 . Mahajan, S.M., Nikol’skaya, K.I., Shatashvili, N.L. and Yoshida, Z., Astrophys. J . 576, L161 (2002). 6. S.M. Mahajan, N.L. Shatashvili, S.V. Mikeladze and K.I. Sigua., Astrophys. J . 634,,419 (2005). 7. Grotrian, W., Natuwiss. 27,214 (1939); Edlen, B., Zeztsch~ftf. Astrophys. 22 30 (1942). 8. Woo, R., Habbal, S.R. and Feldman, U., Astrophys. J. 612,1171 (2004); Habbal, S.H. and Woo, R., Astrophys. J. 549,L253 (2001); Lin, H., Penn,M.J. and Tomczyk, S., Astrophys. J. 541,L83 (2000); Chertok, I.M., et al., Astrophys. J . 567,1225 (2002); Zhang, J., White, S.M. and Kundu, M.K., Astrophys. J . 527,977 (1999);
271
9.
10.
11.
12.
13.
Habbal, S.H., Woo, R. and Arnaud, J., Astrophys. J. 558,852 (2001); Ofman, L. and Davila, J.M., Astrophys. J . 553,935 (2001); Granmer, S., Field, G.B. and Kohl, J.L., Astrophys. J. 518,937 (1999); Lin, H. Astrophy. J . 446,421 (1995). Blandford, R.D. and Payne, D.G., Month. Not. R. Astr. SOC.199,883 (1982); Doshcek, G.A., et al., Astrophys. J . 54,559 (2001); Grall, R.R., et al., Nature 379,429 (1996); Ofman, L., et al., "IPS Observations of the Solar Wind Velocity and the Acceleration Mechanism", in The 31st ESLAB Symposium on Correlated Phenomena at the Sun, Heliosphere and in Geospace, ed. A. Wilson, ESTEC, Noordwijk, The Netherlands, 22-25 September 1997, 361, (1997); Ryutova, M. and Tarbell, T., Phys. Rev. Lett. 90, 191101 (2003); Goodman, M.L., Space Sci. Rev. 95,79 (2001); Uchida Y., et al., Publ. Astron. SOC.Japan 53,331 (2001). P.D. Mininni, D.O. Gomez and S.M. Mahajan. Astrophys. J. 567,L81 (2002); P.D. Mininni, D.O. Gomez and S.M. Mahajan, Astrophys. J . 584, 1120 (2003); P.D. Mininni, D.O. Gomez and S.M. Mahajan. Astrophys. J . 619, 1019 (2005). S.M. Mahajan, and S. Yoshida, Phys. Rev. Lett. 81,4863 (1998); Z.Yoshida, S. Ohsaki and S.M. Mahajan, Phys. Plasmas 11,3660 (2004). Mahajan, S. M. and Krishan, V., MNRAS 359,L27 (2005); Walen, C., Ark. Mat. Astron. Fys. 30A (15), l(1944); Ark. Mat. Astron. Fys. 31B (3), 1 (1945). Mahajan, S.M., Shatashvili, N.L., Mikeladze, S.V and Sigua, K.I., Phys. Plasmas 13,062902 (2006).
272
Fig. 1. Contour plots for the vector potential A (flux function) in the typical arcade-like solar magnetic field.
Fig. 2.
2 -z
plane for a
The distribution of the radial component V, (with a maximum of 3 0 0 k m l s at
t = 0) for the symmetric, spatially nonuniform velocity field.
273 t=43s
n/no
Fig. 3. Hot coronal structure formation: Initial parameters: the flow To = 3eV and
n o = 4 . lo8 cm-3 , the initial background density = 2 . lo8 cme3, Bmas(xO, zo = 0) = 20 G. The primary heating is completed in (2 - 3) min on distances 10000 km. The heating is symmetric and the resulting hot structure is heated t o 1.6 . lo6 K . Much of
the primary flow kinetic energy has been converted t o heat via shock generation.
*:
.&
.... ......... ..... .......... ...... ..... Q4
-02
00
02
04
t = 201s
t = 472s
Pig. 4. Hot Coronal Structure Creation: The interaction of an initially asymmetric, spatially nonuniform primary flow (just the right pulse) with a strong arcadelike magnetic field Bmaz(z0, zo = 0) = 20 G. Downflows and the imbalance in primary heating are revealed.
274
Y c
Fig. 5 .
2D Beltrami ABS Field with periodic boundary conditions.
275
P
1 -5.4 0.1
'
0.2
0.201
0.0398
0.04
Fig. 6. Relation between the roots: a) No catastrophe initial conditions; b) Catastrophe initial conditions.
276
Energy
0.25
t
I
1
0.15 '1, 0,025 0.3 0,035 0,04
Fig. 7. Energy versus control parameter: a) No catastrophe initial conditions; b) Catastrophe initial conditions.
277
~
Fig. 8. Solar Atmosphere: Almost all initial magnetic energy (short scale) is transferred to flow.
278
..
b
.,
,
I
-5
,
.
4
.
1
(b) Fig. 9.
Root coalescence: No separation between roots at the transition.
279
0,6
I
I 0,wo
0,001
0,002
2.1
0,003
0.w
0,WO
.
, 0,001
.
, . 0,002
, . 0,003
,
.
0,004
2-1
Fig. 10. Flow acceleration (1D simulation): 3 sets of curves labelled by (YO for parameters versus height (2- 1). 1 - 2 - 3 correspond to: QO = 0.000013; 0.005; 0.1.
Fig. 11. Plots for (a) Blowup distance and (b) velocity versus
a0
280
Fig. 12. Initial characteristics of Magnetic field and primary flow for the flow ameleration 2.5D dynamical simulation.
281
n
IVI
T
Fig. 13. Acceleration of plasma flow due t o magnetofluid coupling. Acceleration is significant in the vicinity of magnetic field maximum with strong deformation of field lines; there is an energy redistribution due to magnetofluid coupling and dissipation effects. A part of flow is trapped in the maximum field localization area, accumulated, cooled and accelerated; accelerated flow reaches speed greater than 100k m l s in less than 100 sec. Accelerated flow follows to the maximum magnetic field localization areas - RD!
282
5x10’-
0I
0
1000
0
lo00
t
2000
3000
2000
3000
5x10’-
0I
t
0
0
.
1000
t
0 2000
1 3000
2000
3000
1.o 0.8
0.6
5x10’.
0.4
0.2 0
lo00
0.0
t
2000
3000
0
1000
t
Fig. 14. While acceleration flow passes through a series of quasi-equilibria.
ZERO-DIMENSIONAL MHD MODELLING OF TWO GAS-PUFF STAGED PINCH PLASMA WITH FINITE-P EFFECT ARSHAD M. MIRZA
Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan
F. DEEBA, K. AHMED and M. Q. HASEEB Department of Physics, C O M S A T S Institute of Information Technology, Islamabad, Pakistan. The implosion dynamics of two gas-puff staged pinch plasma is investigated using zero-dimensional MHD code in the presence of pressure gradients. A modified snow-plow model has been used to describe the dynamics of staged pinch plasma. Our numerical .results demonstrate that fusion parameters can be achieved for an optimum choice of density ratio of the test to driver gas and kinetic to magnetic pressure ratio.
1. Introduction Z-pinch device is one of the earlier approaches to controlled fusion. In this device the plasma is produced by applying a high voltage pulse across an anode-cathode gap of cylindrical geometry. The plasma is imploded by the azimuthal magnetic field (Be)produced by the axially flowing discharge current ( I z ) .The plasma is heated by joule heating and confined through the self-generated azimuthal magnetic field. Earlier low density pinches driven by low voltages were found highly unstable against the magnetohydrodynamic (MHD) instabilities, namely the sausage ( m= 0), kink ( m = 1) and Rayleigh-Taylor (R-T) modes'. These instabilities disturbed the discharge, long before the desired density and temperature could be reached2. For stabilizing the Z-pinch, axial magnetic fields3 and conducting shells have been used. In a Z-pinch device the sausage mode of instability grows slower than the hydromagnetic R-T instability when the shell radius is large compared to the perturbation wavelength4. The R-T instability manifests itself during the early stages of implosion and limits the energetics of the final
283
284
compression. This has motivated an intensive theoretical as well as experimental research to study R-T instability for various dynamic systems such as Z-pinches, imploding liners etc. Efforts have been made to mitigate the R-T instability. Recently, some attention has also been given to the sheared axial flow effect, the finite Larmor radius effect and the sheath curvature e f f e ~ t However, ~ > ~ . in some multilayer gas-puff experiments, the spatial instabilities appearing in the outer liner during the acceleration phase seems to be quenched on striking with the inner liner. In fact, it has been shown that, in order to suppress a spatial Rayleigh-Taylor instability, it is necessary that the mass of the outer shell be somewhat lower than the mass of the inner one. In this way the shock waves, appearing in the outer shell in its collision with the inner gas-puff, will quench the R-T instability7. To overcome this problem, Rahman et aL8, proposed an alternative scheme of staged pinch in which an annular Z-pinch plasma implodes onto a trapped axial magnetic field and compresses it to several MG in a rise time much shorter than the Z-pinch current rise time. The fast variation of the trapped magnetic field induces a large current on the surface of a solid D-T fiber near the centre of the device. The fiber breaks down and forms a &pinch. This combined Z-8 staged pinch configuration has been remarkably shown to be more stable than the usual Z- or 9-pinches. Apart from fusion, the staged pinch has applications in photoresonant and X-ray lasers, which in turn can be used as drivers for inertial confinement fusion (ICF) scheme. The dynamics of staged pinch plasma with entrained axial magnetic field, studied by Rahman et aL8, assume a very thin annular plasma shell. However , experiments have shown that large diameter thin shell implosions are highly unstable against MHD instabilities (in particular R-T instability is the dominant one), which strongly affect the plasma parameters and so the radiation yield at the final stage of implosiong. To suppress the R-T instability, Rostoker et proposed to spin the gas-puff in the manner that was previously demonstrated for a plasma gun experiment, or by simply adding a cusped magnetic field to the original fields of the Z-pinch. Mirza et a1.l investigated the implosion dynamics of a high-density @-pinchplasma driven by a spinning annular gas-puff. It was shown that the R-T instability can indeed be suppressed by spinning the outer Z-pinch plasma, although fusion conditions cannot be reached. Nasim et a l l 2 have investigated the outer dynamics of double gas-puff Z-pinch plasma with different forms of applied current profiles by using modified snow-plow model13. In this paper, we have extended the said work and investigated the dynamics of outer two gas-puff as well as inner &pinch D-T fiber plasma using
285
a modified snow-plow model with kinetic pressure effects. The inclusion of a pressure gradient term in the outer two gas-puff Z-pinch dynamic equations introduces the usual plasma P-term (the ratio of kinetic pressure to the magnetic pressure) which delays and reduces the maximum compression. On the other hand, compression at an earlier time can be achieved by taking large mass density ratio of the test to driver gas. Our O-D code cm-3 density, 8 keV ion temperature, t N 0.25 nsec, yieldpredicts 6 x ing the Lawson criterion parameter n7 1014 sec/cm3 for DT-fiber plasma with a = 0.1 and /3 = 0.025. Therefore, two gas-puff staged pinch can be used for a controlled thermonuclear fusion device14.
-
2. Staged pinch dynamics The set of MHD equations that describe the dynamics of two gas-puff staged pinch plasma based on modified snow-plow mode112-14 can be expressed as
dT T da (y - 1) x 10-22a2 + pa - pb - Pc] , (3) dt a dt 2Tono where R(= T / T O ~ is ) the normalized radius, c1 = I~/(lOOrnor;) the measure of external force on the pinch per unit mass, c2 = (5roB0"/10)~the degree to which the axial field impedes the implosion, a = (p2/pl)/[(l ( T ~ / T O ) ~ (p2/p1)(~1/7-0)~)], P(= 8nPo/B,") the ratio of kinetic to magnetic pressures and M [ = npl(r; - T : ) np2(r? - ?)] is the mass per unit length accumulated in the current sheath during the implosion. In the two gas-puff model, we have assumed that the driver gas of mass density p1 is < r < TO and the test gas of mass density located in a cylindrical shell p2 is located in T < T I . Equations (1) and (2) represent the equations of motion for the outer Z-pinch and inner 0-pinch fiber plasma respectively, and Eq. ( 3 ) is the energy balance equation for the D-T fiber plasma. Here Po represents the Ohmic heating term, Pa the a-particle self-heating term, Pb the bremsstrahlung and P, the cyclotron radiation loss term as given in Ref. [ 141, whereas y is the thermodynamical specific heat ratio, and we have used y = 5 / 3 in our O-D code. In these equations the fiber radius a and the temperature T are normalized to their initial values a0 and TO,no being the
+
- = - 2 ( y - 1)--
+
+
286
initial density of the fiber plasma in units of describe the dynamics of a staged pinch.
~ m - Equations ~ . (1)-(3)
3. Numerical results and discussion
Using some typical parameters of UCI-experiments, we have numerically solved Eqs. (1)-(3) by using zero-dimensional MHD code with 10 = 10 MA, TO = 4 cm, t o = 50 nsec, mo = 38 pg/cm, TO= 20 eV, BO= 20 kG, mo = 38 pg/cm, a0 = 0.02 cm, no = cm-3 for different values of a and for p in the range 0 6 ,6 6 0.050. Fig. 1 displays the current profile and the radial trajectory for the outer Z-pinch radius R for various values of a and p. We found that for fixed value of a(= O.l), finite ,B effect always delays the maximum compression and small values of p < 0.1 give higher compression. This indicates that two gas-puff devices can be used for controlled thermonuclear fusion. On the other hand, for large ,B case, the maximum compression occurs a t later times. Similarly, large a with fixed p gives fast compression. This indicates that the present device can also be used for X-ray lasers15. The compression of the axial magnetic field generates an induction current in the DT-fibre plasma and as a result the implosion transfers Z-pinch kinetic energy to the magnetic field and then to the 8pinch. Figure 2 displays the fiber-plasma radius, density and temperature as a function of time for different values of /3 with a = 0.1. It is evident from the graphs that without the kinetic pressure term ( p = 0), one can obtain high density (n 1025~m-3)and high temperature (5" 100 keV) plasma with a = 0.1. However, any finite-/3 value with fixed a = 0.1 seems to reduce the maximum compression, and leads to lower temperatures and densities. For example, for p = 0.025, the maximum density (Fig. 2(b)) a t peak compression is about 6 x cm-3 and the temperature is about 8 keV. On the other hand, for p = 0.050, the maximum density (Fig. 2(c)) a t peak compression is about 4 x 1021cm-3 with a temperature of the order of 0.09 keV. From these results one may conclude that the two gas-puff staged pinch can be used for controlled thermonuclear fusion with very small p and high a values.
-
-
Acknowledgments This work was partially supported by the Quaid-i-Azam University Research Fund (2005-2006) and the Pakistan Science Foundation Project No. PSF/Res/C-QU/Phys( 130).
287
References 1. E. G. Harris, Phys. Fluids 5,1057 (1962). 2. R. Carruthers and M. Davenport, Proc. Roy. Phys. SOC.B70,49 (1957). 3. G. Bateman, MHD Instabilities (MIT Press, Cambridge, MA, 1978). 4. N. F. Roderick and T. W. Hussey, in Proc. 2nd Int. Conf. on Dense Zpinches, Laguna Beach (1989), ( A I P Conf. Proc. 195,eds. N. R. Pereira, J. Davis and N. Rostoker, American Institute of Physics, Melville, NY), p.157. 5. U. Shumlak and N. F. Roderick, Phys. Plasmas 5, 2384 (1998). 6. M. R. Douglas, C. Deeney and N. F. Roderick, Phys. Rev. Lett. 78, 4577 (1997). 7. R. B. Baksht et al., in Proc. of 3rd Int. Conf. on Dense Z-pinches, London (1993), ( A I P Conf. Proc. 299,eds. M.G. Haines and A. Knight, AIP, Melville, NY), p.365; A. M. Mirza and F. Deeba, Physica Scripta 70,265 (2004). 8. H.U. Rahman, P. Ney, F.J. Wessel, A. Fisher and N. Rostoker, in Proc. of 2nd Int. Conf. on High-Density Pinches, Laguna Beach (1989), ( A I P Conf. Proc. 195,eds. N. R. Pereira, J. Davis and N. Rostoker, AIP, Melville, NY), p.351. 9. N. F. Roderick and T. W. Hussey, in ibid., p.157. 10. N. Rostoker, G.G. Peterson and H. Tahsiri, Comments on Plasma Phys. and Control. Fusion 16,129 (1995). 11. A. M. Mirza, M.Y. Yu and I. Ahmad, Plasma Phys. Control. Fusion 40, 393 (1998). 12. M. H. Nasim, M. Salahuddin and A. M. Mirza, J. Plasma Physics 53,135 (1995). 13. M. Y. Yu and Xu Xue-Ji, Contrib. Plasma Phys. 30,403 (1990). 14. F. Deeba, K. Ahmed, M. Q. Hasseb and A. M. Mirza, Physica Scripta 72, 399 (2005); Mod. Phys. Lett. B19, 1095 (2005). 15. J. P. Apruzese, J. Davis and K. G. Whitney, J . Appl. Phys. 53,4020 (1982).
288
1.o
0.8
-
0.6
K
0.4 0.2 0.0 0
10
20
30
40
50
60
70
t (nsec)
Fig. 1. Outer pinch current and outer pinch radius as a function of time, for a fixed value of a = 0.1 and different values of p. Curves 1-3 for p = 0,0.025, 0.050, respectively.
289
10'
10'
+. 100
10, 55.0
55.1
55.2
55 3
554
55.5
51.2
51.0
57.4
57.6
51.8
58.0
58.2
I( n w
10'
.
,
.
,
.
,
. 10)
3
Fig. 2. Plots of the normalized fiber radius a, number density n(1OZ2~ m - and ~ ) temperature T (keV) versus time with a = 0.1 and for different values of p. Figs. 2(a) to 2(c) for = 0, 0.025, 0.050, respectively.
DOES QUASI-NEUTRALITY REMAIN VALID IN PAIR-ION PLASMAS? H.SALEEM Physics Research Division, PINSTECH, P. 0. Nilore, Islamabad, Pakistan. A theoretical analysis of the quasi neutrality approximation in pair-ion plasmas is presented. The ion acoustic wave is studied using a kinetic model. It is shown that a small concentration of electrons in the perturbed pair-ion plasma can violate the quasi-neutrality in the limit 1 << X&,k2 (where AD, is the electron Debye length). The damping is reduced and hence the acoustic wave can be easily excited. The results are discussed in comparison with experimental and theoretical findings. Moreover, it is suggested that the fullerene plasma recently produced in a laboratory was not a pure pair-ion plasma contrary to the claim. This investigation can be very useful for further research on pair-ion plasmas.
1. Introduction Research on pair-ion plasmas has started very r e ~ e n t l y l -after ~ the claim of a few scientist^^>^ that pure pair-ion fullerence (C,',) plasmas have been produced in laboratories. It has been shown that, in such plasmas, three kinds of electrostatic waves can propagate parallel to the external magnetic field5. These waves are the ion plasma wave (IPW), the ion acoustic (or ion thermal) wave (IAW) and the intermediate frequency wave (IFW). The IAW observed in the experiment5 has a frequency larger than the ion thermal wave frequency i.e. c',k
< w , where c', =
(e)'
has been defined in
reference [5] as the speed of the IAW. On the other hand, the IAW fre-
(g) I
quency is defined as c, = and c', << c, because Ti << T, is required for IAW to exist. It has been shown that the real frequency of the IAW becomes larger in a pair-ion-electron (pie) plasma compared to the case of ordinary electronion ( e i ) plasma, corresponding to the same density concentration of positive ions. Therefore the experimental observation, c:k < w , indicates that the produced pair-ion fullerence plasma contains a significant concentration of
290
291
electrons3. In this investigation the linear dispersion relation of IAW in the ( p i e ) plasma has been obtained using multi-fluid equations assuming the plasma to be quasi-neutral. Very recently it has been shown that the quasi-neutrality approximation breaks down in the perturbed state of (pie) plasmas when the electron number density is reduced6. The decreasing number density of electrons 1
1-
in the limit 1 << X i , k 2 (where AD, = 4.rm e 2 is the electron Debye 2: length) reduces the Landau damping of IAW. Therefore these waves can be easily excited in such systems and probably this was the situation in the experiment reported in reference [ 5 ] . In this work we shall show how the frequency of IAW increases and why quasi-neutrality does not remain a valid approximation in these systems in ( p i e ) plasmas.
(
2. IAW with Quasi-Neutrality
Let the magnetic field be constant along the z-axis and consider the plasma to be homogeneous. The fluid equation of motion of jth-species can be written as,
mjnjatvj= n j q j (E + vj x Boz)- V p j ,
(1)
where subscript j = denotes the positive and negative ions. We assume E = - 0 p and p j = njTj. The above equation yields,
and
where R j = a. The continuity equation can be written as, m3
atnj
+ nojVl.vjl + nojdzvjt = 0.
(4)
Equations (2-4)give,
{w2(w2 - Rj) - v$jk2w2
+ v+jk$R3}nj - -k,w noj9j
2
2p
mj
-no'*' (w2 - R;)kzp = 0.
(5)
mj
Writing Eq. ( 5 ) for j = f and then subtracting one equation from the other, we obtain,
292
-(ny
+ n!!)-klw mi q
2
2 'p
- (ny
+ n!?)xkZ(w2 - R;)p mi
= 0,
(6)
where Ri = qBo/mi is the ion gyro frequency. Here, the superscript 0 denotes the equilibrium quantities. The magnitude of the charge on both ions is q and they have equal mass mi. The temperatures of both ions have also been assumed to be equal, i.e. T+ = T- = Ti and, hence, we define the ion thermal velocity as ' U T ~= (yiTi/mi)1/2. Here, yi is the ratio of specific heats for the adiabatic ions. The Poisson equation reads, €0 2 (n+- n-) = --V 'p. (7) 4 We want to find a criterion to determine the percentage concentration of electrons in the system. For this purpose we have to investigate the IAW frequency and, therefore, we shall also need an electron Boltzmann density distribution
n,
= noeexp
(g).
The set of equations (6-8) yields a few simple but interesting results which have been discussed in reference [7]. We write here only the linear dispersion relation for a purely parallel propagating wave, the IAW. Assuming quasi-neutrality along with k i = 0 and Bo = 0, Eqs. (6) and (8) yield w 2 = --(Noc:k:) 4
+ v$&, no++n'?
(9) I+€
120
where cs = (Te/mi)ll2, ZIT^ = (Ti/mi)1/21 NO = noe - 1--E 7 € =n+a l l and ny denotes the equilibrium number density of jth species ( j = e, -).
+,
3. Break-down of Quasi-Neutrality
Recently it has been shown6 that quasi-neutrality is not a good approximation to study waves in such systems because the X%,k2 term can become very large. This term is also an important factor in determining both the real frequency of the IAW and it's Landau damping. Let us consider the linear dispersion relation of low frequency electrostatic waves propagating parallel to the external magnetic field in a hot ion
293
plasma using the kinetic model as follows6,
whereZ. - -,AD. fikuTj
3 -
2W p ,j wp3. wjl 9; ) 1 / 2 and qj(Tj)are the charge - ( 4 7m
(temperature) of the j-th species, respectively. The IAW exists in an (ei) plasma in the limit VT& < w < v ~ , k . T o investigate the linear dynamics of IAW in ( p i e ) plasmas, we use the similar approximation i.e V T f k < w < V T , ~ . For an analytical solution we use the limit lZel << 1 and 1 << lZ*lto obtain,
The real part of this equation yields, w =w,
'(
l+--), 3v$+ k2 Lof w 2 N$
f
w," = No
cs2+k2 1 A&k2
+
'
+
Here c,+ = (T,/m+) is the ion acoustic speed corresponding to positive ions. Using Eq.(ll), the real frequency can be approximated as,
for
3 4 k 2 L* *$ < 1.
Assuming w = w, - ~yalongwith y << w,, the damping rate turns out to be,
294
It is obvious from this equation that the damping of IAW depends upon concentration ratios and temperatures ratios of different species. The magnitude of y will be different in the limits T- < T+ and T+ < T- . But this difference in magnitude will not be qualitative as long as T* << Te remains valid. If the temperature of the ions is not much smaller than the electron temperature, i.e. if the limit T& 5 Te holds, then we can not use the assumption 1 << \&\and \Zel << 1. The investigation of this situation is out of the scope of the present work. However when the difference between the ion and electron temperatures decreases, the damping of IAW caused by resonance ions increases6. Apart from the particular experiment5, our aim is to point out how the damping rate and real frequency of IAW is affected because of noe # 0 in (pie) plasmas. In our opinion, the experimental observation of IAW in Fig.2 of reference [5] is an indication of the fact that 1 << X&,k2 can be a common situation in ( p i e ) plasmas. In deriving the above relations we have assumed q+ = q- = 1. Otherwise w, and y will become functions of these parameters as well. Such effects are similar to the case of other multi-component plasmas like dusty plasmas and ( p i e ) plasmas and can be found in the existing literature. We want to point out that IAW can be easily excited in ( p i e ) plasmas in the limit 1 << X&,lc2. Now to focus our attention on this particular point, we assume T+ = T- = Ti and m+ = m- = m similar to the approximations used in theoretical calculations of reference [5].But we also assume Te # 0, noe # 0 and T* << T, for an analysis of IAW dynamics in ( p i e ) plasmas. Then Eq.(ll) can be written as,
(1
+ X g e k 2 ) + --No Te Ti (2,
{ -(-
; ;2
+ --NoZie-z:)} Te Ti
+ -)42:3 + iJ;; = 0.
The real part of this equation yields the real frequency as, WT(k)CvW,
(
yy)
1+--
,
c2k2 where w, = No l+i$, k2 ' In the limit XLek2 << 1, Eq.(16) reduces to w 2 = Nocak 3v$,k2. In the limit 1 << XLek2 for n , ~<< no+ and
+
295 = w, - ~
assuming w
w,(k)
N
yone , obtains 1 n+ o X2 I) k 2+ ]. 2X&,k2 ) + 2noe No
(1f E)W&
(17)
Since 1 < E if noe # 0 and noe < n:, therefore w , becomes larger than the case of (ei) plasmas for the same density of n:. Let Age = E ~ X & , + where
XLe+
=
(-*)47rno e
and el
=
n+ 2. Note
that +A :
= X:,+in
(ei)
plasmas because n,o = n: and Eq.(17) in this case becomes the same as Eq.(4.2.4.4) of reference [6]. However, in (pie) plasmas the result is different from the case of (ei) plasmas due t o E # 0 and 1 < €1. Therefore in ( p i e ) plasmas we may have wpi < w,. The imaginary part of the frequency can be written as y = y$ ,'y where
+
and
Equations (20) and (21) are the same as Eq.(4.2,4.6) of reference [6] which represent the damping rate of IAW in ( e i ) plasmas where NO = 1. It seems important t o point out that corresponding t o a wavelength for which X&,k2 = X&,+k2 < 1 in (ei) plasmas, we may have 1 < X&,k2 because the inequality 1 << €1 may hold due t o noe << Therefore the quasi-neutrality approximation must not be used in ( p i e ) plasmas. F'urthermore, the use of the Poisson equation has also shown that in ( p i e ) plasmas we may have wpi < w, because 1 < E . For the case of longer wavelengths, i.e. X&,k2 << 1, the damping rate of IAW can be very large in (pie) plasmas compared to ( e i ) plasmas because we may have 1 << NO and hence yi << 7' and ye << y.: In this case, one can not excite IAW easily. On the other hand, if 1 << X&,k2 holds which can be the case when 1 << € 1 (even if X&,+k2 5 l), then the damping rate of IAW can reduce significantly in ( p i e ) plasmas because 1 < NO << XLek2 can be
i,
nt.
296
generally valid. In this situation one may find 7; << -yi and 7,' < 7,. Therefore the IAW can be excited easily in (pie) plasmas if the condition 1 < NO<< X&,k2 is satisfied. In the limit 1 << X&,k2, one may express the above relations as,
and f
Yi(k)cv
-*
4. Discussion
Since e 2 k " ~ i << 1, E 5 1, or E << 1,therefore 7,' 5 7i can be the case. That is, corresponding to the same value of X&,+k2 we have 7,' 5 7i because wFi < w,' . But 7,' < 7, always holds due to (1 - e 2 ) < 1. Therefore we conclude that in the limit 1 << X'$,,k2, the damping rate of IAW decreases if the electron concentration decreases and E nears 1. Since w, depends upon T, which is an unknown in the observations of reference [ 5 ] ,therefore we can not estimate the value of No from their Fig. 2 by using the values of w and k . Furthermore, to find the value of X&,+k2 , we again need the value of T,. Therefore in the present work we can not predict any estimate of E or No in this experiment which can give the value of the ratio
a.
4
We have presented an analysis of the ( p i e ) plasmas. First it has been pointed out that quasi-neutrality is not a good approximation in the case of (pie) plasmas where the inequality 1 5 X&,k2 can hold in general due to noe << n: even if X&,+k2 < 1. Second, the real frequency of the wave can be larger than ion plasma oscillation i.e. wpi << w, because 1 < E in Eq.(17) provided that 1 << X&,k2 is satisfied. Third, the IAW has a smaller value of the damping factor in (pie) plasmas compared to ( e i ) plasmas corresponding to the same value of X&,+k2 if 1 << € 1 holds. Therefore this wave can be easily excited in (pie) plasmas in the limit 1 << X&,k2. Fourth, for X&,k2 << 1, the damping of the IAW is larger in (pie) plasmas compared to ( e i ) plasmas corresponding to the same values of X&,+k2 and hence this wave can not be easily excited. It is not appropriate to assume a plasma to be completely free from electrons. Moreover the pair-ion plasma is not exactly similar to electron-positron ( e p ) pair plasmas. Due to lighter mass
297
of ( e p ) plasma particles electromagnetic phenomena are very important in such systems. On the other hand the (pi) plasmas can be dominantly electrostatic. This investigation can be very useful for future research on pair-ion plasmas.
References 1. J. Vranjes and S.Poedts, Plasma Sources Sci. Technol. 14,485 (2005). 2. P.K.Shukla and M. Khan, Phys. Plasmas 2, 014504 (2005). 3. HSaleem, J.Vranjes, and S. Poedts, Phys. Lett. A 3504,375 (2006). 4. W.Oohara and R. Hatakeyama, Phys. Reu.Lett. 91,205005 (2003). 5. W.Oohara and R. Hatakeyama , Phys. Reu.Lett. 95, 175003 (2005); Phys. Scr. T 116,101 (2005). 6. HSaleem, Phys. Plasmas 13,044502 (2006). 7. A.I. Akhiezer, I.A. Akhiezer, R. V. Polovin, A.G. Sitenko, and K. N. Stepanov, Plasma Electrodynamics, uol.1, translated by D. ter Haar (Pergamon Press 1975).
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Relativity, Astrophysics and Cosmology
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INVARIANCE UNDER COMPLEX TRANSFORMATIONS AND ITS RELEVANCE TO THE COSMOLOGICAL CONSTANT PROBLEM GERARD 'T HOOFT * and STEFAN NOBBENHUIS t Institute f o r Theoretical Physics, Utrecht University, Leuvenlaan 4584 CC Utrecht, The Netherlands and Spinoza Institute Postbox 80.195, 3508 TD Utrecht, The Netherlands E-mail: * [email protected], t S. J . [email protected]
In this paper we study a new symmetry argument that results in a vacuum state with strictly vanishing vacuum energy. This argument exploits the well-known feature that de Sitter and Anti- de Sitter space are related by analytic continuation. When we drop boundary and hermiticity conditions on quantum fields, we get as many negative as positive energy states, which are related by transformations to complex space. The paper does not directly solve the cosmological constant problem, but explores a new direction that appears worthwhile. Reprint of the article: Class. Quantum Grav. 23 (2006) 3819-3832. @Institute of Physics Publishing Limited 2006 Reprinted with the permission of Institute of Physics Publishing. m. iop. org/journals/cqg
1. Introduction The cosmological constant problem is one of the major obstacles for both particle physics and cosmology. The question is why is the effective cosmological constant, R e f , , defined as R e f f = R+87rG(p) so close to zero a . As is well known, different contributions to the vacuum energy density from particle physics would naively give a value for ( p ) of order M i which )
=Note that using this definition we use units in which the cosmological constant has dimension GeV2 throughout.
301
302
then would have to be (nearly) cancelled by the unknown ‘bare’ value of
A. This cancellation has to be precise to about 120 decimal places if we compare the zero-point energy of a scalar field, using the Planck scale as a cutoff, and the experimental value of pvac = ( p ) A/87rG, being 10-47GeV4. As is well known, even if we take a TeV scale cutoff, the difference between experimental and theoretical results still requires a finetuning of about 50 orders of magnitude. This magnificent fine-tuning seems to suggest that we fail to observe something that is absolutely essential. In a recent paper, one of us gave a categorization of the different proposals that have occurred in the literature and pointed out for each of them where the shortcomings are.
+
In this paper we discuss in more detail a scenario that has been introduced in [l],based on symmetry with respect to a transformation towards imaginary values of the space-time coordinates: x p -+ i x p . This symmetry entails a new definition of the vacuum state, as the unique state that is invariant under this transformation. Since curvature switches sign, this vacuum state must be associated with zero curvature, hence zero cosmological constant. The most striking and unusual feature of the symmetry is the fact that the boundary conditions of physical states are not invariant. Physical states obey boundary conditions when the real parts of the coordinates tend to infinity, not the imaginary parts. This is why all physical states, except the vacuum, must break the symmetry. We will argue that a vanishing cosmological constant could be a consequence of the specific boundary conditions of the vacuum, upon postulating this complex symmetry. We do not address the issue of non-zero cosmological constant, nor the so-called cosmic coincidence problem. We believe that a symmetry which would set the cosmological constant to exactly zero would be great progress. The fact that we are transforming real coordinates into imaginary coordinates implies, inter alia, that hermitean operators are transformed into operators whose hermiticity properties are modified. Taking the hermitean conjugate of an operator requires knowledge of the boundary conditions of a state. The transition from x to is requires that the boundary conditions of the states are modified. For instance, wave functions that are periodic in real space, are now replaced by waves that are exponential expressions of x , thus periodic in ix . But we are forced to do more than that. Also the creation and annihilation operators will transform, and their commutator
303
algebra in complex space is not a priori clear; it requires careful study. Thus, the symmetry that we are trying to identify is a symmetry of laws of nature prior to imposing any boundary conditions. Demanding a, where a, may be real or imaginary, invariance under x, -+ x, violates boundary conditions at @ + 0 0 , leaving only one state invariant: the physical vacuum.
+
2. Classical Scalar Field
To set our notation, consider a real, classical, scalar field @(x) in D spacetime dimensions, with Lagrangian
c = -;(a,@y
- V ( @ ( x ),)
V ( @= )
+ XQ4 .
(1)
Adopting the metric convention (-+++) , we write the energy-momentum tensor as
Write our transformation as x, = iy, , after which all coordinates are rotated in their complex planes such that y, will become real. For redefined notions in y space, we use subscripts or superscripts y , e.g., 8; = ia, . The field in y space obeys the Lagrange equations with
cY -- -c
= -;(a,y@)2+v(@) ;
(4)
+
Tlv = -Tp = dE@(iy)dyY@(iy) gpvLy(@(iY)) . The Hamiltonian in y -space is
H
Tto = ;IIp
-(iD-1
+ $(L?y@)z
-
)Hy
1
y-1
H -
dD-1 Y G o ;
V ( @ ,) IIy(y) = iII(iy) .
(5)
(6) (7)
If we keep only the mass term in the pot(entia1, V ( @ = ) irnza2,the field obeys the Klein-Gordon equation. In the real Z-space, its solutions
304
can be written as @(z,t ) =
/
II(z, t )
/dD-'p
po =
=
dD-'p (a(p)ei(P5) p o (-ia(p)ei(pz) def
djqG7 ,
(pz) =
+ ia*(p)e-i(Pz)
1?'.? - p o t ,
where a ( p ) is just a c-number. Analytically continuing these solutions to complex space, yields: @ ( i y , i ~=)
I
dD-'q ( aY (4 )ei(qY)+ &
I I y ( y , ~= ) i I I ( i y , i ~= ) /dD-'qqo qo =
d
(-iay(q)ei(qY)
m,
+ i&,(q)e-Z('JY)) ; def
(qy) =
c.y'- QoT.
(12)
(13)
The new coefficients could be analytic continuations of the old ones,
but this makes sense only if the a ( p ) would not have singularities that we cross when shifting the integration contour. Note, that, since D = 4 is even, the hermiticity relation between a y ( q ) and k Y ( q ) is lost. We can now consider solutions where we restore them:
while also demanding convergence of the q integration. Such solutions would not obey acceptable boundary conditions in x -space, and the fields would be imaginary rather than real, so these are unphysical solutions. The important property that we concentrate on now, however, is that, according to Eq. ( 5 ) , these solutions would have the opposite sign for TPv.
Of course, the field in y-space appears to be tachyonic, since m2 is negative. In most of our discussions we should put m = 0 . A related transformation with the objective of Twy4 -Tpvwas made by Kaplan and Sundrum in [2]. Non-Hermitian Hamiltonians were also studied by Bender et al. in for example [3, 4, 5, 61. Another approach based on similar ideas which tries to forbid a cosmological constant can be found in [7].
305
3. Gravity
Consider Einstein’s equations:
R pv - 12gpvR- Agpv
=
-8nGTpv.
(16)
Writing
xp = i y p
=
i(q,T ) ,
gEv(y)
--+
gpv(x = i y ) ,
(17)
and defining the Riemann tensor in y space using the derivatives d,Y , we
see that
RL,, = - R p v ( i y ) .
(18)
Clearly, in y-space, we have the equation
REv - igi,,RY
+ AgEv = +8nGTPv(iy)= -8nGT;’.
(19)
Thus, Einstein’s equations are invariant except for the cosmological constant term.
A related suggestion was made in [8]. In fact, we could consider formulating the equations of nature in the full complex space z = x iy , but then everything becomes complex. The above transformation is a oneto-one map from real space !R3 to the purely imaginary space S 3 , where again real equations emerge.
+
The transformation from real to imaginary coordinates naturally relates deSitter space with anti-deSitter space, or, a vacuum solution with positive cosmological constant to a vacuum solution with negative cosmological constant. Only if the cosmological constant is zero, a solution can map into itself by such a transformation. None of the excited states can have this invariance, because they have to obey boundary conditions, either in real space, or in imaginary space. 4. Non-relativistic Particle
The question is now, how much of this survives in a quantum theory. The simplest example to be discussed is the non-relativistic particle in one space dimension. Consider the Hamiltonian P2 2m where p = -id/dx. Suppose that the function V ( x ) obeys
H = - + V ( Z ),
V ( 5 )= -V(ix)
,
V ( x )= X 2 % ( X 4 ) ,
(21)
306
with, for instance, &(x4) = e-xz4 . Consider a wave function ing the wave equation HI$) = E l $ ) . Then the substitution x = iy
,
p = -ip,
I+(.)
obey-
,
gives us a new function I+(y)) obeying
H,l$(Y)) = -EI@(Y))1 Thus, we have here a symmetry transformation sending the hamiltonian H into - H . Clearly, I+(y)) cannot in general be an acceptable solution to the usual Hamilton eigenvalue equation, since I$(y)) will not obey the boundary condition l+(y)I2 4 0 if y -+ f o o . Indeed, hermiticity, normalization, and boundary conditions will not transform as in usual symmetry transformations. Yet, this symmetry is not totally void. If V = 0 , a state I$o) can be found that obeys both the boundary conditions at x 4 3x0 and y -, f o o . It is the ground state, +(x) = constant. It obeys both boundary conditions because of its invariance under transformations x 4 x a , where a can be any complex number. Because of our symmetry property, it obeys E = -E , so the energy of this state has to vanish. Since it is the only state with this property, it must be the ground state. Thus, we see that our complex symmetry may provide for a mechanism that generates a zero-energy ground state, of the kind that we are looking for in connection with the cosmological constant problem.
+
In general, if V(x) # 0 , this argument fails. The reason is that the invariance under complex translations breaks down, so that no state can be constructed obeying all boundary conditions in the complex plane. In our treatment of the cosmological constant problem, we wish to understand the physical vacuum. It is invariant under complex translations, so there is a possibility that a procedure of this nature might apply.
As noted by Jackiw , there is a remarkable example in which the potential does not have to vanish. We can allow for any well-behaved function that depends only on x4 = y 4 . For example, setting m = 1,
v(.)
= 2x6 - 3x2 = x2(2x4 - 3),
(24)
with ground state wavefunction exp(-x4/2) , indeed satisfies condition (21), which guarantees zero energy eigenvalue. Note that this restricts the transformation to be discrete, since otherwise it crosses the point x = &y
307
where the potential badly diverges. Boundary conditions are still obeyed on the real and imaginary axis, but not for general complex values, see figure 1.
Fig. 1. Region in complex space where the potential is well-defined; the shaded region indicates where boundary conditions are not obeyed.
Moreover, as Jackiw also pointed out, this example is intriguing since it reminds us of supersymmetry. Setting again m = 1 for clarity of notation, the Hamiltonian 1
H = -07 2
+ iW’)(p - iW’),
(25)
with W the superpotential and a prime denoting a derivative with respect to the fields, has a scalar potential
v = -21(W’W’ - W”).
(26)
If W satisfies condition (21), the Hamiltonian possesses a zero energy eigenfunction e-w , which obeys the correct boundary conditions in x and y . The Hamiltonian in this example is the bosonic portion of a supersymmetric Hamiltonian, so our proposal might be somehow related to supersymmetry. We need to know what happens with hermiticity and normalizations. Assume the usual hermiticity properties of the bras, kets and the various operators in x space. How do these properties read in y space? We have
y
x=x+
p=pt,
= -yt
Py = -P, t
7
308
but the commutator algebra is covariant under the transformation:
[p, XI = -i [I,Y’ Y1 = -i
=
-idlax ,
py = -idlay.
(28)
Therefore, the wave equation remains the same locally in y as it is in x , but the boundary condition in y is different from the one in x . If we would replace the hermiticity properties of y and py in Eq. (27) by those of x and p , then we would get only states with E 5 0 . 5. Harmonic Oscillator An instructive example is the x -+ y transformation, with x = i y , in the harmonic oscillator. The Hamiltonian is P2+ m w 1 H=-+ x 2 2 2m for which one introduces the conventional annihilation and creation operators a and u t :
.=E(.+%) ,
H = U(U+U
u t = E ( x - $ )
;
+ i) .
(30) (31)
In terms of the operators in y -space, we can write uy =
E(
y+
mw ’”)
,
Cy = - i ~,
uy = -iat
ay . =
g(”’)
H = -w(CYay
y-G
+ 51 ) .
= -uf
; (32)
(33)
If one were to replace the correct hermitian conjugate of ay by Cy instead of -Cy , then the Hamiltonian (33) would take only the eigenvalues H = -Hy = w ( - n - $) . Note that these form a natural continuation of the eigenstates w ( n $) as if n were now allowed only to be a negative integer.
+
The ground state, (0) is not invariant. In x -space, the y ground state would be the non-normalizable state exp(+$mwx2) , which of course would obey ‘good’ boundary conditions in y -space. 6. Second Quantization
The examples of the previous two sections, however, are not the transformations that are most relevant for the cosmological constant. We wish to
309
turn to imaginary coordinates, but not to imaginary oscillators. We now turn our attention to second-quantized particle theories, and we know that the vacuum state will be invariant, at least under all complex translations. Not only the hermiticity properties of field operators are modified in the transformation, but now also the commutation rules are affected. A scalar field @(z) and its conjugate, II(z) , often equal to &(x), normally obey the commutation rules
[rI(Z,t), a(.”, t ) ]= -i63(Z- Z/) ,
(34)
where the Dirac deltafunction 6(x) may be regarded as
(35) in the limit X T a. If 5 is replaced by iy’, with y’ real, then the commutation rules are
[n(ig,t), @(iy’!,
t ) ]= -i63(i(y’- GI)) ,
(36)
but, in Eq. (35) we see two things happen:
(i) This delta function does not go to zero unless its argument z lies in the right or left quadrant of Fig. 2. Now, this can be cured if we add an imaginary part to A , namely X 4 -ip , with p real. Then the function (35) exists if x = reie , with 0 < 8 < 4 7 r . But then, (ii) If z = i y , the sign of p is important. If p > 0 , replacing x = i y , the delta function becomes d(iy)
= ,/+e-ipyz
-+ -ib(y)
,
which would be +i6(y) had we chosen the other sign for p
(37)
.
We conclude that the sign of the square root in Eq. (35) is ambiguous. There is another way to phrase this difficulty. The commutation rule (34) suggests that either the field @(a, t ) or n(Z, t ) must be regarded as a distribution. Take the field 11. Consider test functions f(Z), and write
II(f, t ) Ef / f ( ? ) I I ( Z , t ) d 3 8 ;
[II(f,t ) , @(Z,t ) ]= -zf(Z). (38)
As long as Z is real, the integration contour in Eq. (38) is well-defined. If, however, we choose x = i y , the contour must be taken to be in the complex plane, and if we only wish to consider real y , then the contour must be along the imaginary axis. This would be allowed if II(Z, y) is holomorphic
310
Fig. 2. Region in complex space where the Dirac delta function is well-defined, ( a ) if X is real, ( b ) if p is real and positive.
Fig. 3. Integration contour for the commutator algebra (38), ( a ) and ( b ) being two distinct choices.
for complex 2 , and the end points of the integration contour should not be modified. For simplicity, let us take space to be one-dimensional. Assume that the contour becomes as in Fig. 3a. In the y space, we have
n(f, t ) ef
s_,
00
[Wf, t ) , q i y , t)l = -if(@),
f(iy)Wiy)d(iy);
(39)
so that
[ W i Y , t ) , @(id, t)l
=
-S(Y
- Y’)
.
(40)
Note now that we could have chosen the contour of Fig. 3b instead. In that case, the integration goes in the opposite direction, and the commutator algebra in Eq. (40) receives the opposite sign. Note also that, if we would be tempted to stick to one rule only, the commutator algebra would receive an overall minus sign if we apply the transformation II: + iy twice. The general philosophy is now that, with these new commutation relations in y -space, we could impose conventional hermiticity properties
311
in y-space, and then consider states as representations of these operators. How do individual states then transform from x -space to y -space or vice versa? We expect t o obtain non-normalizable states, but the situation is worse than that. Let us again consider one space-dimension and begin with defining the annihilation and creation operators u ( p ) and at@) in x -space:
po =
JW,
def
(px) =
5. z - pot,
Insisting that the commutation rules [a@),at(p')] = S ( p - p ' ) should also be seen in y-space operators: [ay(q), ky(q')l = S ( 4 - 4')
we write, assuming p o
=
>
(46)
-iqo and II = -id@/di- for free fields,
so that the commutator (46) agrees with the field commutators (40). In most of our considerations, we will have to take m = 0 ; we leave m in our expressions just to show its sign switch.
312
In x-space, the fields @ and T are real, and the exponents in Eqs (47)-(51) are all real, so the hermiticity relations are a t = ay and litY = ciy . As in the previous sections, we replace this by liY= a;
.
(52)
The Hamiltonian for a free field reads a 3
H
=i
[
dy ($II(iy)2 - $(&,@(i~))~
+ im2@(iy)2)=
J-00
-i/dqqo
+ $)
(ciy(q)ay(q)
=
-i/dqqo(n+
i).
(53)
Clearly, with the hermiticity condition (52) ,the Hamiltonian became purely imaginary, as in Section 2. Also, the zero point fluctuations still seem to be there. However, we have not yet addressed the operator ordering. Let us take a closer look at the way individual creation and annihilation operators transform. We now need to set m = 0, po = [pi, qo = I qI . In order to compare the creation and annihilation operators in real space-time with those in imaginary space-time, substitute Eqs. (47) and (48) into (44), and the converse, to obtain
The difficulty with these expressions is the fact that the x - and the y -integrals diverge. We now propose the following procedure. Let us limit ourselves to the case that, in Eqs. (50) and (51), the y-integration is over a finite box only: IyI < L , in which case a y ( q ) m will be an entire analytic function of q . Then, in Eq. (54), we can first shift the integration contour in complex q -space by an amount i p up or down, and subsequently rotate the x -integration contour to obtain convergence. Now the square roots occurring explicitly in Eqs. (54) and (55) are merely the consequence of a choice of normalization, and could be avoided, but the roots in the definitions of p o and qo are much more problematic. In principle we could take any of the two branches of the roots. However, in our transformation procedure we actually choose qo = -ipo and the second parts of Eqs. (54) and (55) simply cancel out. Note that, had we taken the other sign, i.e. qo = ipo , this would have affected the expression for @ ( i y , 27) such that we would still end up with the same final result. In general, the x -integration
313
yields a delta function constraining q to be k i p , but qo is chosen to be on the branch -ipo , in both terms of this equation ( qo normally does not change sign if q does). Thus, we get, from Eqs. (54) and (55), respectively, 4 P ) = i1/2a,(4)
a,(q)
= i-%(p)
0 - ‘ 0
, ,
4 = iP, 4 - ZP
p = -iq,
(56)
7
p 0 = -iq
0
,
(57)
so that u ( p ) and a,(q) are analytic continuations of one another. Similarly, J ( p ) = i1/2 iiy(q)
,
iiy(q)
= i-1’2
at ( p ) , p
=
-iq,
p o = -iqo
.( 5 8 )
There is no Bogolyubov mixing between a and at . Note that these expressions agree with the transformation law of the Hamiltonian (53). Now that we have a precisely defined transformation law for the creation and annihilation operators, we can find out how the states transform. The vacuum state 10) is defined to be the state upon which all annihilation operators vanish. We now see that this state is invariant under all our transformations. Indeed, because there is no Bogolyubov mixing, all N particle states transform into N particle states, with N being invariant. The vacuum is invariant because 1) unlike the case of the harmonic oscillator, Section 5, creation operators transform into creation operators, and annihilation operators into annihilation operators, and because 2) the vacuum is translation invariant. The Hamiltonian is transformed into -i times the Hamiltonian (in the case D = 2 ); the energy density 2’00 into -2‘00 , and since the vacuum is the only state that is invariant, it must have 2’00 = 0 and it must be the only state with this property.
7. Pure Maxwell Fields This can now easily be extended to include the Maxwell action as well. In flat spacetime:
S=-
1
J
d3x - F f i V ( ~ ) F p U ( x ) , F,, = a,A, ’
4
- &A,.
(59)
The action is invariant under gauge transformations of the form AP(4
+
AP(Z)
+ a,c(s).
(60)
Making use of this freedom, we impose the Lorentz condition a,AP = 0 , such that the equation of motion d,FP”” = 0 becomes OA, = 0 . As is well known, this does not completely fix the gauge, since transformations like
314
(60) are still possible, provided 05 = 0 . This remaining gauge freedom can be used to set V . A = 0 , denoted Coulomb gauge, which sacrifices manifest Lorentz invariance. The commutation relations are
[ E i ( zt, ) , A j ( d ,t ) ]= id$(?
-
."),
(61)
where
is the momentum conjugate to Ak , which we previously called II , but it is here just a component of the electric field. The transverse delta function is defined as
such that its divergence vanishes. In Coulomb gauge, equation CIA'= 0 , and we write
A'
satisfies the wave
where q p , A) is the polarization vector of the gauge field, which satisfies Z.$= 0 from the Coulomb condition V . x = 0 . Moreover, the polarization vectors can be chosen to be orthogonal Z(p, A) . Z(p, A') = 6 ~ x 1and satisfy a completeness relation
The commutator between the creation and annihilation operators becomes
in which the polarization vectors cancel out due to their completeness relation. In complex space, the field A , thus transforms analogously to the scalar field, with the only addition that the polarization vectors Z,(p) will now become function of complex momentum f . However, since they do not satisfy a particular algebra, like the creation and annihilation operators, they do not cause any additional difficulties. The commutation relations between the creation and annihilation operators behave similarly as in the scalar field case, since the second term in the transverse delta function
315
(63), and the polarization vector completeness relation (65), is invariant when transforming t o complex momentum. Thus we find
and again
TOOflips sign, as the energy-momentum tensor reads: T,,
= -F,,F,”
+ -41F f f f i F L 2 P ~ p , .
(68)
In terms of the E and B fields, which are given by derivatives of A , , Ei = Foi , Bk = i E i j k F j k , we have:
Too =
fr ( E 2+ B 2 )
-Too,
(69)
which indicates that the electric and magnetic fields become imaginary. A source term, J P A , , can also be added to the action (59), if one imposes J P + - J p , in which case the Maxwell equations a,FPv = J’ are covariant. Implementing gauge invariance in imaginary space is also straightforward. The Maxwell action and Maxwell equations are invariant under A,(x, t ) + A,(z, t) d,c(z, t ) ). In complex spacetime this becomes
+
A,(iy, i ~ 4 ) A,(iy, i ~ ) i a , ( y , ~ ) t ( i yi , ~ )
(70)
and the Lorentz condition
In Coulomb gauge the polarization vectors satisfy
with imaginary momentum q . Unfortunately, the Maxwell field handled this way will not be easy to generalize to Yang-Mills fields. The Yang-Mills field has cubic and quartic couplings, and these will transform with the wrong sign. One might consider forcing vector potentials to transform just like space-time derivatives, but then the kinetic term emerges with the wrong sign. Alternatively, one could suspect that the gauge group, which is compact in real space, would become non-compact in imaginary space, but this also leads to undesirable features.
316
8. Relation with Boundary Conditions
All particle states depend on boundary conditions, usually imposed on the real axis. One could therefore try to simply view the x -+ is symmetry as a one-to-one mapping of states with boundary conditions imposed on f x --$ m to states with boundary conditions imposed on imaginary axis fix 4 m . At first sight, this mapping transforms positive energy particle states into negative energy particle states. The vacuum, not having to obey boundary conditions would necessarily have zero energy. However, this turns out not to be sufficient. Solutions to the Klein-Gordon equation, with boundary conditions imposed on imaginary coordinates are of the form:
written with the subscript ‘im’ to remind us that this is the solution with boundary conditions on the imaginary axis. With these boundary conditions, the field explodes for real valued x -+ f m , whereas for the usual boundary conditions, imposed on the real axis, the field explodes for ix + f m . Note that for non-trivial a and 6 , this field now has a nonzero complex part on the real axis, if one insists that the second term is the Hermitian conjugate of the first, as is usually the case. This is a necessary consequence of this setup. However, we insist on writing ii = at and, returning to three spatial dimensions, we write for @im(z,t ) and IIim(z,t ) :
Po =
Jm ,
def
+
(px) = p . Z - p o t ,
(74)
and impose the normal commutation relations between a and at :
Using Eqn. (75), the commutator between becomes:
aim and
r I i m at equal times,
[Qim(Z),rIim(Z)]= S ( 3 ) ( Z - Z’),
(76)
317
which differs by a factor of i from the usual relation, and by a minus sign, compared to Eqn. (40). The energy-momentum tensor is given by
and thus indeed changes sign, as long as one considers only those contributions to a Hamiltonian that contain products of a and at :
However, the remaining parts give a contribution that is rapidly diverging on the imaginary axis
but which blows up for f x 00. Note that when calculating vacuum expectation values, these terms give no contribution. ---f
To summarize, one can only construct such a symmetry, changing boundary conditions from real to imaginary coordinates, in a very small box. This was to be expected, since we are comparing hyperbolic functions with their ordinary counterparts, sinh(z) vs. sin(x) , and they are only identical functions in a small neighborhood around the origin.
9. Conclusions
It is natural to ascribe the extremely tiny value of the cosmological constant to some symmetry. Until now, the only symmetry that showed promise in this respect has been supersymmetry. It is difficult, however, to understand how it can be that supersymmetry is obviously strongly broken by all matter representations, whereas nevertheless the vacuum state should respect it completely. This symmetry requires the vacuum fluctuations of bosonic fields to cancel very precisely against those of the fermionic field, and it is hard to see how this can happen when fermionic and bosonic fields have such dissimilar spectra. The symmetry proposed in this paper is different. It is suspected that the field equations themselves have a larger symmetry than the boundary conditions for the solutions. It is the boundary conditions, and the hermiticity conditions on the fields, that force all physical states to have positive energies. If we drop these conditions, we get as many negative energy as positive energy states, and indeed, there may be a symmetry relating positive energy with negative energy. This is the most promising beginning of an
318
argument why the vacuum state must have strictly vanishing gravitational energy. The fact that the symmetry must relate real to imaginary coordinates is suggested by the fact that De Sitter and Anti-De Sitter space are related by analytic continuation, and that their cosmological constants have opposite sign. Unfortunately, it is hard to see how this kind of symmetry could be realized in the known interaction types seen in the sub-atomic particles. At first sight, all mass terms are forbidden. However, we could observe that all masses in the Standard Model are due to interactions, and it could be that fields with positive mass squared are related to tachyonic fields by our symmetry. The one scalar field in the Standard Model is the Higgs field. Its self interaction is described by a potential V I ( @ = ) ;A(@+@ - F 2 ) 2 ,and it is strongly suspected that the parameter X is unnaturally small. Our symmetry would relate it to another scalar field with opposite potential: V2(@2)= - K ( @ 2 ) . Such a field would have no vacuum expectation value, and, according to perturbation theory, a mass that is the Higgs mass divided by d .Although explicit predictions would be premature, this does suggest that a theory of this kind could make testable predictions, and it is worth-while to search for scalar fields that do not contribute to the Higgs mechanism at LHC, having a mass somewhat smaller than the Higgs mass. We are hesitant with this particular prediction because the negative sign in its self interaction potential could lead to unlikely instabilities, to be taken care of by non-perturbative radiative corrections. The symmetry we studied in this paper would set the vacuum energy to zero and has therefore the potential to explain a vanishing cosmological constant. In the light of the recent discoveries that the universe appears to be accelerating , one could consider a slight breaking of this symmetry. This is a non-trivial task that we will have t o postpone to further work. Note however , that our proposal would only nullify exact vacuum energy with equation of state 20 = -1. Explaining the acceleration of the universe with some dark energy component other than a cosmological constant, quintessence for example, therefore is not ruled out within this framework. The considerations reported in this paper will only become significant if, besides Maxwell fields, we can also handle Yang-Mills fields, fermions, and more complicated interactions. As stated, Yang-Mills fields appear to lead to difficulties. Fermions, satisfying the linear Dirac equation, can be
319
handled in this formalism. Just as is the case for scalar fields, one finds that mass terms are forbidden for fermions, but we postpone further details to future work. Radiative corrections and renormalization group effects will have to be considered. To stay in line with our earlier paper, we still consider arguments of this nature to explain the tiny value of the cosmological constant unlikely to be completely successful, but every alley must be explored, and this is one of them.
References 1. S. Nobbenhuis, Found. Phys. 36, 2006; gr-qc/0411093. 2. D.E. Kaplan and R. Sundrum, A symmetry for the cosmological constant, hep-th/0505265. 3. C.M. Bender, Int. J. Mod. Phys. A20, 4646 (2005). 4. C.M. Bender, H.F. Jones and R.J. Rivers, Phys. Lett. B265, 333 (2005), hep-th/0508105. 5. C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998), physics/9712001. 6. Z. Ahmed, C.M. Bender and M. Berry, Reflectionless potentials and P T symmetry, quant-ph/0508117. 7. G. Bonelli and A.M. Boyarsky, Phys. Lett. B490, 147 (2000), hepth/0004058. 8. R. Erdem, Phys. Lett B621, 11 (2005), hep-th/0410063. 9. R. Jackiw, private communication 10. Supernova Search Team Collaboration, A.G. Riess et.al., Astron. J. 116, 1009 (1998), astro-ph/9805201. 11. Supernova Cosmology Project Collaboration, S. Perlmutter et.al., Astrophys. J 517, 565 (1999), astro-ph/9812133. 12. Supernova Search Team Collaboration, A.G. Riess et. al., Astrophys. J. 607, 665 (2004), astro-ph/0402512.
RICCI COLLINEATIONS IN BIANCHI I1 SPACETIME UGUR CAMCI Department of Physics, Art and Science Faculty, Canakkale Onsekiz Mart University, 17100 Canakkale, Turkey E-mail: [email protected]. tr In this study we classify the Ricci collineations (RCs) of Bianchi I1 spacetime according to the degenerate and non-degenerate cases of the Ricci tensor. It is shown that we have thirteen possibilities to be considered for the degenerate case, and found that there are generally infinitely many RCs whereas some cases give finite dimensional Lie algebras of the RCs which have three, four or five RCs. For the non-degenerate Ricci tensor cases, the Lie algebra of the obtained RCs are finite dimensional, in which the number of RCs is also three, four or five.
1. Introduction
A one-parameter group of conformal motions generated by a conformal Killing vector (CKV) form'
< is defined by L [ g a b = 2'$(x)gab, or in component
gab,c
+ sac<$ -k gcbJ:a
=
(1)
where Lt is the Lie derivative operator along the vector field <, and the indices a , b, c, ... run from 0 to 3, and "d'represents differentiation with respect to x a , and '$ = '$(xa)is a conformal factor. If $J;ab # 0, then the CKV is said to be proper. Otherwise, 6 reduces to the special conformal Killing vector (SCKV) if $J; abO, but $,a # 0. Other subcases are homothetic vector (HV) if = 0 and Killing vector (KV) if $J = 0. In most situations of physical interest, we have spacetime symmetries which further reduce the number of unknown functions2. The Einstein field equations (EFEs), Gab = R a b - Z1R g a b = & T a b , are a set of coupled non-linear partial differential equations for the ten unknown functions gab in the case of exterior equations (i.e. when the Ricci tensor R a b vanishes) plus other unknown functions such as the mass-energy density and pressure in the case of the interior equations (i.e. when the energy-
320
321
momentum tensor Tabis different from zero). Since the Einstein tensor Gab contains the Ricci tensor Rab, it is important to look at its symmetries. Beside the metric tensor gab appearing from the left hand side of the EFEs, the Ricci tensor R,b is an important tensor in that side. Then, the wellknown symmetry of the Ricci tensor is called the Ricci collineation (RC) defined' by JxRab = 0 , or
Rab,cXc+ RacX:b
f
RcbX:a = 0 ,
(2)
where X = X a & is the vector field generating the RC symmetry. The other important quantity appearing on the right hand side of the EFEs is the energy-momentum tensor Tab.Recently, much interest has been shown in the study of matter collineations (MCs) defined by EXTab = 0. The MCs and the RCs of the FRW metric have been studied by Camci and Barnes3. Tsamparlis and Apostolop~ulos~ have determined the RCs of Bianchi I space-time in the case of non-degenerate Ricci tensor. Camci and his collaborator^^^^ have classified the RCs of Kantowski-Sachs, Bianchi I and I11 spacetimes. A family of RCs of Bianchi 11, VIII, and IX spacetimes have been discussed by Yavuz and Camci7. The RCs and MCs of locally rotationally symmetric spacetimes are presented in reference [8]. Recently, we have classified the RCs in perfect fluid Bianchi V spacetimeg. Here we provide a complete RC classification of the Bianchi I1 spacetime according to the nature of Rab. In recent years, some important results about the Lie algebra of RCs have been given", which are the following: a. The set of all RCs on a manifold M is a vector space, but it may be infinite dimensional and may not be a Lie algebra. If the Ricci tensor R,b is non-degenerate, i.e. det(Rab)# 0 , the Lie algebra of RCs is finite dimensional. If the Rab is degenerate, i.e. det(R,b) = 0, we cannot guarantee the finite dimensionality of the RCs. b. If the R,b is everywhere of rank 4 then it may be regarded as a metric on manifold. Then, it comes out as a standard result that the family of RCs is, in fact, a Lie algebra of smooth vector fields on manifold M of finite dimension I 10 (and # 9). The paper is organized as follows. In the next section, we will give the Bianchi I1 spacetime and RC equations derived from that spacetime. In section 3, we will obtain a general classification of degenerate RCs for Bianchi I1 spacetime. In section 4, we will find all possible RCs for nondegenerate Ricci tensor of Bianchi I1 spacetime. Finally, in section 5, we
322
conclude with a brief summary of the results obtained.
2. Spacetime and Ricci Collineation Equations The line element for the spatially homogeneous Bianchi I1 spacetime is of the form2>12
ds2 = -dt2
+ A2dx2+ B2 [dy - zdzI2+ C2dz2,
(3)
where the metric functions A, B , and C are the functions of t only. This spacetime corresponds to the diagonal Bianchi I1 spacetime, and admits a group of isometries G3, acting on spacelike hypersurfaces, generated by the spacelike KVs
The Lie algebra has the following non-diagonal commutators:
The non-vanishing components of the Ricci tensor R,b are given by
R11 R23
R1, R22 R2, = - x R ~ , R33 x2R2 + f ,
where Rl(t),R2(t) and f ( t ) are defined as
R1=A2 R2
= B2
A
(
B
AB
AC
AB
BC
where the dot denotes derivative with respect to t. For the Bianchi I1 spacetime (3), using the non-zero Ricci tensor components (6)-(8), we can write the RC equations (2), generated by an arbitrary
323
vector field X a ( t ,x,y, z ) , in terms of R a ( t ) as follows:
+ 2RoXp, = 0 , k l X o + 2RlX,\ = 0 , R2X0 + 2R2F>,= 0 , (x2R2 + f) X o + 2xR2 (X’ - F,z)+ 2 f X;”z= 0 , z g 2 X o + R2 (X’ F,z + zF,,) - f X i = 0 , RoX: + RiX,iO, RoXP, + R2F,t = 0 , &XP, - x R ~ F + J f X: = 0,
RQXO
-
+
+ +
RiX,; R2 (F,z X 3 ) = 0 , RiX,; - x R (F,z ~ X 3 ) f X: = 0,
+
where we have defined F as F E X 2 - x X 3 . Then we find that det(Rab) = R1 R2 f . Therefore, we will study the RCs according to whether det(Rab) = 0 (degenerate case) or det(Rab) # 0 (non-degenerate case). Further, in this paper we will take the proper RCs to denote an RC which is not a KV, or a HV, or a SCKV. If the Rat, is non-degenerate, the standard results on KVs to deduce that the maximal dimension of the group of RCs in a pseudo-Riemannian manifold of dimension n is n(n 1 ) / 2 are valid. Thus, for spacetimes, the maximal dimensions of the group of RCs are 10. Therefore, the possible number of proper RCs for Bianchi I1 spacetime are only 1 , 2 , 3,4 and 7. Further, it is explicitly seen that the case of no proper RCs is possible.
+
3. Ricci Collineations for Degenerate Ricci Tensor Cases
For the degenerate Ricci tensor of Bianchi I1 spacetime, we have the following possibilities: (DO) all of the Re, (to,1 , 2 ) are zero; (Dl)-(D4) one of the Re and f are nonzero; (D5)-(D10) two of the Re and f are nonzero; (Dl1)-(D14) three of the Re and f are nonzero. Case (DO) corresponds to the vacuum in which every vector is an RC. Now, the solutions of the RC equations for Cases (D8) and (D11) are given the following. The remaining degenerate cases are summarized in Table 1 .
324
Case (D8). Ri
# 0 , & = 0 = f (i = 1,2). In this case, we have X
F - XF,, - -R:-a(~F,,, c1
+ F,,.)]
8,
+ F,,,)] a,,
+(xF,, + F,.)a, -
(22)
where Ri # 0, R2 = clRy ( a # 0 ) and F = F ( x ,y, z ) , and the following equations must be satisfied
x2F,,,
If
R1
and
+ F>,, + 2xF9,, = 0 ,
are arbitrary functions, then the RC vector is found as
R2
= IAl
+ Al,x] <(l)
-
A l , x t ( 2 ) -k a O t ( 3 )
(25)
where a0 is a constant, A1 = A l ( x ) , and E ( i ) l ~(i=1,2,3) are given in (4). When R1 is arbitrary and R2 = c2 (constant), the following RC is obtained
X
=
2R1
--Ao,,dt Ri -W 0 , Z
+ Ao& + [ZAO+ A1 - z(zAo,Z + Ai, )] d X
+ 4,)a,,
Y
(26)
where A0 = Ao(x),A1 = A1(x), and Rl # 0. Thus, we see that all subcases of this case give infinitely many RCs. Case (D11). Ri # 0 # f , & = 0. In this case, there exists an interesting situation where we have found the finite number of RCs in most of the subcases. When R1 # c1R2, f # C&1, Rl # 0 or R I C I Rf~#, C 3 R 2 or R1 = c1, # 0 or R1 = c1, R2 = c2, f # 0 , then the obtained RCs are only KVs given in (4). In some subcases we have found the following proper RC in addition to the KVs given by (4)
f
x = € 1 za, + -21 ( € 1 z2 where
€1
and
€2
-
€222)
a, - €2Xdz,
(27)
are constants related with the appeared constraints. If
c2f = clR1,Rl # 0 or f = clR1 and R2 = c2 or R1 = c1 and f = c2, then the constants € 1 and €2 take respectively the values €1 = c1 and € 2 = c2 or € 1 = c1 and € 2 = 1 or €1 = 1 and € 2 = c1/c2, where c1 and c2 are nonzero constants. When R1 = c1, R2 = c2 and f = cg, we find R2 =
infinitely many RCs as follows X(1) = < ( 1 ) ,
X ( 2 ) = E ( 2 ) l X(3) = t ( 3 )
325 Table 1. The RCs of Bianchi type I1 spacetimes in degenerate Ricci tensor cases. In this table c 1 . c ~and cs are non-zero constants related with the components of the Rab; ao,ai,az,ag,a4 and a5 are integration constants, and we have used the transformation d7 = m d t in s@mecases.
D1 DZ
D3
D4
D5 D6
D7
D9
D10
D12
D13
D14
326
where ((11, Q2) and J(3) are KVs given by (4);€1 = l/cl and ~21/c3. For the case R1 # 0, R1 = c2 fl/(o-l) and R2 = c l R f , it follows from the solution of RC equations (12)-(21) that the proper RC is
--at2Ri + xax + pyay + (P - 1 ) Z a ,
X(4) =
Ri
(29)
where /3 is a constant and different from one or two, and the Lie algebra is given by [ x ( i ) x(4)] , =
= x(i),
[x(2),
(P - 1)x(2),[x(3), x(4)] = x(3). (30) = c1R1, f = C3R1 and R1 # 0, the number of RCs
[x(2), x(4)] =
When p = 1, i.e., R2 becomes infinite, in which the components of RCs are given by
where F = F(x, y, z ) , G = G(x, y, z ) , and the following constraint equations have to be satisfied
k
+ ClF,XX) = 0, k G xF,y F , z + ( G , y y + ClF,zy) = 0, x G , y y + G , y z + c1 ( F , x z + x F , x y + F , y ) = 0. G,z + X G , , - - ( F , y y c1
-
-
C1
For R1 = c2 f and R2
= clR:,
X(4) =
i.e.
--at2Ri R1
x(5)clzdx
P = 2, the proper
(31)
(32)
(33)
RCs are given by
+ xax + aya, + za,,
+ -21 (E1z2
-
€2x2)a y
€2Xd~,
-
(34)
where €1 = 1 and €2 = c2. The corresponding Lie algebra has the following non-vanishing commutators:
[X(1)7 X(4)] = 2X(i)7
[Xp)7 X(3)] = X(I),
[X(3) X(4)] = x(3)
[x(3) , x(5)]= -c2x(2).
9
3
[X(2),X(4)] = X(2),
(35)
If R1 = c1 and R2 = af , where a is a constant, then we get X(4) =
where R z
--at2R2 R2
+yay
+ za,,
(36)
# 0, and the Lie algebra is given by , [Xp), x(3)]= x ( i ) ,
[X(i) x(4)] = x(i)
[X(2),X(4)] = x(2).(37)
327
If RI = b/f and R2 = c2, where b is a constant, then we have
where tators
R1
# 0, and the Lie algebra has the following non-vanishing commu-
[x(z), x(3)] =x(i)
[ X ( z ) x(4)] , = x(2).
(39)
Thus, we have a finite number of RCs in most of the subcases of this case, even if the Ricci tensor is degenerate.
4. Ricci Collineations for Non-degenerate Ricci Tensor Cases In this section, we consider the non-degenerate cases of Rab, i.e. det(Rab) # 0, admitted by Bianchi I1 spacetime. Now, we define a set { & ( t ) , f ( t ) } for functions Re and f , where C = 0,1,.2. Thus, we have the following possibilities: (NDl)-(ND4) three elements of the set are zero; (ND5)(ND10) two elements of the set are zero; (NDll)-(ND14) one element of the set is zero; (ND15) all elements of the set are zero. Before giving the solutions of these cases, we write the constraint equations appearing in the classification as
where c takes values 1 or co (a constant related with Ro), and a ,/? are separation constants. If E = 1, then we use the transformation d r = m d t . Otherwise, i.e. when E = co(# l),then & becomes a constant (= C O ) , and the conformal time 7 is equivalent to the physical time t. If a2 and p2 are zero, then it follows from the constraint equations (40) that R1cle2’7‘ and R2 = c z e 2 p 7 , where 17 and p are integration constants. The possible cases and the number of RCs for Bianchi I1 spacetime are given Table 2 . For the cases (ND2), (ND4), (ND5), (ND7), (ND8), (ND10) and (ND14), the KVs is only found from RC equations, which are non-proper RCs given by (4). In cases (ND3), (ND6) and (ND9), and some subcases of (ND11; where Ro = co = E , a2 = O D 2 , f = ~ 3 e ’ ’ 7 ~ # , p 217) and (ND13; where E = 1, a2 = 0 , R2 = c2, f = c3e2q7), we have only one proper RC X(4)=
;l)
z - - a, -ax + -2 (z2 ;az, X C1 c1 -
328 Table 2. The possible cases and the number of RCs of Bianchi I1 space~ the times for non-degenerate Rab. Here, we have used C O , C ~ , C Z , Cas non-zero constants related with the components of Ricci tensor. Case ND1 ND2 ND3 ND4 ND5 ND6 ND7 ND8 ND9 NDlO NDll
ND12 ND13 ND14 ND15
fl of RCs
Constraints
Ro # 0, Ri
2-A2
= c i , R2 = ~ 2 fc3, , C2 # Ro = CO , R2 = C Z , f = c3
Ri # 0,
R 2 # O h = C O , R1 = c1, f = c3 f # 0, Ro = CO, Ri = ci, R2 = c2 Rk # 0, R2 = c2, f = c3 (k = 0 , l ) Rj
# 0, Ri
= ci f = ~3 ( j = 0 , 2 )
&I # 0 # f, RI = c i , R 2 ~ 2 Ra # 0, Ro = co, f = c3 (i = 1 , 2 ) Ri # 0 # f , RO= co, R2 = c2 R2#O#f,Ro=co,Ri=ci R3 # 0 # f , Ro = co = E, a2 = op2 R1 = cle2qt, R2 = c2e2qt, f = c3e2qt,p = 21) R1 = cle2qt, R2 = c2e2Pt, fc3e2qt, p # 21) &I # 0, R1 = c l , R2 = c2e2Pr, f = c3e2Pr E = 1, p2 = o &I # 0, R1 = cle2qr, R2 = c2, f = c3e2qr E = 1, a2 = o Rt # 0, f = c3 (e = 0 , 1 , 2 ) ) Ro = CO, Ri = ~ 1 R2 , = c2, f = c3
5 3 4 3 3 4 3 3 4 3 5 4 4
4 3 5
where c1 and c3 are non-zero constants, and C2 # E A 2 . For the latter case, the Lie algebra has the following non-vanishing commutators
If C2 = ?A2, then the vector field (41) is not a proper MC but actually a KV and the Bianchi I1 spacetime is reduced to the LRS spacetime (see e.g. Refs. 2 and 13) For the case (ND12; where E = 1, p2 = 0, R1c1, f = c3e2pT),a proper RC is found as x(4)=
aT
- pya,
-
pZaZ
(43)
with the non-vanishing commutators
In the cases (NDl), (ND11; where Ro = E = CO, a' = 0 = p2, f = c3e2vt, p = 217), and (ND15), we have found two extra RCs. In the cases
329
(ND1) and (ND15), one of these extra RCs is given by (41) and the other one is
X(5) = a,
(= -at) 1
m
(45)
for [NDl), which is a proper RC, or X(5) = at for (ND15), which is a KV. For the case ( N D l l ) , in addition to the fourth RC given by (41), the fifth RC is obtained as X(5) =at - v x a , -
277Ydy
-
vzaz,
(46)
where 77 = p/2. For the last case, the non-vanishing commutators of the Lie algebra are given by
1 TX(3). c3 Furthermore, for the case ( N D l l ) , it has to be noted that the vector fields (41) and (45) are not proper RCs but they are KVs, and we have -x(2),
A = fieqt,
B
= kzeZqt,
C = fieqt.
(47)
where kz is a constant of integration.
5. Conclusions and Discussion In this paper, we have solved the RC Eqs.(12)-(21) for Bianchi I1 spacetime (3), and obtained all possible RCs according to the degenerate or non-degenerate Rat,. We have found that if the R,b is degenerate (section 3), then there are many cases of RCs for the Bianchi I1 spacetime with infinite degrees of freedom except for most of the subcases of ( D l l ) , the Lie algebra of RCs are finite dimensional, in which there are one or two proper RCs given by (27), (29), (34), (36) and (38). When the Rab is nondegenerate, section 4, we have obtained finite dimensional Lie algebras of RCs which are three, four and five. Therefore, the number of proper RCs in non-degenerate Rat,cases are one or two,which are given by (41), (43) and (45). In some cases of sections 3 and 4, the results are given in terms of Ro and some integration constants together with differential constraints related to the components R1, Rz, and f which must be satisfied. Also, in any case of degenerate or non-degenerate cases of the Ricci tensor, we have also obtained different constraint equations. If we could solve these constraint equations, it would be possible to find new exact solutions of EFEs.
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References 1. G. H. Katzin, J. Levine, and W. R. Davis, J. Math. Phys. 10617 (1969). 2. D. Kramer, H. Stephani, M. A. H. MacCallum, and E. Herlt, Exact Solutions of Einstein Field Equations, (Cambridge Univ. Press, Cambridge, 1980). 3. U. Camci and A. Barnes, Class. Quant. Grav.19393 (2002). 4. M. Tsamparlis, and P. S. Apostolopoulos, J. Math. Phys.417543 (2000). 5. U. Camci, I. Yavuz, H. Baysal, I. Tarhan, and I. Yilmaz, Int. J . Mod. Phys. D10 751 (2001). 6. U. Camci and I.Yavuz, Int. J. Mod. Phys. D l 2 89 (2003). 7. I. Yavuz, and U. Camci, Gen. Rel. Grav. 28 691 (1996). 8. M. Tsamparlis and P. S. Apostolopoulos, Gen. Rel. Grav. 36 47 (2004). 9. U. Camci and I. Turkyilmaz, Gen. Rel. Grav.36 2005 (2004). 10. J. Carot, J. d a Costa, and E. G. L. R. Vaz, J. Math. Phys. 35 4832 (1994). 11. G. S. Hall, I. Roy, and E. G. L. R. Vaz, Gen. Rel. Grav. 2 8 299 (1996). 12. S. R. Roy and S. K. Banerjee, C1ass.Quantum Grav. 14 2845 (1997). 13. G. F. R. Ellis and M. A. H. MacCallum, Commun. Math. Phys. 12, 108 (1969).
ON THE GAUSS-BONNET GRAVITY NARESH DADHICH IUCAA, Post Bag
4, Ganeshkhind, Pune
411 007, INDIA E-mail:nkd@iucaa. ernet. an
We argue that propagation of the gravitational field in the extra dimension is motivated by physical realization of second iteration of self interaction of gravity and it is described by the Gauss-Bonnet term. The most remarkable feature of Gauss-Bonnet gravity is that at high energy it radically transforms radial dependence from inverse to proportionality as the singularity is approached and thereby makes it weak. A similar changeover also occurs in the approach t o the singularity in loop quantum gravity. It is analogous to Planck’s law of radiation where a similar change occurs for high and low energy behavior. This is how it seems t o anticipate in qualitative terms and in the right sense the quantum gravity effect in 5 dimensions where it is physically non-trivial. The really interesting question is, could this desirable feature be brought down to the 4-dimensional spacetime by dilatonic coupling to the Gauss-Bonnet term or otherwise?
The most distinguishing feature of gravitation is that it is universal and hence links to everything that physically exists including massive as well as massless particles and above all with itself. Its linkage to massless particles can only be negotiated through curved space2. That means gravity must curve spacetime and its dynamics has thus t o be entirely determined by the spacetime curvature, the Ftiemann curvature tensor. We have no freedom to make any prescription, Newton’s law should follow from the Ftiemann curvature. It indeed does through the Bianchi differential identity satisfied by the curvature tensor. It leads to the Einstein equation which contains the Newton’s law in the limit2. The Einstein equation so obtained naturally contains the so called cosmological constant A in a natural way as a constant of integration without any reference to cosmology. It comes on the same footing as the matter tensor in the equation and is indeed a new constant of the theory. It really indicates the distinguishing feature of gravitation that here the spacetime
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332
background was not fixed as was the case for the rest of physics but was dynamic describing gravitational force. A is the measure of this property. The Einstein equation is valid in all dimensions where Ftiemann curvature is defined,i.e. n 2 2. It is well-known that in dimensions < 4, it is not possible to realize the free field dynamics. Thus we come to the usual 4-dimensional spacetime. That means 4 dimensions are necessary for description of gravitational field. The question is, are they sufficient too? Let us now turn to the property of self interaction of gravitation which can be evaluated only by an iteration process3 a. The spacetime metric is the potential for the Einstein equation which contains its second derivative and square of the first derivative. It thus contains the first order iteration through the square of the first derivative. The natural question that arises is, how do we stop at the first iteration? We should go to second and higher orders as well. The basic entity at our disposal is the Ftiemann curvature, so should we square it and add to the usual Einstein-Hilbert action of the Ricci scalar? This will also square the second derivative which is the highest order of derivative. If the highest order of derivative does not occur linearly in an equation, then there will be more than one equation, and the question of having a unique solution does not arise. The property that highest order of derivative occurs linearly is known as quasilinearity. Is it then possible to have higher powers of first derivative yet with the second derivative remaining linear? Yes, the differential geometry offers a particular combination, known as the Gauss-Bonnet (GB) given by RabcdRabcd- 4RabRab R2, which ensures the quasi-linearity character of the resulting equation. This particular combination cancels out the square of the second derivative. However it turns out that this term makes no contribution in the Einstein equation for dimension < 5. We are thus forced to go to the extra 5th dimension for the physical realization of second iteration of self interaction of gravity. This is an important conclusion we have reached simply by hooking onto the iterative realization of self interaction.
+
Now the question arises, where does this iteration process of going to higher dimensions stop? If all the matter fields are confined to 3-brane/space, the 5-dimensional bulk is completely free of matter and aThere is a long and distinguished history of deriving the Einstein equation from Newtcnian gravity through perturbative inclusion of self interaction (see [4]).However the self interaction we are referring t o here is the inherent property of the Einsteinian gravity and which is evaluated in the dynamic curved spacetime framework.
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hence it is homogeneous and isotropic in space and homogeneous in time, and thereby maximally symmetric. It is therefore of constant curvature, an Einstein space with vanishing Weyl curvature. That means there is no more free gravity to propagate any further in the higher dimension. The iteration chain thus naturally terminates at the second iteration in 5-dimensional bulk for matter fields living on the 3-brane. In how many dimensions should matter live has however to be determined by the dynamics of matter fields. The gravitational dynamics in the 5-dimensional b ~ l k ~including > ~ , the Gauss-Bonnet terms, is described by
GAB= Q H A B - A g A B , where GAB= RAB- i R g A B , and
HAB = -2 (RRAB- 2 R ~ c R g- 2 R C D R ~ ~ ~ ~ 1 +RzDERBCDE) T g A B (R2
+
-
~
R
+~R C D~ E FR R ~~~ ~ ~~ ) . (1)
Here a is the parameter coupling the Einstein-Hilbert action with the GB term. It is easy to see that the condition of constant curvature solves this equation to give an Einstein space with redefined A given by
which in the first approximation reduce t o A and -A- $ < 0 for a > 0. The first case with +ve sign has the a 0 limit leading to the Einstein case. It is flat when A = 0, which means GB contribution has no independent existence. It comes only as a correction riding on A. In the second case with -ve sign, there is no Einstein limit and the effective X is always negative leading to Ads. It exists on its own even when A vanishes and cannot be switched off. This suggests that its source is not sitting in the bulk. These two cases clearly indicate that they refer t o two different situations, and what could they be is what we consider next. --f
The GB contribution could arise in two different ways. One is when we study the most general action giving rise t o the quasi-linear equation for gravitation in 5-dimensional spacetime. In this case GB represents the higher order correction and Einstein gravity results in the a -+ 0 limit.
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This is the case for the +ve sign solution. Second, is when the GB term is produced by the second order iteration of the self interaction of the gravitational field whose source is sitting in the 3-brane. It is purely free gravity leaking into the bulk from the brane that is the source of the GB term in the bulk and which cannot be switched off in the bulk. This is the case for the -ve sign solution. As argued earlier, bulk spacetime could either be dS or Ads. Since it is free gravity that propagates in the bulk and has negative energy density, hence it would generate AdS rather than dS. This demands that the GB parameter a must be positive t o give Ads. We thus end up with a scenario similar t o the Randall-Sundrum braneworld model (RS)7 purely from classical considerations without any reference t o string theory. Here AdS bulk is not an assumption but follows from the property of gravity. It is therefore no surprise that AdS bulk thus sourced through GB term will also localize gravity on the brane718, and of course it will have no a 4 0 limit in the bulk. These are the two different situations indicated by the f solutions and they get further resolved when a mass point is introduced. Introduction of a mass point in this setting is described by the wellknown Boulware-Deser s o l u t i ~ given n ~ ~ by ~~ ds2 = -Adt2
+ A-'dr2 + r2dRi,
(3)
f
(4)
where
r2 A = 1 - -[-1 2a
Here M is the mass term which has dimension of L 2 , and the two solutions are distinguished by f signs. Let us term the +ve sign solution for which the limit a -+0 exists as the bulk solution (BS) while the -ve sign one, which has no a -+ 0 limit, is denoted as the brane-bulk solution (BBS). The term under the radical sign must be positive, which will be so for a M > 0, ah > 0. For the BS, M > 0 and consequently a > 0 will correspond to the usual attractive gravity in the bulk while it would be repulsive for the BBS case unless we reverse the sign of M and a. Note that the metric is nowhere
+ &.
singular and as r -+ 0 it tends to A 4 1- (f@) In the limit r = 0, A # 1 and hence it is not flat but represents a spacetime asymptotically approximating to a global monopole with a solid angle deficit''>l2. The approach to the limit is however through Ads. When M = 0, the limiting space is Minkowski flat. Our main aim is to probe GB gravity and hence we shall now set A = 0, which does not play any critical role.
335
We define the equivalent Newtonian potential, Q, = ( A - 1)/2, which leads to a gravitational force given by
For large r this approximates to the familiar 5-dimensional Schwarzschild for BS while for BBS it is anti-Schwarzshild-Ads unless both M , Q are -ve, when it would be Schwarzschild-dS. For smaller r , it goes as -5 f O ( T ~which ), shows that the approach t o the centre r = 0 is always through Ads. This demonstrate the remarkable effect of GB contribution which transforms the radial dependence of gravity, from inverse to proportional. This is why the singularity structure is radically altered13. The central singularity is however weak because the Kreschmann scalar (square of Riemann curvature) diverges only as rP4. That means energy density will diverge as T - ~which on integration over the volume will vanish as T + 0. This is because a t the singularity the metric approximates t o that of a global monopole12 for which this is the characteristic behavior. Thus GB contribution, which would be dominant a t high energy as singularity is approached, results in smoothening and weakening of the singularity. This is done not by gravity altering its sense, attraction to repulsion, but by its behavior transforming from inverse square to proportional to r. The BS solution has the Einstein limit A = 1- $,( M = m2)which is the 5-dimensional Schwarzschild solution. Note that, in the first approximation, there is no GB contribution and further the higher order contribution comes as riding on M . It is Minkowski flat when M = 0, hence GB contribution has no existence of its own and it comes only as a riding correction. It has horizon at r: = m2 - Q which will exist only if m2 2 a , else it will be a naked singularity. Here Q behaves like electric charge in the Reissner - Nordstrgm solution for a charged black hole. Thus the singularity structure of the BS solution would be similar to the Reissner - Nordstrom. It is quite interesting that asymptotically Q has no effect while at the horizon it behaves like a “charge”. Though GB contribution comes in this case only as rider yet its effect becomes dominant as the horizon is reached and it radically changes the horizon and singularity structure13. The BBS solution with M = m2,a > 0, for large r approximates to
336
+
+ g,
A =1 $ which is AS-Ads. Note that the mass point is repulsive. We could however reverse the situation by taking A4 = -m2, Q < 0, then it would be S-dS. Here the GB contribution comes from gravity leaking from the brane into the bulk and that produces a spacetime of negative constant curvature, which is Ads. That is why it will always exist on its own and can not be switched off unless one switches off gravity entirely in the brane. Clearly, there is no horizon and there is only a weak naked singularity.In this case, the background is set up by the gravitational field leaking from the brane into the bulk which should generate an Ads and hence Q must be positive. The fact that the addition of a mass point in this setting produces repulsive gravity is the most remarkable and intriguing feature which we do not quite understand. Our main purpose here was to bring forth and highlight the critical role GB contribution plays. It is however non-trivial only in 5 or higher dimensions. GB gravity arises in two different ways. On the one hand, for n > 4 dimensions, it should be included in the most general action leading to second order quasi-linear equation. It is thus a higher order correction which can not stand all by itself but rides on matter and A in the higher dimensional spacetime. On the other hand, GB term could be sourced by free gravity leaking from the 3-brane into the bulk as second iteration of self interaction. This exists all by itself and generates an Ads in the bulk. It can not be switched off t o give Q 4 0 limit simply because its source is not sitting in the bulk but instead in the brane. The bulk is free of matter and hence it is a maximally symmetric space of constant curvature which is negative because it is solely produced by free gravitational field having negative energy density. That is why bulk spacetime has t o be an Ads and not dS. Note that it is not an assumption but follows from the basic character of the gravitational field. On the other hand, in the RS model Ads bulk is required for localization of gravity7is. Further Ads is also favoured in a very recent investigation of geodesics and singularities in higher dimensional spacetime14. Also note that we have obtained RS model like scenario purely from classical consideration without any reference t o string theory. We are driven to the 5-dimensional bulk simply by the physical realization of second order iteration of self interaction of gravity. What the second iteration essentially does is to produce a constant negative curvature in the bulk. A spacetime of constant curvature however solves the equation (1). The most interesting case is the BBS where there is a gravitational shar-
337
ing of dynamics between brane and bulk. In this case, there never occurs a horizon irrespective of whether we have the AS-Ads with M > 0, Q > 0 or S-dS with A4 < 0, Q < 0. In the braneworld gravity, Ads bulk is required for localization of gravity on the brane. It has recently been shown that a black hole with sufficiently large horizon on the bulk will delocalize gravity by sucking in zero mass gravitons15. A mass point in the GB setting presents a variety of possibilities as there occurs no horizon at all for BBS and even for BS it could be avoided for m2 < a. The absence of horizon altogether in the BBS case is perhaps indicative of the fact that localization of gravity on the brane would continue to remain undisturbed by the introduction of a mass point in the bulk. This is perhaps because our interpretation of the BBS is solely guided by the dynamics of gravitational field. In this way, GB could therefore play a very important and interesting role in localizing as well as stabilizing braneworld gravity 16. The most distinguishing and characteristic feature of the GB gravity is the negative constant curvature background which manifests as Ads, and its dominance over the mass at high energy as r 4 0 is approached. Asymptotically as r 4 00, the field goes as rm3for BS and as Ads r - 3 for BBS. At the other end, r -+ 0, it goes proportional to r. At high energy, gravity effectively changes its radial dependence from inverse to proportionality. This is what is responsible for the smoothening and weakening of the singularity (Similar indications also emerge when we consider dust collapse in the GB setting17). This makes a crucial difference in gravitational dynamics at high and low energy. It is something analogous t o Planck's law of radiation which has similarly different behavior at high and low energy. In loop quantum gravity, apart from gravity turning repulsive, there also occurs a similar change at high energy as the singularity is approached both in cosmology and black hole; density transforming from inverse power t o positive power of the scale factor and radius respectively1s-20. The GB term, which also arises as one loop contribution in string theory21)22,seems t o anticipate some aspects of quantum gravity effects at least qualitatively. Thus it could rightly be considered as intermediate limit of quantum gravity. It could be thought of, in another way, as a pointer to quantum gravity effects. In the context of loop quantum gravity, we should rather ask for GB gravity as its intermediate limit and so Ads rather than flat space. That is the limiting continuum spacetime to loop quantum gravity t o be rather 5-dimensional Ads than 4-dimensional flat space. This is the suggestion which naturally emerges and hence deserves serious further consideration.
+
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Very recently there has been an attempt to see a connection between loop inspired and braneworld cosmology23. It is shown that the effective field equations in the two paradigms bear a dynamical correspondence. There appear t o be a resonances of this in some other calculations as well24. Such a bridge between the two approaches to quantum gravity is quite expected and most desirable as the two refer t o complementary aspects. In this perspective, the GB term could also be seen as indicative of a similar bridge between the two approaches. It is quite rooted in the string paradigm through the first loop contribution as well as in the braneworld paradigm. It mimics features similar to that of loop quantum calculations in the high energy regime when the singularity is approached. Our paradigm makes a very strong suggestion for the intermediate semi-classical limit t o the loop quantum gravity as AdS 5-dimensional spacetime rather than 4-dimensional flat spacetime. This is a clear prediction. There have been several considerations of higher order terms including GB and GB coupled to dilaton in FRW cosmology (see for example [25, 261). There, higher order terms act as a matter field in the fixed FRW background simply modifying the Friedman equation. It is only a prescription while here we have a true second order quasi-linear equation t o be solved to determine the spacetime metric. The two situations are quite different. The former is an effective modification of the Einstein’s theory while the latter is the natural generalization demanded by the dynamics of gravity. It turns out that GB thus has a determining say a t high energies. However all this happens in 5 dimensions where GB attains a non-trivial physical meaning. It certainly points in the right direction that quantum gravity effects would at the very least weaken the singularity if not remove it altogether. The most pertinent question is, could this desirable feature of weakening of the singularity be brought down t o 4 dimensions through dilaton scalar field coupling to the GB term27y28or otherwise? Very recently, a new black hole solution has been found2’ in which effects of GB and KaluzaKlein splitting of spacetime are manifest in 4 dimensions. What happens is that GB weakens the singularity and regularizes the metric while KaluzaKlein modes generate the Weyl charge as was the case for one of the first black hole solutions on the Randall-Sundrum brane described by a charged black hole metric3’. It is remarkable that the new solution asymptotically does indeed approximate t o the black hole on the brane.
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Acknowledgements I wish to t h a n k Atish Dabholkar and Parampreet Singh for some clarifying comments and references.
References 1. 2. N. Dadhich, Subtle is the Gravity, gr-qc/0102009; Universalization as a Physical Guiding Principle, gr-qc/0311028. 3. N. Dadhich, Probing Universality of Gravity, gr-qc/0407003; Universality, Gravity, the enigmatic Lambda and Beyond, gr-qc/0405115; M. Sami and N. Dadhich, Unifying Brane World Inflation with Quintessence, hep- th/04050 16. 4. S. Deser, Gen. Relativ. Grav. 1, 9 (1970); T.Padmanabhan, gr-qc/0409089. 5. N. Deruelle and J. Madore, On the quasi-linearity of Einstein-Gauss-Bonnet Gravity Equation, gr-qc/0305004. 6. D. Lovelock, J . Math. Phys. 12,498 (1971). 7. L. Randall and R.Sundrum, Phys. Rev. Lett. 83,4690 (1999). 8. J. Garriga, T. Tanaka, Phys. Rev. Lett. 84,2778 (2000). 9. D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). 10. J. T. Wheeler, Nucl. Phys. 268,737 (1986). 11. M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341 (1989). 12. N. Dadhich, K. Narayan and U. A. Yajnik, Pramana 50,307 (1998). 13. T. Torii and H. Maeda, Spacetime Structure of Static Solutions In GaussBonnet Gravity: neutral case, hepth/0504127. 14. E. Anderson and R. Tavakol, Geodesics, the Equivalence Principle and Singularities in higher dimensional General Relativity and Braneworlds, grqc/0509054. 15. S.S. Seahra, C. Clarkson and R. Maartens, Class. Quant. Grav. 22, L91 (2005). 16. N. Dadhich, T Naskar, S.R. Chodhary and M. Sami, Work in progress. 17. N. Dadhich, S.G. Ghosh, D. W. Deshkar, Work in progress.. 18. M. Bojowald, Elements of Loop Quantum Cosmology, in 100 Years of Relativity - Space-time Structure: Einstein and Beyond, ed. A. Ashtekar (World Scientific), gr-qc/0505057. 19. M. Bojowald, Phys. Rev. Lett. 95, 061301 (2005), gr-qc/0506128. 20. M. Bojowald, R. Goswami, R. Maartens and P. Singh, Phys. Rev. Lett. 95, 091302 (2005), gr-qc/0503041. 21. B. Zwiebach, Phys. Lett. B156,315 (1985). 22. A. Sen, How does a fundamental string stretch its horizon 1 , hepth/0411255. 23. E. J. Copeland, J. E. Lidsey and S. Mizuno, gr-qc/0510022. 24. Parampreet Singh, Private Communication. 25. S. Nojiri and S. D. Odintsov, hepth/0006232; S. Nojiri, S. D. Odintsov and S. Ogushi, hep-th/0205187.
340 26. M. Sami, N. Savchenko and A. Toporensky, hepth/0408140; S.Tsujikawa, M. Sami and R. Maartens, astro-ph/0406078. 27. D.G. Boulware and S. Deser, Phys. Lett. B175,409 (1986). 28. N. Dadhich and T. Naskar, Work in progress. 29. H. Maeda and N. Dadhich, Kaluza-Klein black hole with negatively curved extra dimensions in string generated gravity models, hep-th/0605031. 30. N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Phys. Lett. B487,l(2000).
GEOMETRY AND SYMMETRY IN GENERAL RELATIVITY GRAHAM HALL Department of Mathematical Sciences, King’s College, University of Aberdeen Aberdeen AB.24 3UE, U.K. E-mai1:g. hallomaths. abdn.ac. uk
The purpose of this paper is to present a short survey of the mathematical and geometrical ideas behind the theory of symmetry in Einstein’s general relativity. Although most proofs are omitted, the essential ideas and definitions are given and discussed and the main results stated.
1. Introduction The study of symmetry in general relativity has always been an important aspect of Einstein’s gravitation theory. In practice one first describes the symmetry in question in local geometrical terms and then reformulates it as a system of differential equations involving the space-time geometry and a certain vector field. Although these differential equations turn out to be rather useful for calculation, their interpretation is not always transparent. The geometrical description, on the other hand, whilst clumsy for certain operations, is clear and often has great merit regarding quick and elegant solutions to certain problems. In this talk I will describe the geometrical features of various symmetries in general relativity and try to show how they arise naturally and simply from a certain collection of local diffeomorphisms on space-time. I will also describe in a geometrical fashion the important concepts of orbit and isotropy as they arise in symmetry theory. The general approach will be to explain the basic ideas in a somewhat informal setting, but without loss of rigour. Proofs will largely be omitted but references for them will be provided. The basic idea behind Einstein’s general relativity theory will be assumed. In particular, ( M , g ) will denote a space-time where M is a 4-
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dimensional connected smooth Hausdorff manifold and g a smooth metric on M with Lorentz signature (-,+,+,+). The unique symmetric LeviCivita connection on M arising from g is denoted by V and its associated Christoffel symbols in some (any) coordinate system by Fgc. The type (1,3) curvature tensor arising from V is denoted by Riem and has components R a b c d . The associated Ricci tensor is denoted by Ricc and has components R,b = R C a c b and the Ricci scalar is R = R , b g a b . A covariant derivative with respect to V is denoted, in a component representation, by a semi-colon, whilst a partial derivative is denoted by a comma. The symbol L denoteds a Lie derivative. In component notation, square brackets denote the usual skew-symmetrisation of indices.
2. The Geometry of Symmetry The essential idea of a symmetry in space-time is of a local diffeomorphism f : U -+ V where U and V are open subsets of M and f is a bijective map with f and f - l smooth. This map must then be related to some geometrical object on M in a special way, thus becoming a %ymmetry” of it. For example, if U is a coordinate domain of M with coordinates x” then f naturally leads to V being a coordinate domain with coordinates y” = xa o f-’ (since M and f are smooth). One could now pronounce f a symmetry of some tensor T on M if the components of T at each p E U and in the coordinate system xa are numerically the same as the components of T at f ( p ) E V in the coordinate system y”. The role here played by f is crucial. It is through f that one tries to give U and V “similar” coordinate systems and then the tensor component equality for T is stipulated at the f-related points p and f ( p ) . Of course, one need not insist on equality of these tensor components at .€p and f ( p ) ;one may, for example, substitute proportionality or some other relationship, depending on the symmetry required. In more technical language, the above equality of components of T at p and f ( p ) , for each p E U is equivalent to f *T = T, where f * is the pullback o f f [l,21. The idea expressed in the previous paragraph should, perhaps be termed a local symmetry since the domain U off need not be M . Indeed, the assumption that f should have domain M is rather restrictive both mathematically and physically. But retaining the concept of local symmetry would, should the symmetry be required everywhere on M , require many such maps (so that the union of the domains such as U equals M ) . The situation is clearly
343
starting to get clumsy! In practice one proceeds to obtain the collection of local diffeomorophisms such as f in another way (see, for example, [3]). Let X be a smooth vector field defined globally on M and let p E M . If W is the domain of a coordinate system containing p and with coordinates xa then the curve in W represented by x a ( t )for t in some open interval I of R containing 0 and satisfying d x a / d t = X " for t E I and also the condition x " ( 0 ) = p , is called an integral curve of X starting f r o m p . The domain I is important and depends crucially on the components X " of X in W . However, a theorem from the theory of differential equations [4]says that for each p E M there exists an open neighbourhood U of p and a real number E > 0 such that through any q E U there exists an integral curve of X starting from q and with domain ( - E , E ) . Thus for any fixed t E ( - E , E ) and for any q E U there is an integral curve of X starting from q whose domain contains t. Thus one has a map pt whose domain is U and which maps q to the point on this integral curve with parameter t. This map can be shown to be a local (smooth) diffeomorphism with domain U and range V = pl(U) [2]. The collection of local diffeomorphisms so constructed is such that the union of their domains is M and such that they combine (functionally) together in a pleasant way. Each such map pt is called a local flow or local dzffeomorphism of X. The previous paragraph suggests that the local flows of vector fields may make good candidates for "symmetry" maps. If, for example, each map pt of a vector field X is a symmetry of the metric tensor g , so that each pt satisfies p,"g= g (i.e. each pt is a local isometry) then Cxg = 0, and conversely [l]. The equivalence of the symmetry assumptions on the maps vt and the condition Cxg = 0 means that reference to the local flows may now be omitted and the symmetry condition can then be rewritten in the more useful form as Killing's equations LXg =0
Xa;b + Xb;a = 0
(1)
A vector field satisfying (1) is called a Killing vector field. The symmetry encoded in (1) may be viewed in another way. Suppose at some p E M , X ( p ) # 0. Then one may always choose a coordinate system x , with domain some open subset U of p such that, on U , the components of X satisfy X " = (1,0,0,0) = ST [4].Thus, when restricted to U , X = d / d x ' and the integral curve of X through a point q E U with coordinates ( a ,b , c , d ) , with a , b,c, d E R is t -+ ( t + a, b , c , d ) . When this information is substituted into (1) one easily finds that dgab/dxl = 0,
344
that is, the metric components are independent of the coordinate X I . This, together with the observation that the integral curves of X are curves along which x2,x3 and x4 are constant, that is along which only X I changes, reveals quite clearly how d / a x l ( = X) is a "direction of symmetry" for g. There is an obvious and easily proven converse to this result . Suppose that in some coordinate domain U with coordinates x a the metric components gab are independent of x'. Then d / d x l is a Killing vector field (on U ) . Other types of symmetries may be described in a similar way (for full details see [ 5 ] ) .For example, the condition that X be a conformal vector field is equivalent to the statement that each local flow pt associated with X, and with domain U is a conformal diffeomorphism (that is, p,*g = o g for some smooth function o : U -+ EX). If o is a constant function for each q t associated with X, then X is called homothetic. The condition that X be a projective vector field is that each local flow pt associated with X maps any (piece of a) geodesic in U t o a (piece of a) geodesic in pt(U). If each such map, in addition, preserves any affine parameter, then X is an a f i n e vector field. If one denotes the set of all Killing (respectively, conformal, homothetic, projective and affine) vector fields on M by K ( M ) (respectively, C ( M ) , N ( M ) , P ( M and ) d(M)) then it can be shown (see e.g. [ 5 ] )that each of these sets is a finite dimensional vector space and, in fact, a Lie algebra under the Lie bracket operation, that is, if X and Y are in any one of these algebras, then so also is the Lie bracket [X,Y ]of X and
Y. The above results can be collected together into a useful general result regarding these symmetries. Recalling that (CXg)ab = Xa;b Xb;a, we can decompose Xa;b into its symmetric and skew symmetric parts in any coordinate domain as
+
Xa;b = 1/2hab
+ Fab
(2)
where F is the bivector associated with X and where hab = hba =
(3)
Then X is projective if and only if in any coordinate domain
+ gac'd)b -k gbc'd)a
hab;c = 2gab'd)c
(4)
for some global closed 1-form $. It then follows that X is a f i n e $ =0 on M in (4) (which is equivalent to the global type (0,2) tensor field h being covariantly constant on M), that X is conformal h = 4 g , with 4 a smooth function: M + lR in (4),that X is homothetic it is conformal
345
and, in addition, q5 is a constant function on M (and proper homothetic if q5 # 0) and that X is Killing h = 0 in (3). It is clear that K ( M ) c X ( M ) C C ( M ) and that X ( M ) c d ( M ) C P ( M ) .
*
3. Symmetry Orbits Let ,C be one (any) of the above Lie algebras of (symmetry) vector fields on M. Let X E C and let pt be a local flow of X with domain U . Let V = cpt(U) and suppose Y E C and that +s is a local flow of Y whose domain W intersects V . Then there is an open subset U' of U on which the map +s o pt is defined, mapping p E U' to +,(cpt(p)). In this way, and given x1 , . . . , XI,E c with local flows ptl, . . . , ptk, one can consider maps (where defined) of the form [6]
for all appropriate choices of k, X I , . . ., xk and (tl, ...tk) E R,under the usual rules of composition and inverse. Denote by G the collection of all such maps. Now, for p E M , let 0, denote the subset of M defined by q E 0, f(p) = q for some f E G. The set 0, is called the orbit of p associated with C (or G); those points of M which can be LLreachedll from p by following integral curves of members of C according to maps like ( 5 ) . Such orbits are important, both physically (since they suggest for a given symmetry how a convenient coordinate system for representing the metric, etc, may be chosen) and mathematically (by encoding within them the geometry of C). To make this precise one must firstly describe the structure of such orbits and secondly relate the orbits geometrically to C. I will mention four results regarding such orbits (for details see [ 5 ] ) .The first one points out that no assumption of any Lie (or other group) action on M has been made. If the symmetry in question is assumed to arise from a Lie group action, there is a well known standard theory in place to Here, only the existence of the deal with the problems presented below [4]. finite-dimensional Lie algebra L is assumed and which, in general, is not assumed to arise from any such Lie (or other) group action. (The necessary and sufficient condition that C does, in fact, arise from a Lie group action is that C is a finite dimensional Lie algebra and that each vector field X E C is a complete vector field, that is, its maximal integral curves through any point of M are defined on the whole of R [7] (see also [4]).)
346
The second result is that each orbit associated with C is a submanifold of M [6, 8, 91. Such a result is, of course, geometrically satisfying and, from the practical viewpoint, useful for constructing coordinate systems which are “attached” to an orbit. The third result shows how L is related to its associated orbits by saying that each such orbit is an integral manifold of C [8]. To see exactly what this means, let p E M and define C, = { X ( p ) : X E C} so that C, is that subspace of the tangent space T p M to M at p formed by members of C evaluated at p . This third result then says that, at each p E M , C, and the subspace of T p M tangent to the orbit submanifold through p coincide. Thus the orbit through p has the same dimension as Cp. The fourth (rather technical) result is that if f is a differentiable map from some manifold into M , whose range lies in an orbit associated with 13, then f is differentiable as a map into this orbit submanifold [9]. This result may fail if the orbit is replaced by a general submanifold of M , and is related to the topology of the orbit. It is remarked that the dimensions of the subspaces C, (and hence the dimensions of the orbits associated with C) will, in general, depend o n p. If the orbits (and hence the subspaces C p ) are of constant dimension over M , the Frobenius theory applies and the second and third of the above results have easier proofs (see e.g. [4]).Also, in this case, some of the restrictions regarding the finite-dimensionality of C can be removed. One final remark concerns the actual geometry of the orbits themselves. If N is a submanifold of M of dimension 1,2 or 3, then its “nature” (spacelike, timelike or null) will, in general, vary from point to point of N . If N is an orbit under the Killing algebra K ( M ) , however, this nature is constant over N . In addition, if N is not null, it has an induced metric g’ on it from the space-time metric g and the Killing vector fields in K ( M ) are tangent to N and can easily be shown to give rise to Killing vector fields of the metric g’ (see, e.g. [lo]. Similar results arise in the case of the Lie algebras N ( M ) and C ( M ) [5]. The details here depend on the last result of the previous paragraph. A further investigation here suggests an advantage in the introduction of the concept of the “dimensional stability” of an orbit. Roughly speaking, an orbit N is called dimensional stable if orbits ‘‘close to” N have the same dimension as N . Dimensionally stable orbits are the most “well-behaved”. Further details can be found in [lo, 51.
347 4. Isotropy
Returning to the Killing vector theory studied in section 2, let X E K(M) and write Killing’s equations (1) in the equivalent form Xa;b = Fab (= -Fba) where F is a global smooth skew-symmetric second order tensor field on M called the (Killing)bivector of X and which is defined in any coordinate system by this equation. This equation, together with its covariant derivative and the use of the Ricci identity on X , gives
Xa;b = Fabr
Fab;c = RabcdXd
(6)
The equations in (6), with the geometry of M assumed given, are a closed set of first order differential equations for the components of X and F. To see this let z a ( t ) be a curve in M and contract the first equation in (6) with ib= dzb/dt to get
(Xa,b- r & X c ) i b = Fabib
-
dX,/dt = D a
(7)
and similarly contract the second equation in (3) with i cto get dF,b/dt = Dab, where D , and Dab are defined along the curve and depend only on X,, Fab, t and, of course, the given geometry of M. Thus if X , and Fab are given a t some (any) p E M, Picard’s theorem guarantees that they are uniquely determined along some open segment of this curve containing p . Now suppose that q is any other point in M. Since M is a manifold which is connected, it is also path connected and so there is a smooth path c from p to q which is a connected subset of M. If Y is another global Killing vector field on M such that the values of Y, and of its bivector agree with those of X , and its bivector at p, let A and B be the subsets of the smooth path c on which the values of X and Y agree and disagree, respectively. The above argument shows that A is open and an elementary continuity argument shows that B is open also. Since this path is connected, one of A and B is empty. But p E A and so B is empty. It follows that X and Y agree all along the path and hence a t q. Since q was arbitrary, X and Y are equal everywhere and so X = Y on M (see e.g. [51) .Thus, each global Killing vector field on M is uniquely determined by the values X,(p) and Fab(p)a t some (any) point p E M. It follows that dimK(M) I 4 6 = 10. A similar analysis on the other Lie algebras gives dim N(M) 5 4 6 1 = 11, dimC(M) I 4 6 4 1 = 15, d i m d ( M ) I 4 6 10 = 20 and d i m F ( M ) I 4 6 10 4 = 24.
+ + + + + +
E M and consider the subset K; of K(M) defined by KE = K(M) : X ( p ) = 0}, that is, the subset of Killing vector fields which
Now let p
{X
E
+ +
+ + +
348
vanish at p . It is easily checked that K; is a vector subspace of K ( M ) and, in fact, a Lie subalgebra of K ( M ) , called the isotropy subalgebra of K ( M ) at p . Now define a linear map y from K; to the vector space of all type (0,2) skew-symmetric tensors (bivectors) at p by mapping X E K; to its Killing bivector at p . Now this latter vector space at p is a Lie algebra under the commutation of matrices and this map is easily checked to be a Lie algebra homomorphism. The uniqueness argument of the previous paragraph shows that y is injective since if X maps to the zero bivector at p (and recalling that X ( p ) = 0) then X = 0 on M . Since the vector space of bivectors at p , under matrix commutation, may be taken as the Lie algebra of the Lorentz group (the Lorentz algebra, see e.g. [5]),one sees that the isotropy algebra at p is isomorphic to a subalgebra of the Lorentz algebra, this subalgebra, in general, depending on p . Finally, consider the linear map w from K ( M ) to the subspace K ( M ) , = { X ( p ) : X E K ( M ) } of T p M given by X X ( p ) . [This latter subspace is the vector space C, of section 3 with C = K ( M ) . ] The kernel of w is K; and so, by elementary linear algebra, dim K ( M ) = dim K ( M ) , +dim K;. Referring back to section 3 one now sees that the dimension of the Killing orbit 0, through p satisfies dim0, dimK; = dimK(M) for each p E M (for details, see [5]). ---f
+
If X E K; then X ( p ) = 0 and the associated local flows each fixp (cpt(p) = p for all appropriate t ) . Thus p is called a zero or a fixed point of X. The isotropy algebra assumes importance at such points. It is, perhaps, best described in terms of exponentials of the matrices ( X a , b)* = (Fab), [2] but, for the present purposes, the following restricted but simpler calculation will suffice [5]. Noting that, since X is a Killing vector field, LX Ricc = 0, one evaluates this equation at the zero p of X to find
Now the matrix Fab(p) is not zero (unless X = 0 on M ) and so, since Fab(p) is skew-symmetric, it has even rank, 2 or 4. If the rank is 2, Fab(p)may be written as dash] where r, s E T,M. In this case, F(p) is called simple and the 2-dimensional subspace (2-space) of T,M spanned by r and s is, in fact, uniquely determined by F(p) and called the blade of F at p . If the rank is 4,F(p) is called non-simple and a pseudo-orthonormal tetrad (u, x , y, z ) exists at p for which -uaua = xaxa = yaya = zaza = 1and in which Fab(p) may be written as Q dazbI /3 x[ayb]( a ,p E R,(Y # 0 # p). The orthogonal pair of 2-spaces of T,M spanned by u and z and by x and y are uniquely determined by F(p) and are referred to as its canonical blades [ll].
+
349
It turns out [5] from (8) that if F(p) is simple, the blade of F(p) is an eigenspace of the Ricci tensor at p , that is, each non-zero member of this blade is a n eigenvector of the Ricci tensor with the same eigenvalue. If F(p) is non-simple, the same applies to each canonical blade (but with different eigenvalues in general) [12]. Thus each non-zero member of the isotropy algebra contributes an eigenvalue degeneracy to the Ricci and hence, (through the Einstein field equations) to the energy-momentum tensor. In this way the algebraic type of the energy-momentum tensor is restricted by any such isotropy in an easily determined way [5]. For example, the spatial isotropy usually insisted upon in cosmology (for example, in the FRW models), shows, according to the above argument, that the energy momentum tensor has three independent spacelike eigenvectors with the same eigenvalue, together with a timelike eigenvector. Thus it must be of the perfect fluid type (the details about the actual eigenvalues being determined by the physics.) Similar remarks apply to the Weyl tensor C . For this tensor one has CXC = 0 and evaluating this equation at a zero p of X one can check that if the Petrov type a t p is I,II or 111, then F ( p ) = 0 and so such Petrov types admit no isotropy of this nature. If, however, F ( p ) satisfies this equation and is non-simple, or simple with a blade which is non-null, then the Petrov type is D or 0 whereas if F(p) is simple with a null blade, the Petrov type is N or 0 [13, 51. It can also be shown that if the isotropy algebra a t p has dimension three or more, then C ( p )= 0. Thus the cosmological models mentioned above are conformally flat. One can also consider, in a similar fashion, isotropy subalgebras for C ( M ) , X ( M ) , P ( M ) and d ( M ) [14, 51. One useful result which follows from this is that if X is a proper homothetic vector field on M which vanishes at p E M then, a t p , the Petrov type is 111, N or 0 and the Segre type of the energy-momentum tensor is either {(31)},((211)) or 0 and, in each case, all eigenvalues vanish [5].The plane wave metrics are examples of this situation with Petrov type N or 0 and energy-momentum tensor of Segre type {(211)}, with zero eigenvalue, or 0.
As simple examples of some practical uses of the above theory let p E M and let 0, be the Killing orbit through p . If dimK(M) > dim0, then, from a formula above, K; is non-trivial for each q E 0,. Thus the Petrov type and the Segre type of the energy-momentum tensor are seriously restricted a t each q E 0, as described above. As another example, suppose M admits a proper homothetic vector field (so that K ( M ) c X ( M ) but
350
K ( M ) # N ( M ) ) .Let p E M and let 0, and 0; be, respectively, the K ( M ) and N ( M ) orbits through p . If 0, = 0; then each q E 0, has a non-trivial proper homothetic isotropy (i.e. there exists a proper homothetic vector field which vanishes at q ) . This severely restricts the Petrov and energymomentum algebraic types at each q E 0,. To see the above (bracketed) result, let X I , ...X, E K ( M ) be such that X1(q), ...X , ( q ) span the subspace of T,M tangent to 0,. Now let Y E Z ( M )\ K ( M ) . If Y(q)= 0, the result follows. If Y(q)# 0 then, since Y(q)lies in the Killing orbit at q , there exists CY~...Q,E R such that Y(q)= C a i X i ( q ) . So consider the vector field Z = Y - CoiXi on M . Clearly, Z E Z ( M ) \ K ( M ) and Z ( q ) = 0 and the result follows. In particular, if the space-time is transitive (which is equivalent to dimK(M), = 4 for each p E M or to the statement that M is the only Killing orbit [15, 6, 91) then a proper homothetic vector field could exist on M only if the very restrictive algebraic conditions set out above hold a t every point. Thus, for example, the Godel space-time can not admit a proper homothetic vector field (since it is a perfect fluid with Segre type (1, (111))). There are many other applications of these results [5].
5. Local Killing (and other Symmetry) Vector Fields In this short section it is simply remarked that, although the discussion so far given has involved local symmetry transformations on the space-time manifold, they have always been assumed to have arisen from Killing vector fields defined globally on M . However, the practicality of the physics suggests that each observer would only be able to determine a n open neighbourhood U on which these local symmetry transformations applied and which would thus arise from Killing vector fields o n U. Thus, only a certain Killing algebra K(U)would be defined, but, of course, such a neighbourhood U would thus be assumed to exist for each observer, that is, for each p E M . The question is: given that each observer in our universe could determine such a neighbourhood U and algebra K(U),under what conditions are the members of K(U)for each of these observers merely restrictions, to U , of global Killing vector fields on M ? Clearly, one would need dim K(U) to be the same for each such U . Unfortunately this is not enough and, as expected, the topology of M is involved. However, if M is simply connected, the result follows. A similar result holds for the other symmetries mentioned earlier [16, 171.
351
6. Curvature Symmetry Now suppose that the geometrical object whose symmetries are desired is the curvature tensor Riem. Thus one seeks those global vector fields on M satisfying CxRiem= 0, referring to them as curvature collineations. The collection (clearly a vector space) of all such vector fields on M is then labeled CC(M). However, in this case, difficulties arise [18, 51. In the above studies of the algebras IC(M),%(Ad), d ( M ) , C ( M ) and P ( M ) ,the smoothness of the vector fields in the appropriate algebra was assumed. In fact, it is sufficient for the first three of these algebras to assume that its members are C1. Their smoothness then follows. For the last two, the same result follows from the assumption that its members are C 2 . For CC(M), one would clearly want its members to be differentiable. But of which degree of differentiability? Its members may, in fact, be shown to be of any desired degree of differentiability and some degree of differentiability may be lost on forming the Lie bracket. If we insist on M being a smooth manifold, the interpretation of symmetry given in section 2 only makes sense if each member of CC(M) is smooth (since we insist on the functions y a being smooth). So let us re-define CC(M) to be the vector space of global smooth vector fields on M satisfying CxRiem= 0 (this is clearly now a Lie algebra since smoothness is preserved under the Lie bracket). Another difficulty which arises is that CC(M) may be infinite dimensional. This means that the possible integral submanifold nature of its orbits is, so far as the author knows, unknown. Now the Lie algebras IC(M),'FI(M),and d (M) are contained in CC(M). It turns out that there is a good reason to define a member of CC(M) to be proper if it is not a member of d ( M ) [19]. It also turns out that, generically, the Lie algebras CC(M) and d ( M ) are equal. (In fact, CC(M) and N ( M ) are generically equal [20, 51.) Thus it is generic that proper curvature collineations do not exist. But many interesting special cases arise outside this generic class. The following example is instructive. Consider the space-time ( M , g ) where M is R4 and g is the metric [21, 18, 51 ds2 = -dt2
+ d,,dX"dx'
(9)
Here, Greek indices take the values 1,2 and 3 and d is a positive-definite metric defined on the hypersufaces of constant t (which are just copies of R3)and the dag are independent o f t . Then the vector field X = f ( t ) a / a t is a curvature collineation for any choice of the function f. The infinitedimensionality of CC(M) is obvious since, for no positive integer n is the set of members of CC(M) given by X, t X , . . ., t n X linearly dependent
352 This is a consequence of the fact that if a polynomial in a single variable vanishes on some interval of R it vanishes everywhere on R. The full set CC(M) depends on the nature of the metric h. It may be thought that the infinite-dimensionality of CC(M) and the existence of proper members of this Lie algebra could be equivalent conditions. But this is not the case as the following example shows. Again let M = R4 and consider the metric on M given by [19] ds2 = f ( t , z)dt2
+ k ( t ,z)dz2 + ~ ( z ) e ~ ’ / ~+(ddy 2z)~
(10)
where f , k and u are smooth functions with j’k < 0 and u > 0 on M and where these functions are chosen so that in each of the induced geometries of the t , z and z, y spaces, the curvature tensor is non-zero. For this spacetime, CC(M) has finite dimension ( 6) I but d / d y is in CC(M) but not in
4M). Fortunately, many more examples regarding the algebra CC( M ) have appeared [22, 23, 24, 25, 261. A problem with similar difficulties is the study of those (smooth) vector fields X satisfying LXRicc= 0 (and the associated problem studying C x T = 0 where T is the energy momentum tensor). These “Ricci c~llineations’~ and “energy momentum collineations” , respectively, have also been studied recently [27, 28, 291.
7. Acknowledgements The author wishes to thank the organisers of the meeting in Islamabad for their hospitality and financial support during his stay in Pakistan. In particular, he thanks Dr Ghulam Shabbir, Prof Muhammad Sharif, Dr Abdul Kashif and Prof Ashgar Qadir for their kindness and for many useful discussions. He also thanks Dr John Pulham, Dr Azad Siddiqui and Prof Victor Varela for help in preparing the manuscript.
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353 5. G. S. Hall, Symmetries and Curvature Structure in General Relativity, (World Scientific, 2004). 6. H. J. Sussmann, Trans. Am. Math. SOC.180, 171 (1973). 7. R. S. Palais, Mem. Am. Math. SOC.22, (1957). 8. R.Hermann, International Symposium on Non-Linear DifferentialEquations and Non-Linear Mechanics, (Academic Press, New York 1963), p. 325. 9. P. Stefan, J. London. Math. SOC.180, 544 (1980). 10. G. S. Hall, Class. Quant. Grav. 20,3745 (2003). 11. R. K. Sachs, Proc. Roy. London. A264, 309 (1961). 12. G. S. Hall and C. B. G. McIntosh, Int. J . Theor. Phys. 22, 469 (1983). 13. J. Ehlers and W. Kundt, in Gravitation; and Introduction to Current Research, ed L.Witten, (Wiley, New York, 1962), p. 49. 14. G. S. Hall and M. T. Patel, Class. Quant. Grav. 21, 4731 (2004). 15. W. L. Chow,Math. Ann. 117,98 (1939). 16. K. Nomizu, Ann. Maths. 72, 105 (1960). 17. G. S. Hall, Class. Quant. Grav. 6,157 (1989). 18. G. S. Hall and J. d a Costa, J. Math. Phys. 32, 2848 and 2854 (1991). 19. G. S. Hall and Lucy MacNay, Class. Quant. Gmv. 22,5191 (2005). 20. G. S. Hall, Gen. Rel. Grav. 15,581 (1983). 21. G. H. Katzin, J. Levine and W. R. Davies, J . Math. Phys. 10, 617 (1969). 22. G. S. Hall and G. Shabbir, Class. Quant. Grav. 18,907 (2001). 23. G. S. Hall and G. Shabbir, Gravitation and Cosmology 9,134 (2003). 24. C. B. G. McIntosh, J. Math. Phys. 23, 436 (1982). 25. E. G. L. R. Vaz and C. D. Collinson, Gen. Rel. Grav. 15,661 (1983). 26. A. Bokhari, A. R. Kashif and A. Qadir, J . Math. Phys. 41,2167 (2000). 27. U. Camci and A. Barnes, Class. Quant. Grav. 19, 393 (2002). 28. J. Carot and J. d a Costa, Fields Institute Communcations 15, 179 (1997). 29. M. Sharif and S. Aziz, Gen. Rel. Grav. 35, 1093 (2003).
GRAVITATIONAL COLLAPSE IN QUANTUM GRAVITY VIQAR HUSAIN Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB E3B 25’9, Canada. and Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 8Y5,Canada. E-mail: [email protected] We give a review of recent work aimed at understanding the dynamics of gravitational collapse in quantum gravity. Its goal is to provide a nonperturbative computational framework for understanding the emergence of the semi-classical approximation and Hawking radiation. The model studied is the gravity-scalar field theory in spherical symmetry. A quantization of this theory is given in which operators corresponding to null expansions and curvature are well defined. Together with the Hamiltonian, these operators allow one t o follow the evolution of an initial matter-geometry state to a trapped configuration and beyond, in a singularity free and unitary setting. Keywords: Quantum gravity, Hawking radiation, gravitational collapse, black holes.
1. Introduction One of the outstanding problems in theoretical physics is the incomplete understanding at the quantum level of the formation, and subsequent evolution of black holes in a quantum theory of gravity. Although a subject of study for over three decades, it is fair to say that, in spite of partial results in string theory and loop quantum gravity, there is no widely accepted answer to many of the puzzles of black hole physics. This is largely because there has been no study of quantum dynamical collapse in these approaches. Rather, progress has focused mainly on explanations of the microscopic origin of the entropy of static black holes from state counting. A four-dimensional spacetime picture of black hole formation from matter collapse, and its subsequent evolution is not available in any approach to quantum gravity at the present time.
354
355
This paper summarizes an attempt to address this problem in the context of Hawking's original derivation of black hole radiation: spherically symmetric gravity minimally coupled to a massless scalar field. This is a non-linear 2-d field theory describing the coupled system of the metric and scalar field degrees of freedom. Gravitational collapse in the classical thebut its full ory in this model has been carefully studied quantization has never been addressed. Hawking's semi-classical calculation3 uses the eikonal approximation for the wave equation in a mildly dynamical background, where the dynamics centers on the surface of a star undergoing collapse. Its essential content is the extraction of the phase of the ingoing mode from an outgoing solution of the scalar wave equation as a classically collapsing star crosses its Schwarzschild radius. According to this calculation, emitted particles appear to originate near the event horizon. This means that an emitted particle observed by a geodesic observer is transplankian at the creation origin due to the gravitational redshift (which is infinite at the horizon). Its back reaction is therefore not negligible, bringing into question the entire approximation. It is likely that a complete understanding of quantum evolution in this system will resolve all the outstanding problems of black hole physics in the setting in which they originally arose. The following sections contain a summary of the work described in the 2. Classical theory
The phase space of the model is defined by prescribing a form of the gravitational phase space variables qab and % j a b , together with falloff conditions in r for these variables, and for the lapse and shift functions N and N a , such that the ADM 3+1 action for general relativity minimally coupled to a massless scalar field,
is well defined. The constraints arising from varying the lapse and shift are
356
where ii = iiabqab and R is the Rcci scalar of gab, The falloff conditions imposed on the phase space variables are motivated by the Schwarzschild solution in Painleve-Gullstand (PG) coordinates, which itself is to be a solution in the prescribed class of spacetimes. These conditions give the following falloff for the gravitational phase space variables (for E > 0)
(4) where f a b l gab, hab are symmetric tensors, rab= i i a b / f iand , q In this general setting we use the parametrization
= detqab.
for the 3-metric and conjugate momentum for a reduction to spherical symmetry, where eab is the flat 3-metric and na = xa/r. Substituting these into the 3+1 ADM action for general relativity shows that the pairs ( R ,PR) and (A, PA) are canonically conjugate variables. We note for example the Poisson bracket,
which is the bracket represented in the quantum theory (described below). The falloff conditions induced on these variables from ( 4 ) ,together with those on the lapse and shift functions, ensure that the reduced action,
SR =
2G
/
dtdr (PRR+ PAA+ P44 - constraint terms
is well defined. This completes the definition of the classical theory. At this stage we perform a time gauge fixing using the condition A = 1 motivated by PG coordinates. This is second class with the Hamiltonian constraint, which therefore must be imposed strongly and solved for the conjugate momentum PA. This gauge fixing eliminates the dynamical pair (A, PA),fixes the lapse as a function of the shift, and leads to a system describing the dynamics of the variables ( R ,PR) and ( 4 ,P4)5.The reduced radial diffeomorphism generator, C r e d 3 PA(R,4, PR,p4)
+ p44' + PRR' = 0 ,
(9)
357
remains as the only first class constraint. It also gives the gauge fixed Einstein evolution equations via Poisson brackets, for example $ = { A J d7- NTCred}. 3. Quantum gravity
The quantization route we follow is unconventional in that field momenta are not represented as self-adjoint operators; rather only exponentials of momenta are realized on the Hilbert space. This is similar to what happens in a lattice quantization, except that, as we see below, every quantum state represents a lattice sampling of field excitations, with all lattices allowed. This quantization allows definitions of bounded inverse configurations operators such as 1/z,which for quantum gravity leads to the mechanism for curvature singularity resolution described below. A quantum field is characterized by its excitations at a given set of points in space. The important difference from standard quantum field theory is that in the representation we use, such states are normalizable. A basis state is leixk akPR(Zk),
eiLzcl
bLP+(yr))
la1 . . . U
N ~b;l
. . .b N z ) ,
(10)
where the factors of L in the exponents reflect the length dimensions of the respective field variables, and a k , bl are real numbers which represent the excitations of the scalar quantum fields R and $ at the radial locations { z k } and {yl}. The inner product on this basis is
.
( a l . . U N ~ b;l ,
. . . b N 2 1 a i . . . ahl;b: . . . bhz)= dal,a; . . . bbNz,bh2,
if the states contain the same number of sampled points, and is zero otherwise. The action of the basic operators are given by
fif l a l . . .
a ~bl ~. . ;. b N z ) = L2 x a k f ( Z k ) l a l . . . U N ~ bl; . . . b N 2 ) , ( 1 1 ) k
..
e i x ~ p R ( s ~ ) ~U a~ N .~ bl ;
. . .b N , )
,
= l a l . . . uj
-xj,.
. . u N ~ b; l . . . b N , ) , ( 1 2 )
where aj is 0 if the point z j is not part of the original basis state. In this case the action creates a new excitation at the point xj with value - X j . These definitions give the commutator
ph- ] =
-
% ~ P R ( Z ) -Xf(z)L2eiXPR(s).
(13)
Comparing this with (7), and using the Poisson bracket commutator correspondence { , } H iti[ , ] gives L = a l p , where l p is the Planck length. There are similar operator definitions for the canonical pair (4, P4).
358
3.1. Singularity resolution To address the singularity avoidance issue, we first extend the manifold on which the fields R etc. live to include the point r = 0 , which is the classical singularity in the gauge fixed theory . We then ask what classical phase space observables capture curvature information. For homogeneous cosmological models, a natural choice is the inverse scale factor a ( t ) .For the present case, a guide is provided by the gauge fixed theory without matter where it is evident that it is the extrinsic curvature that diverges at r = 0, which is the Schwarzschild singularity. This suggests, in analogy with the inverse scale factor, that we consider the field variable 1 / R as a measure of curvature. A more natural choice would be a scalar constructed from the phase space variables by contraction of tensors. A simple possibility is
The small r behaviour of the phase space variables ensures that any divergence in ii is due to the 1 / R factor. We therefore focus on this. A first observation is that the configuration variables R(r,t ) and $(r,t ) defined at a single point do not have well defined operator realizations. Therefore we are forced to consider phase space functions integrated over (at least a part of) space. A functional such as 00
R f = l drfR
(15)
for a test function f , provides a measure of sphere size in our parametrization of the metric. We are interested in the reciprocal of this for a measure of curvature. Since R r asymptotically, the functions f must have the falloff f ( r ) r-2-e for R f to be well defined. Using this, it is straightforward to see that 1 / R f diverges classically for small spheres: we can choose f > 0 of the form f 1 for r << 1, which for large r falls asymptotically to zero. Then R f r2 and 1 / R f diverges classically for small spheres. A question for the quantum theory is whether 1/Rj can be represented densely on a Hilbert space as l/hj. This is possible only if the chosen representation is such that R f does not have a zero eigenvalue. If it does, we must represent 1 / R f as an operator more indirectly, using another classically equivalent function. Examples of such functions are provided by Poisson bracket identities such as
-
N
N
N
359
where the functions f do not have zeroes. The representation for the quantum theory described above is such that the operator corresponding to Rf has a zero eigenvalue. Therefore we represent 1/Rf using the r.h.s. of (16). The central question for singularity resolution is whether the corresponding operator is densely defined and bounded. This turns out to be the case. Using the expressions for the basic field operators, we can construct an operator corresponding to a classical singularity indicator: -
l\
/
2
The result is that basis states are eigenvectors of this operator, and all eigenvalues are bounded. This is illustrated with the state
Isao)
leiaOPR(r=O) ),
(18)
which represents an excitation uo of the quantum field Rf at the point of the classical singularity:
if
= (2e4.f (0)ao
(19)
which is clearly bounded. This shows that the singularity is resolved at the quantum level. In particular if there is no excitation of Rf at the classical singularity, ie. a0 = 0, the upper bound on the eigenvalue of the inverse operator is 2/$.
3.2. Quantum black holes The event horizon of a static or stationary black hole is a global spacetime concept. It does not provide a useful local determination of whether one is inside a black hole. The fundamental idea for defining a black hole locally is that of a trapped surface, first introduced by Penrose. One considers a closed spacelike 2-surface in a spacetime, and computes the expansions 8+ and 8- of outgoing and ingoing null geodesics emanating orthogonally from the surface. If 8+ > 0 and 8- < 0, the surface is considered normal. On the other hand if 8+ 5 0 and 8- < 0, the surface is called trapped. This provides a criterion for subdividing a spacetime into trapped and normal regions. The outer boundary of a trapped region may be considered as the (dynamical) boundary of black hole, also known as the “apparent horizon” in numerical relativity. It is a function computed in classical numerical
360
evolutions to test for black hole formation. Similarly, a setting for studying quantum collapse requires an operator realisation of the null expansion “observable”, and a criterion to see if a given quantum state describes a “quantum black hole” . The classical expansions in spherical symmetry are the phase space functions6
+
(2R2AA‘ f PA 4A2RR’). (20) 2A Given phase space functions on a spatial hypersurface C, the marginal trapping horizon(s) are located by finding the solution coordinates r = ri (i = 1. . . n ) of the conditions O+ = 0 and 8- < 0, (since in general there may be more than one solution). The corresponding radii Ri = R(ri) are then computed. The size of the horizon on the slice C is the largest value in the set {Ri}. Since only translation operators are available in our quantization, we define PAindirectly by
O*
= --
Pi =
&
(UA
-
u:) ,
where 0 < X << 1 is an arbitrary but fixed parameter, and UA denotes exp(iXPA/L). This is motivated by the corresponding classical expression, where the limit X -+ 0 exists, and gives the classical function PA.X is perhaps best understood as a ratio of two scales, X = l p / l o , where lo is a system size. As for a lattice quantisation, it is evident that momentum in this quantisation can be given approximate meaning only for X << 1. X is also the minimum value by which an excitation can be changed. Definitions for the operators corresponding to R’ and A‘ are obtained by implementing the idea of finite differencing. We use narrow Gaussian smearing functions with variance proportional to the Planck scale, peaked at coordinate points rk d p , where 0 < E << 1 is a parameter designed to sample neighbouring points:
+
1 exp
[-
( r - rk
-
dp)
21;
Denoting Rfe by R, for this class of test functions we define 1 R ( 7 - k ) := LPE
( R , - Ro) .
Putting all these pieces together, we can construct the desired operators
36 1
which have a well defined action on the basis states. In analogy with the classical case, we propose that a state [Q) represents a quantum black hole if ( Q l d + ( r k ) l Q ) = 0,
and
(QId-(rk)l!P)
< 0.
(25)
for some T k . The corresponding horizon size is given by RH = ( Q [ k ( r k ) l Q ) . This definition is utilised as follows: Given a state with field excitations a t a set of coordinate points { T i } , one would plot the expectation values in Eqn. (25) as functions of (Qlk(?-k)l@), and locate the zeroes, if any, of the resulting graph. The resulting "quantum horizon" location is invariant under radial diffeomorphisms because these act on states t o shift the coordinate locations of field excitations, but leave the expectation values unchanged - the graph is a physical observable. It is straightforward to construct explicit examples of states satisfying these quantum trapping conditions. Some examples appear in [6]. The quantum horizons so determined are not sharp since (0;) # 0. 4. Summary and outlook
The results so far from this approach to understanding black hole formation in quantum gravity are threefold: (i) A quantization procedure which allows explicit calculations t o be carried out, (ii) a test for black holes in a full quantum gravity setting which makes no use of classical boundary conditions a t event horizons, and (iii) singularity free and unitary evolution equations using the Hamiltonian defined in [7]. The main computational challenge is to use the formalism to explicitly compute the evolution of a given matter-geometry state until it satisfies the quantum black hole criteria, and then to continue to the evolution to see if and how Hawking radiation might arise. This work is in progress. References 1. 2. 3. 4. 5. 6. 7.
M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993). C. Gundlach, Phys. Rept. 376 339 (2003). S. W. Hawking, Commun. Math. Phys. 43 (1975) 199. V. Husain, 0. Winkler, Phys. Rev. D71 104001 (2005). V. Husain, 0. Winkler, Class. Quantum Grav. 22, L127 (2005). V . Husain, 0. Winkler, Class. Quantum Grav. 22, L135 (2005). V. Husain, 0. Winkler, Phys. Rev. D (2006); to appear.
CAN
55c0
GIVE
us THE DESIRED PROMPT
EXPLOSION
OF MASSIVE STARS ? Jameel-Un-Nabi Faculty of Engineering Sciences, Ghulam Ishaq Khan Institute Topi-23640, NWFP, Pakistan Core collapse simulators are striving hard to achieve prompt explosion of a collapsing core of a massive star. Various parameters need to be fed into the simulation code before results of such a complex problem emerge. The most important nuclear physics input parameters to such codes include weak interaction rates (electron capture and beta decay rates) of key nuclides. So far, the weak rates fed into the code resulted in an undesired delayed explosion. Simulators attribute this result partly to somewhat suppressed electron capture rates of these key nuclides. Recently I calculated electron capture rates of these key nuclides using the proton-neutron quasi-particle random phase approximation (pn-QRPA) theory in a microscopic fashion. The microscopic results of QRPA certainly yield more enhanced rates for these nuclides. 55C0 is not only present in abundance in presupernova phase but is also advocated to play a decisive role in the core collapse of massive stars. The important question to ask is ‘can QRPA rates contribute to triggering a prompt explosion?’
1. Introduction
Weak interactions and gravity decide the fate of a star. These two processes play a vital role in the evolution of stars. Weak interactions deleptonize the core of a massive star, and determine the final electron fraction (lepton to baryon ratio, Ye)and the size of the homologous core. Electron capture and photodisintegration processes in the stellar interior cost the core energy by reducing the electron density and as a result the collapse of the stellar core is accelerated under its own ferocious gravity. This collapse of the stellar core is very sensitive to the core entropy and to the lepton to baryon ratio, Ye1. These two quantities are mainly determined by weak interaction processes. The simulation of the core collapse depends strongly on the electron capture on heavy nuclides2. When the stellar core attains densities close to l o 9 g/cm3, it consists of heavy nuclei embedded in an electrically neutral plasma of electrons, with a small fraction of drip neutrons and an even smaller
362
363
fraction of drip protons3. At this stage the density of the stellar core is much lower than nuclear matter density and thus the average volume available to a single nucleus is much greater than that of the nuclear volume. Electron capture and beta decay decide the ultimate fate of the star. During the stellar core collapse, the entropy of the stellar core decides whether the electron capture occurs on heavy nuclei or on free protons produced in the photo-disintegration process. Stars with masses > 8 M a , after passing through all hydrostatic burning stages develop an onion-like structure and produce a collapsing core at the end of their evolution and lead to increased nuclear densities in the stellar core4. Electron capture on nuclei takes place in the very dense environment of the stellar core, where the Fermi energy of the degenerate electron gas is sufficiently large to overcome the threshold energy given by negative Q values of the reactions involved in the interior of the stars. This high Fermi energy of the degenerate electron gas leads to enormous electron capture on nuclei and results in the reduction of the electron to baryon ratio Ye. Two fully microscopic approaches, i.e., the shell model and quasi-particle random phase approximation (QRPA), have been used extensively for the large scale calculation of weak rates. The shell model puts more emphasis on interactions than correlations while QRPA gives more weight to correlations. The QRPA approach allows us to perform calculations in a luxurious model space (as big as 7 b ) . I evaluated the weak interaction rates and summed them over all parent and daughter states to get the total rate. A total of 30 excited states in the parent nucleus were considered in this project. The large model space is large enough to handle excited states in parent and daughter nuclei (around 200) which leads to satisfactory convergence of the electron capture rates. Transitions between these states play an important role in the calculated weak rates. All previous compilations of weak interaction rates either ignore transitions from parent excited states due to complexity of the problem, or apply the so-called Brink hypothesis, when taking these excited states into consideration. This hypothesis assumes that the Gamow-Teller strength distribution on the excited states is the same as for the ground state, only shifted by the excitation energy of the state. I did not use Brink’s hypothesis to estimate the Gamow-Teller transitions from parent excited states but rather performed a state-by-state evaluation of the weak interaction rates and summed them over all parent and daughter sates to get the total weak rate. This is the second major difference between this work and previous calculations of electron capture rates. The result is an enhancement
364
of electron capture rates on 55C0compared to the earlier reported rates. Reliability of calculated rates is a key issue and of decisive importance for many simulation codes. The reliability of the pn-QRPA model has already been established and discussed in detail5-'. There the authors compared the measured data of thousands of nuclides with the pn-QRPA calculations and got good agreement. 2. Results and Discussions To save space, I refer for details of the formalism of the pn-QRPA calculations. The weak decay rate from the ith state of the parent to the j t h state of the daughter nucleus is given by
where ( f t ) i jis related to the reduced transition probability Bij of the nuclear transition by ( f t ) i j = D / B i j , where D is a constant and Bij's are the sum of reduced transition probabilities of the Fermi and GT transitions. The phase space integral integrates over total energy and for electron capture is given by fij =
Srn
w J ~ ( w +, w ) F~(+Z, W ) G-dw.
(2)
'w1
In Eq. (2), w is the total energy of the electron including its rest mass, and w, is the total capture threshold energy (rest kinetic) for electron capture. G- is the electron distribution function. The number density of electrons associated with protons and nuclei is p Y e N ~( p is the baryon density, and N A is Avogadro's number),
+
(y)3
1 pYe = 7r2 N A
(G- - G+)p2dp
(3)
Here p = Jw';;z--I is the electron momentum, the equation has units of mol cmV3 and G+ is the positron distribution function. This equation is used for an iterative calculation of Fermi energies for selected values of p Y e and T . Details of the calculations can be found in5. The B(GT*) strength distributions for ground and two excited states at 2.2 MeV, and 2.6 MeV are shown in Fig.1. Note that the GT strength is fragmented over many daughter states. For electron capture, the GT centroid lies in the energy range 7.1 - 7.4 MeV in the daughter 55Fe and more or less in the energy range 6.7 - 7.5 MeV for both excited states. I
365
get good energy resolution in my spectra due to the large model space. The corresponding values for the ground and the two excited states using the shell model'' are 6 MeV and 9 - 10MeV, respectively, and are also shown by an asterisk in Fig. 1. One clearly sees from it that the QRPA centroid is shifted to much lower energies for the excited states. The QRPA code also calculated GT transitions for the states at 0.3 MeV and 0.4 MeV which lie close to the ground state. The centroids for these states are in the ranges 6.6 - 7.4 MeV and 7.2 - 8.1 MeV, respectively. Tkansitions from these low-lying states contributed to the enhancement of the electron capture rates. In some situations] the total GT strength is more important than the GT centroid. The total GT strength for the electron capture on 55C0 is calculated to be 7.4.
1oo
.c
t
*
*
'
*
J
10'
r
P
+
&
lo2
m
lo3
lo'
0
5
10 15 20 25 3 0 0
E (MeV)
5
10 15 20 25
E(MeV)
10 15 20 25 30
E(MeV)
Fig. 1. Gamow-Teller (GT+)strength distributions for electron captures on 55C0. The left panel shows G T strength distribution for the ground state, whereas middle and right panels show GT strength distribution for 1st and 2nd excited states. The energy scale is the excitation energies in daughter 55Fe.
I compare my electron capture rates with those of the shell modell' in Fig. 2. Here p7 measures the densities in lo7 g/cm3. One notes that QRPA rates are two orders of magnitude faster at low temperature as compared to reference". Due to the complexity of the spectroscopy involved, those authors had to switch to approximations like back resonances (the GT back resonance are states reached by the strong GT transitions in the electron capture process built on ground and excited states6>" and Brink's hypoth-
366 loo.
1
.’.<
-
h
-
v)
2
d
---
p, = 4.32.
-_-_-- _-- -.
10.~ 9.0
I
I
9.2
.
I
.
I
9.4 9.6 Log T
.
I
9.8
.
I
10.0 9.2
.
,
.
9.4
,
9.6
.
,
9.8
.
10.0
Log T
Fig. 2. Electron capture rates on 55C0as functions of temperature for different densities (left panel). The right panel shows the results of Ref.ll for the corresponding temperatures and densities. T is temperature in Kelvin, p is the density in lo7 g/cm3, and A,, are the electron capture rates per second.
esis. The enhancement of QRPA electron capture rates at pre-supernova temperatures is due to large GT+ transitions from the low-lying states of the parent nucleus. These states have finite probability of occupation at pre-supernova temperatures. My calculation does not use Brink’s hypothesis and shows that it is a first order approximation and transitions from excited states are often many orders of magnitude higher than those from the ground state6. Low-lying transitions are important at low temperatures and densities and supplement the electron capture rate from the GT resonance if the Q value only allows capture of high energy electrons from the tail of the Fermi-Dirac distribution. One sees from the GT distribution (Fig. 1) that the GT centroid of the QRPA is very close to the shell model centroid, but QRPA centroids for the excited states are at low energy in the daughter nucleus as compared to shell model centroids. Contribution to rates from these states is many orders of magnitude larger than the ground state leading to an overall enhancement of QRPA rates at low temperatures. At supernova temperatures the difference between the two calculations decreases. At higher temperatures and densities the energy of the electron is
367
large compared to the Q value for transitions to the GT centroid. In such conditions the capture rates are no longer dependent on the energy of the GT distribution but rather depend on the total GT strength13. The total GT strength of the QRPA, 7.4, is less than the shell model, 8.7. My rates are still around four times faster than shell model ones at higher densities and temperatures (see also Ref.g). QRPA calculated a 55C0half-life of 1108 sec. (18.47hours), which is in good agreement with the experimental value of 17.53 hours14.
3. Conclusions 55C0is considered a strong candidate among the other F e peak nuclei that play a dominant role in electron capture and hence in the core collapse of a star. At a pre-supernova stage the dynamics of the star is very complex and large numbers of nuclear excited states are involved. The pn-QRPA is a judicious choice for handling this large number of excited states in heavy nuclei in the pre-supernova conditions of the stellar core. QRPA results point to a much more enhanced capture rate for 55C0 as compared to the reported shell model rates and can have a significant astrophysical impact on the core collapse simulations. The reduced capture rates for 55C0in the outer layers of the core from the previous compilations resulted in slowing the collapse and posed a large shock radius to deal with'. What impact will my enhanced rates have on the core collapse simulations? According to Aufderheide et. al.15, the rate of change of lepton-to-baryon ratio (Qe) changes by about 50% alone due to electron capture on 55C0. My results favor a prompt explosion. I am in the process of evaluating the impact of QRPA calculated capture rates on 55C0 in the core collapse calculations and hope to report the results soon.
References 1. H.A. Bethe, G.E. Brown, J. Applegate, J.M. Lattimer, Nucl. Phys. A324, 487 (1979). 2. W R. Hix et. al., Phys. Rev. Lett. 91,201102 (2003). 3. F.K. Sutaria, A. Ray, J.A. Sheikh, P. Ring, Astron. Astrophys. 349, 135 (1999). 4. A. Heger, N. Langer, S.E. Woosley, Ap. J . 528,368 (2000).
5. J.-U. Nabi, H. V. Klapdor-Kleingrothaus, Atomic Data and Nuclear Data Tables 71,149 (1999). 6. J.-U. Nabi, H.V. Klapdor-Kleingrothaus, Atomic Data and Nuclear Data Tables 88, 237 (2004). 7. J.-U. Nabi, H. V. Klapdor-Kleingrothaus, Eur. Phys. J . A5,337 (1999).
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J.-U. Nabi, Ph.D. Thesis, Heidelberg University Germany, 1999. J.-U. Nabi, M.-U. Rahman, Phys. Lett. B612,190 (2005). J.-U. Nabi, M.-U. Rahman, to appear in Phys. Rev. C (2006). K. Langanke, G. Martinez-Pinedo, Phys. Lett. B436,19 (1998). A. Heger, K. Langanke, G. Martinez-Pinedo, S.E. Woosley, Phys. Rev. Lett. 86,1678 (2001). 13. K. Langanke, G. Martinez-Pinedo, Nucl. Phys. A673,481 (2000). 14. P. Moller, J. Randrup, Nucl. Phys. A514,1 (1990). 15. M. B. Aufderheide, I. Fushiki, S.E. Woosley, E. Stanford, D.H. Hartmann, Astrophys. J . Suppl. 91, 389 (1994).
8. 9. 10. 11. 12.
SYMMETRY CLASSIFICATION AND INVARIANT CHARACTERIZATION OF TWO-DIMENSIONAL GEODESIC EQUATIONS A. R. KASHIF
College of Electrical €4 Mechanical Engineering, National University of Sciences and Technology, Peshawar Road, Rawalpindi, Pakistan E-mail: [email protected] F. M. MAHOMED
School of Computational and Applied Mathematics, Centre of Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, W i t s 2050, Johannesberg, South Africa Email: fmahomedOcam.wits. ac.za A. QADIR
Centre of Advanced Mathematics and Physics, National University of Sciences and Technology, Peshawar Road, Rawalpindi, Pakistan and Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 30261, Saudi Arabia Email: [email protected] We present the point symmetry classification and invariant characterization of a system of two geodesic equations. Previously, Aminova and Aminovl attempted the point symmetry classification. However, they did not identify the Lie algebras nor did they provide an invariant characterization, in terms of coefficients, of the geodesic equations. In this paper we do both.
Keywords: Lie symmetries, geodesic equations
1. Introduction
Aminova and Aminov' and Feroze, Mahomed and Qadir2 provided a geometric approach to the solution of differential equations by considering them
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as geodesic equations. This helps in the symmetry classification as well as provides a means to the invariant characterization of second-order quadratically semilinear systems of differential equations that are linearizable3 via invertible maps. The invariant classification can be extended further to systems that are not linearizable in themselves but for which the reduced equations are linearizable4. The former correspond to spaces with zero local curvature tensor, even though they may not have trivial topologies. The latter corresponds to spaces with a non-zero constant curvature tensor. In this paper we present the point symmetry classification and the invariant characterization of a system of two geodesic equations for two functions of one variable. The symmetry classification was attempted by Aminova and Aminov’ but they did not identify the Lie algebras of the groups and nor does their work contain the invariant characterization. These aspects are considered in Section 2. The following are well-known and can be found in standard text books. The Christoffel symbols for a metric tensor, g i j , are given by r j k = igim(gjm,k gkm,j - g j k , m ) - the comma stands for partial derivative and gim is the inverse of gij. In this notation, the system of n geodesic equations is
+
$ +rjk+kk
=0
,
i,j,k=l,
. . . ,n,
(1)
where the dot is the derivative relative t o the arc length parameter t defined by dt2 = gijdxadxj. The Riemann tensor is defined by ~ j kil = , r ij1,k .
-ra.j k , l + r imk r? - r iml r? jk* j1
(2)
2. Symmetry classification and invariant characterization We consider a general system of two geodesic equations for two functions of one variable. For n = 2 equation (1) reduces to two geodesic equations for two functions of one variable
+ 2 b ( s , y ) d y l + c ( 5 ,y)yI2 = 0 , y” + d ( z ,y ) d 2 + 2e(z, y)z’y’ + y)yt2 = 0. d’ + a(5,y)5’2
f(5,
(3)
(4)
The Christoffel symbols in terms of these coefficients are
r:,
=
a,r;2= b,ri2= c,r21= ~ i , r=: ~ e , 1 3= ~ f.
(5)
For Rjkl = 0 , i.e. flat space, the system of geodesic equations is linearizable and this case was fully treated in references [l-31. We review this for
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system (3) and (4). Moreover, we consider the case R:kl # 0 for the system (3) and (4). We denote the metric coefficients by 911 = p ; g12 = q , and g22 = r in the sequel. Also we write the Ricci or curvature scalar as R = k, where k can be a constant or a function depending upon the situation. We now list the two-dimensional geodesic equations together with their symmetry generators as obtained in [l].We list the cases according to the Lie algebra L,., T = 2,3,4,5,15 of dimension T spanned by the generators. 1. L15 case Here the space is flat and the geodesic equations are
x” = 0, y” = 0,
(6)
where ’ stands for differentiation with respect to t here and in the following. 2. L5 case The system of geodesic equations is
x” - g(e1xx’2 + 2enyx’y’ - e 2 ~ y ’ = ~ )0, y”
+ &(elyx’2 - 2elxx’y’
-
e 2 ~ y ’= ~ )0,
(7)
where x and y are functions o f t , prime represents differentiation relative to t, g = 1 5 ( e 1 x 2 e2y2) and e l r e 2 = fl. The symmetry generators are a a a y-)a -e1(2 2 a XI = -,X2 = t-,XB = z(x- g)-,
+
+
at
at
a 2 + y-) + -e2(2 ax ay K
X4 = y(x-
a
ax + ay + K ax a a - e2x-.d - g)-’X5 = elyaY dX BY
(8)
3. L4 case The geodesic system is XI’
+ Z(X’2 f y’2) = 0, y“ + 2Zx‘y‘ = 0,
(9)
where a is a constant. The generators of symmetry are
XI
a
a
at
at
= -,x2 = t-’XS
=
a -’x4 dY
d a ax + y--.ay
= x-
4. L3 case There are two subcases listed. The first set of geodesic equations is 2’’
+ a x ” + i(hyx’y’ - EQY”) = 0, y“ - ;h,xt2 + 2 a ~ ‘ y ‘= 0,
(11)
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where
E =
f l ,Q = constant, h = h(y). The symmetries are
a x1= -,xz at
a
= t--,X3 =
at
-.a
ax
The second system of geodesic equations is 2/’
+ h(x)(x’2 f y’2) = 0, y’l + 2h(x)x’y’ = 0,
where h = h ( z ) .The symmetry operators are
a
a
XI= -,xa
a
=t--,X3 = -
at
at
’
5. L2 case The geodesic equations listed are
dl
+ hz(x/2 - EY12) + 2hyz’y’ = 0,
2”
- hY(Edl - y’2)
+ 2hzx’y’ = 0,
(15)
where E = f l ,h = h(x,y). The symmetry generators, in terms of the geodetic parameter, are
XI
=
a
-,xz at
a
= t-. at
We now discuss each of the cases 1 to 5 listed above as taken from
[l] in detail. We here also consider the invariant characterization apart from working out the algebra structures. Both of these aspects were not considered before in [l]nor in other works. The first case is the flat space case and has symmetry algebra sZ(4) as is well known (see references [l-31). However, the invariant criteria for the reduction of the two geodesic equations (3) and (4)to (6) has only recently been fully worked out in reference [3]. This corresponds to linearization via invertible transformations. There are four conditions on the six coefficients of the geodesic equations for linearizability. These are the invariant criteria which are in terms of the six coefficients that give linearization via invertible maps for the geodesic equations. In the second case the system of geodesic equations correspond t o spaces of constant nonzero curvature k . The symmetry algebra is so(3) $d2 for k < 0 (canonical form being a sphere) and so(p, q)$d2 for k < 0, where p + q = 3 (canonical form being a hyperboloid in 1 or 2 sheets). This was looked a t in references [2,3] recently. Also the invariant criteria corresponding to spaces of constant nonzero curvature, in terms of coefficients, for reduction of two
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geodesic equations (3) and (4) to (7) was considered in detail in reference
[41. The third case equations correspond to spaces of nonzero nonconstant curvature k = 2a/22f2cr,if a # 0, -1. The symmetry algebra here is fourdimensional and is d2 @ dh, where dz is the dilation algebra for the geodetic parameter space and dh is the dilation algebra in the geometric space. The invariant criteria for the reduction of the two geodesic equations (3) and (4) to the canonical form (9) is as follows. There are two cases that arise. For the diagonal metric case, q = 0, we have
be - cd = 0, ay
= bz,
ey = f x , dcy - b f z
+ 2cd(b bdx - day + 2bd(e
(17) -
f) = 0,
-
a ) = 0,
with 1 -Icp = d, - e, ae - bd df - e2. 2 The non-diagonal metric case q # 0 gives 1 -kq = b, - ay be - cd 2 and the single condition
+
+
(18)
( a + 4,.
(20)
+
(b
+ f)x
=
Here k = 2a/z2+20. The fourth case equations again correspond to spaces of nonzero nonconstant curvature. There are two subcases. For the first we have that k = -2~Cexp(-2az){(h’/2h)’+(h’/2h)~} provided (h’/2h)’+(h’/2/~)~ #0 or (h’/2h)’ (h’/2h)2 # 0 = A. The symmetry algebra is d2 @ t l , where tl is the one dimensional translation algebra and d2 the usual algebra. The invariant criteria for reduction of two geodesic equations (3) and (4) to the canonical form (11) yields two cases. They are again (17), (18) or (19) and (20) but with k = -2~Cexp(-2az){(h’/2h)’ (h’/2h)2}. The second subcase has k = (-2h’/A)exp(-2Jh(z)dz) with h’ # 0 or h’ # Cexp(2Jh(z)dz). The algebra here too is d2 @ tl. The invariant criteria for the reduction of the two geodesic equations (3) and (4) to the canonical form (13) yields two cases. They are again (17), (18) or (19) and (20) but with Ic = (-%‘/A) exp(-2 h(z)dz). The fifth case equations correspond to spaces of nonzero nonconstant curvature different from the previous case in that the function k is different.
+
+
374
+
Here we have k = -2(h,, chyy)/I'(h),where r is an arbitrary function of h (actually p = I'(h))with the added conditions that h,, Eh,, # 0 , I' # C(h,,+eh,,), h # o/z or h is not a function of z alone. The algebra for this case is d2. The invariant criteria for the reduction of the two geodesic equations (3) and (4)to the canonical form ( 1 5 ) yields two cases. They are again (17), (18) or (19) and (20) but this time with k = -2(h,,
+
+
Eh,,) /I'(h). 3. Discussion We have provided the symmetry algebra structure as well as invariant criteria for each of the classes of geodesic equations presented in refer. therefore provides us with a complete understanding of twoence [ l ]This dimensional geodesic equations. It will be of interest to classify in a similar manner three-dimensional geodesic equations for symmetries as well as to provide invariant criteria. To that end some work has been done for the linear and nonzero constant curvature cases in references [2,3].
4. Acknowledgments
FM acknowledges a visiting professorship through the HEC of Pakistan during which this work was started at NUST-CAMP. References 1. A. V. Aminova, and N. A. M. Aminov, Projective geometry of systems of differential equations: general conceptions Tensor N S 62, 65 2000. 2. T. Feroze, F. M. Mahomed, and A. Qadir, The connection between isometries and symmetries of geodesic equations of the underlying spaces, to appear, Nonlinear Dynamics. 3. F. M. Mahomed, and A. Qadir, Linearization criteria for a system of secondorder quadratically semi-linear ordinary differential eqautions, in press, Nonlinear Dynamics. 4. F. M. Mahomed, and A. Qadir, Invariant criteria for a system of geodesic equations corresponding to spaces of constant nonzero curvature, in preparation.
GRAVITATIONAL COLLAPSE WITH NEGATIVE ENERGY FIELDS JAYANT V. NARLIKAR Inter- University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, P u n e 411 007, India This paper re-examines the classical problem of the collapse of a dust ball, with the added input of a negative energy scalar field. It is shown that not only is the collapse halted prior to the singularity, but a black hole may not even form. The object bounces at a stage when it is still outside any event horizon.
1. Introduction The problem of gravitational collapse has been part of astrophysical literature from the 1930s. The question of how stars maintain their equilibrium in spite of their strong gravitational force has been addressed by physicists using Newtonian as well as Einsteinian gravity. The problem becomes more difficult in the general relativistic framework as was shown by various theorems on spacetime singularity (for details see: Hawking and Ellis 1973l). As these theorems show, provided certain positivity requirements are satisfied by the energy-momentum tensor of the physical contents of space and time, the star (or any compact massive object) would eventually collapse into a spacetime singularity. It was pointed out in the 1960’s (Hoyle and Narlikar 1964’) that a singularity can be avoided in a gravitationally collapsing object, provided negative energy fields exist. More specifically the C-field, used in the context of the steady state cosmology, has the ability to halt gravitational collapse and make the collapsing object bounce. In the 1960s, the physics community viewed the idea of negative energy fields with suspicion and so this line of investigation did not attract the attention it deserved. Today the situtation is different: negative energies are gaining credibility and respectability amongst theoretical physicists. The notion of phantom fields (see for example, Gibbons (2003)3, Shin’ichi and Odnitsov (2003)4) which are no different from the C-field of Hoyle and Narlikar (1963)5, is used in
3 75
376
cosmology today. It is therefore pertinent to revive the 1960’s concept, both in cosmology and in the study of compact massive objects. In this paper we consider a minor variation on the C-field used by Hoyle and NarIikar5 - a variation that gives rise to the quasi-steady state cosmology (QSSC) proposed in 1993 by Hoyle, et al.617 We will consider a specific question: Does a compact massive object become a black hole if the negative energy scalar field that drives the QSSC is also present? The mathematical framework for this problem has been discussed by Hoyle, Burbidge and Narlikar in 19958, (HBN hereafter), and we outline briefly its salient features. To begin with, it is a theory that is derived from an action principle based on Mach’s Principle and assumes that the inertia of matter owes its origin to other matter in the universe. This leads t o a theoretical framework wider than general relativity as it includes terms relating t o inertia and creation of matter. We will address our problem of gravitational collapse in such a framework, using for our explicit calculation the classic problem of the collapsing dust ball. The dust ball has no internal pressure and so it presents a situation wherein gravity apparently has no opposition t o collapse. Except, that in the present situation the presence of the negative energy field will resist contraction. This is what we will consider explicitly. 2. The equations The equations of general relativity are replaced in the present theory by
with the coupling constant
f defined as f = -
2 372.
(We have taken the speed of light c = 1.) Here T = h/mp is the characteristic life time of a Planck particle with mass mp = The gradient of C with respect to spacetime coordinates zi(i = 0,1,2,3) is denoted by
d m .
Ci. Although the above equation defines f in terms of the fundamental constants, it is convenient t o keep its identity on the right hand side of Einstein’s equations since there we can compare the C-field energy tensor directly with the matter tensor. Note that because of positive f , the C-field has negative kinetic energy. Also, the constant X is negative in this theory.
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As in the old steady state theory, in the QSSC also the expansion is driven by creation of matter. In fact creation of matter provides the source for the C-field and vice versa. We will consider here the implications of this property for creation of matter in explosive fashion. The idea that the universe has new matter being created in explosive fashion had been proposed by V. Ambartsumiang who pointed to the active galactic nuclei and (possibly) expanding clusters of galaxies as instances of minicreation events. Although the steady state theory used the C-field to generate new matter, in the 1960's version, the creation was at a uniform rate all over the universe. The question now arises of why astrophysical observation suggests that the creation of matter occurs in some places but not in others. For creation to occur at the points Ao, Bo, .. ., it is necessary classically that the action should not change (i.e. it should remain stationary) with respect to small changes in the spacetime positions of these points, which can be shown to require
C~(A~)CZ(A~)
= c~(B~)cz(B~) = . . . = mg.
(3)
This is in general not the case: in general at a typical point, the magnitude of C i ( X ) C z ( X )is much less that m$. However, as one approaches closer and closer to the surface of a massive compact body, C i ( X ) C i ( X )is increased by a general relativistic time dilatation factor, whereas mp stays fixed. This suggests that we should look for regions of strong gravitational field such as those near collapsed massive objects. In general relativistic astrophysics such objects are none other than black holes, formed from gravitational collapse. We will now see how the classic notion of black holes gets modified.
3. Collapse and bounce of a dust ball As mentioned earlier, theorems by Penrose, Hawking and others (see Hawking and Ellis 1973l) have shown that provided certain positive energy conditions are met, a compact object undergoes gravitational collapse to a spacetime singularity. Such objects become black holes before the singularity is reached. However, in the present case, the negative energy of the C-field intervenes in such a way as to violate the above energy conditions. What happens to such a collapsing object containing the C-field apart from ordinary matter?
378
We conjecture that such an object does not become a black hole. Instead, the collapse of the object is halted and the object bounces back, thanks to the repulsive effect of the C-field. We will refer to such an object as a compact massive object (CMO) or a near-black hole (NBH). In this section we discuss the problem of gravitational collapse of a dust ball with and without the C-field to illustrate this difference. Consider how the classical problem of gravitational collapse is changed under the influence of the negative energy C-field. First we describe the classical problem. This was first addressed by B. Datt in 19381° and later, independently, by J.R. Oppenheimer and H. Snyder". We write the spacetime metric inside a collapsing dust ball in comoving coordinates (t, r, 8,
$1
as
["
ds 2 = dt 2 - a2(t) 1- ar2
1
+ r2(d02+ sin20d$2) ,
(4)
where r,8,$ are constant for a typical dust particle and t is its proper time. Let the dust ball be limited by r 5 rb. In the original collapse problem we may describe the onset of collapse at t = 0 with a ( 0 ) = 1 and b(0) = 0. The starting density po is related to the constant Q by
The field equations (1) without the C-field then tell us that the equation of collapse is given by a2 =
.(--),
l-a
and the spacetime singularity is attained when a(t)
0 as t -+t s , where
Note that we have ignored the X-term, as it turns out to have a negligible effect on objects of size small compared to the characteristic size of the universe. The collapsing ball enters the event horizon at a time t = tH when rba(tg) = 2GM,
(8)
379
where the gravitational mass of the dust ball is given by 47r ar3 M = -rbpo = 3. 3 2G
(9)
This is the stage when the ball becomes a black hole. When we introduce an ambient C-field into this problem, it gets modified as follows. In the homogeneous situation under discussion, C is a function o f t only. As before, let a(0) = 1, h(0) = 0 and let C at t = 0, be given by p. Then it can be easily seen that the equation (6) is modified to 1-a
where y
= 27rGfp2 >
1-a
0. Also the earlier relation ( 5 ) is modified to
It is immediately clear that in these modified circumstances a ( t ) cannot reach zero, the spacetime singularity is averted and the ball bounces at a minimum value, amin > 0, of the function a ( t ) . Writing p = y / a , we see that the second zero of b ( t ) occurs at amin= p. Thus even for an initially weak C-field, we get a bounce at a finite value of
4t). 4. Is a black hole formed?
But what about the development of a black hole? The gravitational mass of the black hole at any epoch t is estimated by its energy content, i.e., by,
47r M ( t ) = -rb3a3(t) 3 2G Thus the gravitational mass of the dust ball decreases as it contracts and consequently its effective Schwarzschild radius decreases. This happens because of the reservoir of negative energy whose intensity rises faster than that of dust density. Such a result is markedly different from that for a collapsing object with positive energy fields only. From (12) we have the ratio
380
Hence,
-=-{--(l+p)}. dF arf 2p da a2 a We anticipate that p << 1, i.e., the ambient C-field energy density is much less than the initial density of the collapsing ball. Thus F increases as a decreases and it reaches its maximum value at a 2 2p. This value is attainable, being larger than amin. Denoting this with F,,, we get
a rf 4P In general arb2 << 1 for most astrophysical objects. For the Sun, a ~E f 4 x lop8, while for a white dwarf it is 4 x We assume that A, although small compared to unity, exceeds such values, thus making F,, < 1. In such circumstances black holes do not form. We consider scenarios in which the object soon after bounce picks up high outward velocity. From (10) we see that maximum outward velocity is attained at a = 2p and it is given by F,,
-,
N
aLax M
a
-.
4P
As p << 1, we expect a,,, t o attain high values. Likewise the C-field gradient (C in this case) will attain high values in such cases. Thus, such an object after bouncing at amin, will expand and as a ( t ) increases the strength of the C-field falls, while for small a ( t ) increases rapidly according to equation (10). This expansion therefore resembles an explosion. Further, the high local value of the C-field gradient will trigger off creation of Planck particles. This possibility suggests its relevance t o high energy phenomena like gamma ray bursts, and will be discussed in a separate paper. It is worth stressing here that, even in classical general relativity, the external observer never lives long enough to observe the collapsing object enter the horizon. Thus all claims to have observed black holes in X-ray sources or galactic nuclei really establish the existence of compact massive
381 objects and, as such, they are consistent with the NBH concept. To the external observer in classical general relativity a black hole never forms. However in this case, when the negative energy field is present, the black hole does not form for any observer.
5. Conclusions The introduction of a negative energy field (1) permits creation of matter near dense regions, (2) reverses gravitational collapse at finite density, (3) can lead to bounce that develops into an explosive situation, and (4)does not permit the formation of a black hole. These conclusions suggest that negative energy fields have a role to play in nature and may be invoked t o explain some of the so-called violent phenomena in the universe.
Acknowledgement
I wish to thank the organizers of the 12th Regional Conference on Mathematical Physics for their warm hospitality.
6. References
References 1. S.W. Hawking and G.F.R. Ellis, Large Scale Structure of Space-time, (Cambridge University Press, 1973). F. Hoyle and J.V. Narlikar,Proc. R. SOC.A 278,465 (1964). G. Gibbons, hep-th/0302199 (2003). N. Shin’ichi and S.D. Odnitsov, hepth/0304131 (2003). F. Hoyle and J.V. Narlikar, Proc. R. SOC.A 273,1 (1963). F. Hoyle, G. Burbidge and J.V. Narlikar, A p . J . 410,437 (1993). 7. F. Hoyle, G. Burbidge and J.V. Narlikar, 2000, A Diflerent Approach to Cosmology, (Cambridge University Press, 2000). 8. F. Hoyle, G. Burbidge and J.V. Narlikar, Proc. R. SOC.A448,191 (1995). 9. V.A. Ambartsumian, 1961, A . J . 66, 536 (1961); V.A. Ambartsumian, 1965, Structure and Evolution of Galaxies, Proc. 13th Conf. on Physics, University of Brussels, (Wiley Intersource, New York, 1965). 10. B. Datt, 2. Phys. 108,314 (1938). 11. J.R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939). 2. 3. 4. 5. 6.
QUANTUM NON-LOCALITY, BLACK HOLES AND QUANTUM GRAVITY ASGHAR QADIR Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Peshawar Road, Rawalpindi, Pakistan and Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahmn 31261, Saudi Arabia E-mails: [email protected]; [email protected]; aqadirQkfupm.edu.sa It is now generally taken for granted that quantum gravitational effects will appear at Planck scale and no earlier. However, there is no theory of “quantum gravity”. Since Planck scales will not be experimentally accessible in the foreseeable future, purely speculative theories have been proposed, whereby the desired effects could appear at scales before the Planck scale. There has been no experimental support for these theories. On the other hand, Penrose has forcefully argued that the tests of Bell’s inequalities demonstrate nonlocality of Nature at the quantum level that is incompatible with the usual picture of spacetime at meter and and even kilometer scales. He has also argued that the state vector reduction is due to the effect of quantum gravity. In this paper the onset of quantum gravity is probed by confronting the extreme quantum (a photon) by the extreme of a gravitational field (a black hole).
1. Introduction
At the end of the nineteenth century it had been stated that practically all of science was understood and there were only ‘(twosmall clouds on the horizon”. These “clouds” developed into quantum theory (QT) and special relativity. There had also been a minor astronomical problem noted, that Mercury did not follow the orbit required by Newtonian theory. Attempts to explain the discrepancy by appealing to some unseen astronomical body had met with no observational success but due to problems of viewing a planet in an orbit between the Sun and the orbit of Mercury, this difficulty had not been regarded as critical. Following his early success with the restricted,
382
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or special, theory of relativity in 1905, Einstein had been searching for an unrestricted, or general, theory that would deal with arbitrary motion [l]. His theory of 1916 (GR) not only provided the required generalization, it also explained the discrepancy in the orbit of Mercury. At the end of the twentieth century tests of these theories verified them to ever higher accuracy. In the beginning of the twenty-first century, between them, QT and GR, form the basis for our understanding of the structure of matter at the smallest scale and of the Universe at the largest scale. However, at
a fundamental level the two theories seem t o be mutually incompatible! The problems are seen to arise when one tries to provide a consistent quantum theory for gravity. Dirac had developed a procedure for converting any classical field theory to a quantum field theory. This was called Dirac, or canonical, quantization. When applied to electromagnetism the procedure had yielded the meaningless result of infinite probability of events. A method to obtain sensible answers by changing the norm of the function describing the fields, called renormalization, was adopted as a paradigm for quantum field theories. When quantization was applied to gravity, again infinite answers resulted. However, here the renormalization procedure did not help. It was subsequently discovered that there were conditions for a quantum field theory to be renormalizable, which were not met by the Dirac quantization of general relativity. Other procedures of quantization were attempted, but with no better success [2]. In the early days of quantum theory and relativity it was realised that these theories reduced to “Newtonian” theory in the limit that the relevant universal constant, for quantum theory h, for special relativity 1/c and for general relativity G, is taken to be zero. Correspondingly, if units are chosen such that these constants have unit magnitude the differences of the relevant theory from classical theory will be significant. Planck noted that quantities with units of length, time and mass could be constructed from these fundamental constants. (One normally uses “dimensional analysis” to determine the validity of formulae, taking three distinct dimensions: length L ; time T ; and mass M . ) He further pointed out that, at these values, all three theories would be required. These units are now called Planck units [3]. They are: the Planck length lp = the Planck time t p = and the Planck mass m p = which yields the Planck energy Ep = Their values are of the order of 10-33cm, 10-40sec, 10-7gm and 10igGeV, respectively. Though Planck’s argument only says that quantum theory and general relativity cannot be neglected at Planck scales, the view somehow seems
d v ;
d m ;
dw.
m,
384
to have crept into the literature, and thence got embedded in scientific thought, that effects of a theory of quantum gravity will only appear a t these scales. Had it been a linear field theory the expectation would not have been unreasonable. However, general relativity gives a nonlinear field theory of gravity. (Note that it is often regarded as being a theory of gravity, but that is not how Einstein had originally intended it, nor how he seems to have thought of it. It was primarily meant to be a theory of arbitrary motion, but as developed was limited to motion in a gravitational field. Einstein spent most of his later life in trying to remove this limitation from the theory and allow for motion in an electromagnetic field as well, in a “unified field theory” .) Using a weak field approximation of the theory for quantization naturally leads to this expectation. The actual theory must retain its nonlinearity t o give significant general relativistic quantum effects. The problem is that quantum theory, relying as it does on the superposition principle, is necessarily linear. This is only one of the manifestations of the incompatibility between the world-views of the two theories. 2. Incompatibility in Tests of Bell’s Inequalities There had been a long debate between Bohr and Einstein regarding the foundations of quantum mechanics [4]. It was Einstein’s contention that the quantum theory, as formulated a t the time, was an incomplete description of Nature and a better theory (probably incorporating his general relativity) would be required. Bohr argued that it was not that quantum mechanics was incomplete, but that a t the fundamental level Nature could not be described as Einstein expected. His arguments were epistemological and based on logical positivism. It depended on the belief that there is no physical reality to be described other than the final outcome of some experiment. Any question that asks what happened between the start and end of the experiment must be ruled out as being metaphysical. The very process of measurement changes what is being observed. To establish that physical reality is a well-defined concept; Einstein, Podolsky and Rosen defined it [5]. They went on t o try t o demonstrate that quantum theory is incomplete by showing that it would require fasterthan-light signals, which are disallowed by special relativity. The essence of the argument is that if a composite system is broken into two, one can know the value of the momentum of one particle with certainty without interfering with it, by measuring the momentum of the other component and appealing to the conservation of momentum. The argument was inconclusive because of the Heisenberg uncertainty principle applied to the momentum, which
385
is a continuous variable [6]. Bohr maintained that no information could be extracted from such “experiments” [7]. To avoid the uncertainty Bohm suggested using angular momentum, which can only take discrete values [8].The assumption was that each of the particles would carry their spins and observation would determine what the spin of one particle is so that the spin of the other could be deduced. This would then be verified by experiment. He went on to generalize it for any “local hidden variable” theory. This suggestion was developed further by Bell [9] to construct a system of inequalities for the probability of outcomes for particles carrying “local hidden variables”, based on a “classical picture” that will be violated if quantum theory is valid. Though “common sense” would have led one to expect that quantum theory would be ruled out, the surprising fact is that when Aspect attempted to test for the inequalities [lo] it turned out that quantum theory is correct and “common sense” wrong! It has been claimed that this test proved that Einstein was wrong and Bohr right. I fail to see how. Bohr claimed that the question posed in the so-called “EPR-paradox” could not be meaningfully asked. Since Aspect’s experiment answered it, it could certainly be meaningfully asked. The fact is that Einstein had, once again, made a fundamental contribution, but this time his expectation was proved wrong. The bigger surprise is that the test was conducted at meter scales. Contrary to expectations, that quantum mechanics applies only at molecular and atomic scales, it was here seen to apply at macroscopic scales. While Einstein’s argument that quantum effects could be used for superluminal signalling turned out to be wrong, one gets around the apparent violation of causality only by appealing to “non-locality’’. That the hidden variables are non-local means that quantum entities cannot be thought of as points but as finite sized. In fact, the size of the object can be of meter scale, not Planck scale! As Penrose insists, this shows that the usual spacetime picture has broken down for the purposes of this experiment [2]. It is worth mentioning that the experiment has now been performed at kilometer scales. 3. Reduction of the Wave Function
Bell pointed out a crucial problem in quantum theory [ll]as it now stands to do with what is ‘called “the measurement problem”. Quantum theory, as given by the Schrodinger or Heisenberg equations, is linear and time reversible. This fact is inconsistent with the fact that measurement does not allow reversibility. According to the standard theory, the process of
386
measurement is described by the action of a projection operator, which is nonlinear and gives the “reduction of the wave function”. It is this “reduction” that provides the non-locality required when measurement of the hidden variable of one particle gives the hidden variable of the other. The wave function, which represents a probability amplitude, may be regarded as a single entity till the measurement is made and one of the possible outcomes becomes a certainty, while the other outcomes become impossible. At this stage the wave function is said to “collapse” and the composite system instantaneously becomes a collection of the constituents. The standard interpretation of quantum mechanics takes the measurement to occur when “the quantum object encounters a classical measuring device” [12]. Bell points out that there are no separate “classical” devices. If the quantum theory is in fact complete the “classical” measuring device is just a collection of quantum entities. To complete the theory, at least the “measuring device” must be provided a fully quantum description. Otherwise one has only entered the first step of an endless regress. Penrose, following the general philosophical views of Einstein, proposes that the non-linearity in the “reduction”, or “collapse”, of the wave function may arise from the non-linearity of general relativity [13]. This is an attractive idea. It avoids the introduction of extra, mysterious, processes to explain measurement. However, it still fails to provide any answer to the question “where does quantum behaviour end and classical behaviour begin for the purpose of approximating the classical measuring device?” It is worth mentioning for completeness that there is another interpretation, due to Everett [14], that addresses the measurement issue - the “many worlds interpretation”. According to it, each time a measurement is made for which there could be different outcomes all of t h e m occur, but do so in different copies of the world. It is not clear that one has not traded a bad problem for a still worse one. Most people find it difficult to swallow all these worlds. Those who adopt the idea generally use it as a a purely mathematical description of an ensemble of possible worlds, of which one is the “real world”. A more esoteric interpretation is that the consciousness of intelligences threads through these many worlds. 4. Degenerate Stars and the Role of Gravity in
Quantum Theory To test for the above non-locality, one needs to arrange for the delicate composite quantum entity to survive over large enough distances for long enough times. However, the scale at which this incompatibility occurs is
387
not limited to these refined tests of Bell’s inequalities. For degenerate configurations like neutron stars, or white dwarfs, there is non-local collective quantum behaviour at the scale of about 10 kilometers or 10,000 kilometers! In white dwarfs only the electrons form a degenerate Fermi gas while the nuclei behave classically. Neutron stars are almost entirely degenerate Fermi gases, with only a slight amount of matter beyond the Fermi temperature 1151. Note that these objects do not have to be maintained carefully but are remarkably robust. When Einstein introduced general relativity he appealed to Mach’s principle, but later came to regard it as redundant. It is often pointed out that Mach’s principle does not need to be assumed but is a consequence of general relativity. Barbour [16] believes that it states a non-trivial requirement and is hence worth considering further, especially in the context of quantum gravity. The key point is that there is no a priori reason to believe in a spacetime in which there is nothing. As such, for spacetime to exist matter must be present and hence the existence of mass determines the spacetime. At the basic level, then, spacetime is no more than a model to describe dynamics. Different observers (with different local distributions of matter) would construct different models of spacetime in general. One needs an additional assumption that the various models will match. This is Mach’s principle and it could break down at the quantum level. It has generally been taken for granted that quantum theory rules the domain of the very small size and the very low mass or energy, while gravity rules the large scale size, mass and energy. In degenerate stars we see both in different perspectives. Here quantum theory dominates on enormous scales and gravity plays a new role - not in opposition to the quantum but propping it up. Is the presumption that large masses destroy the fragile quantum behaviour wrong? What exactly is the role of gravity vis-a-vis the quantum? Is it irrelevant except at Planck scales, as superstring theory (and similar approaches) insist? Does it destroy quantum coherence as Penrose maintains (as I will discuss shortly)? What about Barbour’s viewpoint using Mach’s principle? Does it instead stabilize the quantum, as it seems to do for degenerate stars? In my opinion there are too many preconceptions on this issue at present and it is necessary to take a fresh look at the matter without them.
388
5. Confronting the Photon with the Black Hole Our purpose here is to probe the Penrose suggestion and look for the onset of simultaneous effects of QT and GR. In particular, one would like to check the validity of Penrose’s expectation that the linear QT is broken down by the non-linear GR. In the conclusion I will give a suggestion of how to view the results given in this section. Wheeler’s view is that in facing a problem one should face it “squarely” [17],i.e. in its most extreme form. The ultimate (and in fact original) quantum is the photon. The most extreme gravitational source is a black hole. It would be best to see both together to highlight the problems of quantum gravity so as to be able to ‘‘confrontthem squarely”. Since black holes are the ultimate phase of gravitational collapse after a star has passed through a partial and a nearly totally quantum degenerate phase, contrary to normal expectations, it could be argued that it would be an even more extreme quantum object than degenerate stars. Holz and Wheeler [18] had made an interesting proposal for observing a black hole by shining light on it and looking at the totally reflected rays (i.e. whose path is bent by an odd multiple of 7r/2). For a Schwarzschild black hole, this will give a primary ring for the first multiple and secondary concentric interior rings for the higher multiples. For a spinning black hole this ring would be distorted [19].More interestingly, it will suffer chromatic aberration [20]. If the light goes in with an angular momentum “counterrotating” relative to the black hole, it will be red shifted and if it goes in “co-rotating” it will be blue shifted. Consider a photon of some wavelength, A, shot towards a rotating black hole of mass M and spin (angular momentum per unit mass) a. The photon will lose energy and angular momentum to the hole if it red-shifts. Consequently, it will “spin up’’ the hole by the corresponding amount. For a rotating black hole the outer horizon (beyond which even light can not escape) is given by Th = GC 2M +
/y-.’
c2 .
(5.1)
In the extreme case that a = G M / c 2 the horizon size is simply the mass in gravitational units (G = c = 1). In this case the distance of closest approach for the totally reflected photon is G M / c 2 for the counter-rotating photon. The closer it approaches the more energy is transferred. Thus, the maximum “spin-up” occurs when the photon passes by at G M / c 2 from the black hole centre. For a spin value of a = the distance of closest
%&,
389
approach is
+
b = (1 2@)GM/c2.
(54
The corresponding wavelength shift for the first ring is given by bX GMa X b2c2 ‘
_ N -
(5.3)
For the nth ring this value is multiplied by (2n - 1). The co-rotating case is also mentioned for completeness. Here energy and angular momentum would be extracted from the black hole and carried away by the photon. The closest approach for the totally reflected photon is 4GM/c2. Here for the above spin value the distance of closest approach is E 4GM b = (1- -)(5.4) 9 c2 . The above formula for the wavelength shift applies here as well, but with a negative sign. The “spinning-up’’ experiment will not work if the photons are shot towards the black hole so that some are co-rotating and some counterrotating. In other words, they must all be shot a t one side of the black hole. For a normal, central black hole of 106Ma, there would be no inherent limit on the accuracy for any reasonable wavelength photon. However, for a smaller black hole one would have to worry about the uncertainty due to its “size”. How small would the black hole have to be for, say, a visible photon of 5 x 10-5cm not to be able to ignore the hole as it goes by? The general requirement for the “spin-up experiment” to work is N
X
< b = GM/c2.
(5.5)
Above this wavelength, the “quantum entity” and the “classical general relativistic object” will not interact. For the photon t o be just able t o “spin-up” the hole it must have a mass greater than about 6 x 1022gm. This is not an unreasonably low value. We are not limited to Planck scales to talk of situations where quantum theory and gravity have to be taken into account. Admittedly, this is also well beyond present technology and it is difficult to see what information it would give about any putative theory of quantum gravity. However, for my present purpose it is enough to be able to demonstrate that the Planck scale is not required for quantum relativistic effects to appear. Let us now cast the above discussion, not in terms of sizes but in terms of energies. Re-phrasing the above requirements, for a photon of energy El
390
we require than
E>
GM
+
c3h d G 2 M 2 - a2c2
(5.6)
A photon of lower energy would not be affected by the hole but a photon of higher energy could spin up the hole. Notice the similarity of this behaviour to the photo-electric effect in that there is a threshold below which nothing happens and above which the effect starts! The fractional increase in energy is given by SE E
-
_ N -
GMa b2c2
-
GJ b2c2’
(5.7)
and of angular momentum by
SJ GE J - bc3’
-N-
For the onset the inequality (5.6) must hold. It can do so easily for any photon for large enough black holes. However, for the effect to be significant the ratios in eqns. (5.7) and (5.8) must be large. As such, it would be greatest for smaller black holes but would disappear if the black hole became too small. In the extreme case of maximum spin of the black hole, the entire photon energy can be handed over to the black hole. Note that according to eqn.(5.6) the greater the mass of the black hole the less the minimum energy for the quantum to interact with the black hole. Larger masses seem to make the quantum entity stabler! Look at eqn.(5.6) in reverse as giving the limiting energy at which the photon can ignore the black hole. The greater the mass of the “classical general relativistic object” the more easily it can be ignored by the“quantum entity”. This seems to be contrary to what Penrose expects. 6 . Conclusion
There can be no claim that the above discussion provides a theory of quantum gravity. However, it does answer a point that is made, or taken for granted, for it, that of the scale of onset of quantum gravity. We see that it is not the Planck scale but arises for different masses and energies at different sizes as Penrose expects. However, its behaviour seems to be opposite to Penrose’s expectation. How did Penrose ignore the quantum nature of white dwarfs and neutron stars? I do not think he did. These objects have a large Ricci curvature and relatively small Weyl curvature. I presume that he would argue that the Ricci curvature can stabilise the quantum but that
391
the Weyl curvature de-stabilises it. He talks of the “single graviton level” and “gravitons” come from the Weyl, not the Ricci, curvature. However, in this thought experiment we are dealing with black holes, which have only Weyl and no Ricci curvature. Clearly, the scale will change with the type of experiment considered. What was considered here was only the most extreme case of both general relativistic and quantum behaviour. Other thought experiments would yield more complicated expressions. It should be stressed that the experiment proposed is only a thought experiment. There is no expectation that it could be actually performed meaningfully. Where did the Planck scale go? The answer is contained in the above analysis. It arises in answer to the question “when will the photon energy equal the rest energy of the black hole, so that both are equally dominant?” The interesting answer is that it is essentially at the Planck energy! More accurately,
E
=h E p .
(6.1)
It is worth remarking that Penrose’s view that any theory that addresses quantum theory and general relativity should also provide for the large scales seen in the tests of Bell’s inequalities is accommodated here. Of course, this is not a theory of quantum relativity but merely an exploration of the limits at which both theories can be used ignoring the other. The quantum measurement problem still needs resolution. The changes that relativity will require yet need exploration. If the usual concept of spacetime has to be modified this provides no inkling of how that is to be done. Acknowledgment I am most grateful to the King Fahd University of Petroleum & Minerals, where this work was started, for hospitality during my visit as Adjunct Professor. I also thank Viqar Hussain and Madhavan Varadarajan for their comments on my conference presentation, which helped me sharpen the presentation here.
References 1. A. Pais, Subtle is the Lord: The Science and Life of Albert Einstein, (Clarendon Press, 1982).
392 2. R. Penrose, The Road to Realtiy: A Complete Guide to the Laws of the Universe, (A. Knopf, 2004). 3. J.A. Wheeler, “From Relativity to Mutability” in The Physicist’s Conception of Nature, ed. J. Mehra, (D. Reidel, 1973). 4. M. Jammer, Conceptual Foundations of Quantum Mechanics, (Addison Wesley, 1984). 5. A. Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47,777 (1935). 6. W.H. Furry, Phys. Rev. 49,393 (1936). 7. N. Bohr, Nature 136,65 (1935); Phys. Rev. 48,696 (1935). 8. D. Bohm, Quantum Theory (Prentice Hall 1951); D. Bohm, Phys. Rev. 85,166, 180 (1952); D. Bohm and Y. Aharonov, Phys. Rev. 108,1070 (1957); D. Bohm and Y. Aharonov, Nuovo Cimento 17 964. 9. J.S. Bell, Physics 1, 195 (1964); J.S. Bell, Rev. Mod. Phys. 38,447 (1966). 10. A. Aspect, Phys. Lett. 54A (1975) 117; Phys. Rev. D14 (1976) 1944; A. Aspect, C. Imbert and G. Roger, Opt. Comm. 34 (1980) 46; A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460; A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49 (1982) 1804; A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49 (1982) 91. 11. J.S. Bell, The Speakable and the Unspeakable in Quantum Mechanics, (Cambridge University Press 1987). 12. J.A. Wheeler, “Law Without Law” in, Quantum Theory and Measurement, eds. J.A. Wheeler and W.H. Zurek, (Princeton University Press 1983). 13. R. Penrose, “The Role of Gravity in the Reduction of the Wave Function” in Quantum Concepts of Space and Time, eds. C.J. Isham and R. Penrose, (Cambridge University Press, 1974); ibid., Gen. Rel. 6 3 Grav. 28,581 (1996). 14. H. Everett, “Relative State Formulation of Quantum Mechanics” in Quantum Theory and Measurement, eds. J.A. Wheeler and W.H. Zurek, (Princeton University Press 1983). 15. R. Ruffini, “Relativistic Astrophysics” in Physics and Contemporary Needs Vol.1, ed. Riazuddin, (Plenum Press 1977). 16. J.B. Barbour, The Discovery of Dynamics: A Study From a Machian Point of View of the of the Discovery and Structure of Dynamical Theories, (Oxford University Press 2001). 17. J.A. Wheeler, Private communication. 18. D.E. Holz and J.A. Wheeler, Astropohys. J. 57,330 (2002). 19. F. DePaolis, A. Geralico, G. Ingrosso, A.A. Nucita and A. Qadir, Astron. and Astrophys. 415,1 (2004). 20. F. DePaolis, G. Ingrosso, A.A. Nucita and A. Qadir, ”Observing Black Holes”, in Proceedings of the 11th Regional Conference on Mathematical Physics, eds. S . Rahvar, N. Sadooghi and F. Shojai, (World Scientific 2005).
HOMOTHETIC AND CONFORMAL MOTIONS K. SAIFULLAH Centre f o r Advanced Mathematics and Physics, National University of Sciences and Technology, Rawalpindi, Pakistan (On leave from Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan) Email: [email protected] The metric tensor plays a central role in Riemannian geometry and its applications to the general theory of relativity. Invariance under Lie transport of the metric tensor defines motions. If it is preserved up to a constant factor we get homothetic motions, and up to a variable factor conformal motions. These symmetries are discussed for some classes of spacetimes with particular reference to the dimensionality of the related Lie algebras. Keywords: Motions; Homothetic Motions; Conformal Motions.
For two differentiable vector fields, X, Y and a differentiable function f on a smooth manifold M ,
[X,Ylf = X F f ) - Y F f )
(1)
is the Lie bracket of X = X"& and Y = Y a & . A Lie algebra, C, is a vector space over some field F equipped with the Lie bracket satisfying linearity
[ax+ bY, Z] = a [ X ,Z] + b[Y,Z]
[X,aY + bZ] = a[X,Y] + b[X,Z],V a , b E R; skew commutativity
[X,YI = -[Y,XI; and the Jacobi identity
[X,[Y,Zll
+ [Y,[Z, XI] + [Z, [X,Y]]= 0.
393
394
The vector fields X, Y are called generators of the Lie algebra. The Lie algebra is said to be finite (say n) dimensional if its basis is finite and is otherwise infinite dimensional. These generators satisfy the fundamental relations of the Lie algebra
[Xi,Xj] = CtjX,, where Ctj = -C;i are the structure constants for which the Jacobi identity gives
The metric tensor g is a dynamical quantity in general relativity. The isometry group G, of ( M , g )is the Lie group of smooth maps of M onto itself leaving g invariant, where m is the number of generators (isometries) of the group. Its Lie algebra consists of continuously differentiable transformations K a & , where K a = K a ( x b )are the components of the vector field, K, known as a Killing vector (KV) along which the Lie derivative of the metric tensor g is zero i.e. E K ( g a b ) = 0. In addition to isometries there are other motions - homothetic motions (HMs) and conformal motions - which are also useful. Conformal motions are directions along which the metric tensor of a spacetime remains invariant up to a scale i.e.
EEgab = &lab.
(2)
They are determined by the arbitrary constants appearing in the vector field = taa/axa when 4 = @ ( t , z , g , z ) . If @ is constant, the equation gives HMs. The study of the symmetry groups of a spacetime is a useful tool in constructing spacetime solutions of the Einstein field equations (EFEs) and also classifying the known solutions according to the Lie algebras, or structure generated by these symmetries. Conformal motions have been found for Minkowski spacetimes’, F’riedmann-Robertson-Walker spacetimes2 and pp-waves3. We present some results regarding the dimensionality of the algebra of motions4y5: Theorem 1 A Riemannian space, V,, admits a maximal group of motions, G,, where m 5 n(n 1 ) / 2 . Theorem 2 A Riemannian space, V,, cannot admit a maximal group of motions G,, where m = n(n 1 ) / 2 - 1. If a spacetime admits a G, as the maximal group of isometries then the HM group H , is at most of order r=m+l.
<
+
+
395
Theorem 3 The set of conformal vector fields o n M is finitedimensional and its dimension is less than or equal to 15. If this maximum number is attained the spacetime is conformally flat. If it is not conformally fiat then the dimension of the set of conformal Killing vectors (CKVs) is
r. The classification of HMs of spherically symmetric spacetimes admitting maximal isometry groups larger than SO (3) was obtained along with their metrics5. The metric for spherically symmetric spacetimes is ds2 = -eu(t,T)&2 + eA(t,r)dT2+ r2df12,
(3)
+
where df12 = do2 sin2 Odp2. For spherically symmetric spacetimes the possible maximal homothety groups Hr are of the order r = 4,5,7,11. The solution of the HM equations for this metric, up to known functions of 6 and 4, comes out to be
Ho = [-r 2 ep-u {sin 0 (gl sin 4 - g 2 cos 4) + g3 cos 0) + g4], H1 = [T2ep-' (sin8 (9; sin $ - g; cos ~ p )+ g; cos e } + g 5 ] , H2 = [- cose (gl sin 4 - g2 C O S ~ + ) g3 sin8 + (c1 s i n 4 - c2 cos 4)] , H S = [- cosece (91 cos4 g2 sin(6) + cote ( C I C O S + ~ c2 sin$) + cg] ,
+
where dot and dash denote derivatives with respect to t and r , respectively. Here cj ( j = 1,2,3) correspond to the generators of SO (3). The metric admitting maximal group of motions, G4, can be written as ds2 = T2(1-a)dt2 - dr2 - r2df12,
(0
#
1)
and the HMs are
+
Ho = (~$0 H 1 = r40,
CO,
H2 = (clsin~-c2cosp), H 3 = cot O(c1 cos 4 + c2 sin 4), where a is a scalar parameter. Thus in this case there are 5 HMs. When CY = 1 and gj # 0, j = 1 , 2 , 3 , we get Minkowski spacetime for which the maximal group of isometries is S 0 ( 1 , 3 ) @ R4, and admits 11 homotheties. A metric admitting a Gs as the maximal isometry group is ds2 = dt2 - 6 (t -
[dr2+ dR2] ,
396
and the corresponding HMs are
H o = (t - 0)$0,
+ -r2 + (2 P)$o, cos 0 a3 H2 = (a1 sin $ - a2 cos $) - sin 8 + (c1 sin $ - c2 cos $), r r cos ec8 H 3 = -(a1 cos$ + a2 sin$) + cot0 (c1 cos$ + cz sin$) + c3. +
H 1 = sinQ(a1sin$ - U ~ C O S $ ) a3cos8
-
-
T
These are 7 HMs and form a finite dimensional Lie algebra. For spacetimes to have SO (3) as the maximal isometry group one must have g j ( t ,r ) = 0. In spherical coordinates, zQ = ( t ,r, 0, 'p) = (x', xi),the metric of static spherically symmetric spacetimes can be written as ds2 = -,zv(T)dt2
+ ,zX(T)dr2 + r2dR2.
(4)
It admits the 4 KVs,
a xo = at XI= sin$-
a +cos$cot8-, a
ao
a X 2 =cos$-
ao
x3 =
a4
-sin$cot8-,
a
84
d
-.
a$
The classification of conformal motions for this metric is given in reference [6]. The group of conformal motions in this case is G4fn where n is either 2 or 11. For the conformally flat case we have G15. In this case the Weyl tensor is zero. The general form of KVs of plane symmetric static spacetimes
has the form'
397
where Ai( t ,x) , K ( t ,x) and L ( t ,2) satisfy some constraint equations. The classification includes the 5-dimensional isometries which are given by the class of metrics
+ dy2) ,
ds2 = evdt2 - dx2 - e2x/xQ(dz’
u”
#0
(6)
and the corresponding KVs
KO
= ZOClV’/2
x
= -c1xo,
1
2
+ c5,
+ +
K = c1y c2 cgz, K 3 = c1z - c3y + c4. There are three metrics with 6 isometries. These are planar analogues of the Bertotti- Robinson metrics and two other similar metrics’. They have p ( x ) = 0 and u (x) = In cosh2 ax or eZax or In cos2 ax, and the corresponding KVs are
K
=
+ (c4 sin at + c5 cos at) d/dx + (cl + C3Z) a m + (cz - c3y) a m ,
[c0 - tanh a x (c4sin at - c5 cos at)]d / d t
+
K = [c0 - (c4t2 c5t)] a/at
+ ( 2 C d +- c5) a/ax + (cl + c3z) a/ay +
(c2 - C3Y)
a/az
K = [c0 - (c4sin at + c5 cos at)tan ax]d / d t + (c4cos at + c5 sin at) d/dx + (cl
+ c 3 4 aiay + (c2 - c3y) a m ,
respectively, where ci are arbitrary constants. Static plane symmetric spacetimes admit either 4,5 , 6 or 10 isometries. Therefore, they can admit either 5, 6, 7 or 11 HMs’. The spacetime admitting 11 HMs is the Minkowski spacetime
ds2 = ( a
+ x)’ dt2 - dx2 - (dy2 + d z 2 ) ,
with HMs
Ho =
Y { (c4 sinh ht + C6 cosh t )x} + a (c4sinht + C6 cosh h) + z ( a +l x ) 2 [ 1 -~
{ (c5 sinh t + c7 cosh t ) x + a (c5 sinh t + c7 cosh t ) }
+ cg cosht) + H1 = [Y(c4 cosh t + C6 sinh t ) + Z (c4 cosh t + sinh t ) ] + (c’ cosht + cg sinht) + p ( a+ x) , H 2 = cgz + yp + (c4 cosht + cgsinht) x + a (cqcosht + cgsinht) + c1, H 3 = -c3y + zP + (c5 cosh t + c7 sinh t )x + a (c5 cosh t + c7 sinh t ) + ( a + x)
(C8
sinht
CO,
C6
CZ.
1
398
A metric admitting
G6
is
and the corresponding HMs are
+~ 4 H2 = + H 3 = - ~ 3 y+ ~ H1 = cat
,
C ~ Z CO,
1 .
The Lie algebra corresponding to the symmetry group is [S0(2) BSR2] @ SO(1,2), where @s stands for the “semi-direct product”, signifying that the generators of the subgroups do not commute. Another metric admitting G6 as the maximal isometry group is
ds2 = ( P x ) ; (dt’
-
dy2 - dz’)
-
dx2
(8)
and the corresponding HMs are
H o = C ~ Y+ C ~ (Zp - 1/2) t H1 = Px, H2 = C ~ + Z y ( p - 1/2) ~ H 3 = - C ~ Y + z ( p - 1/2)
+
CO,
+ 4 +t CI, + ~ 5 +t
~ 2 .
This spacetime admits [S0(1,2) BSR3] @ SO(2) as the maximal isometry group. Another metric admitting Glo as maximal isometry group is the anti-de Sitter spacetime
ds2 = e 2 x / x 0(dt2 - dy2 - d z 2 ) - d x 2 .
(9)
This spacetime admits S 0 ( 2 , 3 ) as the maximal isometry group. The metrics admitting G6 as the maximal HMs group identical to isometries are
ds2 = dt2 - d x 2 - eAz+B ( d y 2 + d z 2 ) . and the HMs are
H1 = -2co/A, H2 =C
+
~ Z COY
+ CI, +
H 3 = -C3y 4-C g Z
C2.
(10)
399
The group determined by the Lie algebra for this spacetime is [ S 0 ( 2 ) gs R2 €3s R]B R. A metric admitting Gq as the maximal group of motions with non-trivial HMs is
ds2 = d t 2 - dx2 -
(dy2
+ dz2).
(11)
The corresponding HMs are
H o = (P - 1/2) t + CO, H 1 = Px,
+
+
H 2 = C 3 Z (P - 1/2) y H 3 = - C ~ Y (P - 1/2) z
+
~ 1 ,
+ cg.
This Lie algebra determines the group as [ S 0 ( 2 ) BSR2 BsR]€3 SO(2). Another metric admitting Gq as the maximal group of motions is
ds2 = ev(")dt2- dx2 - (Px); (dy2
+ d z 2 ).
(12)
The group determined for this spacetime is the same as in the previous case i.e. [SO(2) gsR2 gsR]€3 R. Hence we see that KVs and HMs are special cases of conformal motions with their corresponding Lie algebras forming a subset of the Lie algebra for conformal motions. The work on conformal motions of plane and cylindrically symmetric static spacetimes is underway and will be reported elsewhere.
References 1. Y . Choquet-Bruhat, C. Dewitt-Morrette, and M. Dillard-Bleick, Analysis, Manifolds and Physics (Amsterdam, North-Holland, 1977). 2. R. Maartens, and S.D. Maharaj, Class. Quantum Grav. 3,1005 (1986). 3. R. Maartens, and S.D. Maharaj, Class. Quantum Grav. 8 , 503 (1991). 4. G.S. Hall, Symmetries and Curvature Structure in General Relativity, (World Scientific, 2004). 5. D. Ahmed, M. Ziad, J. Math. Phys. 5 , 38 (1997). 6. R. Maartens, S.D. Maharaj and B.O.J. Tupper, Class. Quantum Grav. 12 , 2577 (1995). 7. A. Qadir and M. Ziad, Proceedings of the VI Marcel Grossmann meeting, eds. T. Nakamura and H. Sato, (World Scientific, Singapore, 1993) p. 1115. 8. T. Feroze, A. Qadir and M. Ziad, J . Math. Phys. 42 , 49471 (2001). 9. S. Kiran, Classifications of Homotheties of Plane Symmetric Static Spacetimes, M.Phil. Dissertation, Quaid-i-Azam University Islamabad, (1997).
PROPER PROJECTIVE SYMMETRIES IN SPACE-TIMES GHULAM SHABBIR Faculty of Engineering Sciences, GIK Institute of Engineering Science and Technology, Topi Swabi, NWFP, Pakistan
A study of proper projective symmetry in the Schwarzschild and ReissnerNordstrom metrics is given by using algebric and direct integration techniques. It is shown that projective vector fields admitted by the above space-times are the Killing vector fields. Keywords: Killing vectors, Projective collineations
1. I n t r o d u c t i o n Throughout this paper M represents a four dimensional, connected, Hausdorff space-time manifold with Lorentzian metric g of signature (-, +, +, +). The curvature tensor associated with g a b , through the LeviCivita connection, is denoted in component form by Rabcd, and the Ricci tensor components are R a b = RCacb. The usual covariant, partial and Lie derivatives are denoted by a semicolon, a comma and the symbol L , respectively. Round and square brackets denote the usual symmetrization and skew-symmetrization, respectively. The covariant derivative of any vector field X on M can be decomposed as
where hub (= h b a ) = Lxgab and Nub (= -Nba) are symmetric and skew symmetric tensors on M , respectively. Such a vector field X is called projective if the local diffeomorphisms & (for appropriate t ) associated with X map geodesics into geodesics. This is equivalent to the condition that hat, satisfies
400
40 1
+,
for some smooth closed 1-form on M with local components +a. Thus is locally a gradient and will, where appropriate, be written as +a = +,a for some function on some open subset of M . If X is a projective vector field and +a;b = 0 then X is called a special projective vector field on M . The statement that h a b is covariantly constant on M is from ( 2 ) , equivalent t o being zero on M and is, in turn equivalent to X being an affine vector field on M (so that the local diffeomorphisms, preserve not only geodesics but also their affine parameters). If X is projective but not affine then it is called proper projective'>'. Further if is affine and h a b = 2 C g a b , c E R, then X is homothetic (otherwise proper affine). If X is homothetic and c # 0 it is proper homothetic whilst c = 0 if it is Killing.
+
x
2. PROJECTIVE SYMMETRY
If X be a projective vector field on M , then from (1) and ( 2 ) 3
Following [4] (see also [5]) the Ricci identity on h gives haeRebcd
+ h b e R e a c d = gac+b;d
- gad'@b;c
+ gbc$a;d
- gbd'$a;c.
Let X be a projective vector field on M so that (1) and ( 2 ) hold and let F be a real curvature eigenbivector at p E M with eigenvalue X E R (so that Rabcd Fcd = XFab at p ) , then at p one has4
Equation ( 3 ) gives a relation between FabandP a b (a second order symmetric tensor) at p and reflects the close connection between h a b , $a;$ and the algebraic structure of the curvature at p . If F is simple then the blade of F (a two dimensional subspace of T,M) consists of eigenvectors of P with the same eigenvalue. Similarly if F is non-simple, then it has two well defined orthogonal timelike and spacelike blades at p , each of which consists of eigenvectors of P with the same eigenvalue6.
3. MAIN RESULTS In this section we will briefly discuss the existence of proper projective symmetry in the Schwarzschild and Reissner-Nordstrom metrics. The situation is well known when the above space-times admit proper affine vector fields7 and proper homothetic vector fields. It is assumed that the space-time under consideration admits no such symmetries and it is not of constant curvature.
402
Consider the Schwarzschild metric in the usual coordinate system
(t,r, 8, p) (labeled by (zo,d ,x 2 ,x 3 ) , respectively) with line element' ds2 = - (1 -
F) + F)-' dt2
(1 -
dr2 + r2 (do2
+ sin2 8dp2) .
(4)
It follows from ref. [8] that the above space-time admits four independent Killing vector fields, which are a cosp- a -cotesinp-, a sinp- a +cotBcosp--,a -.d ae ae av aP The non-zero independent components of the Riemann tensor are m Rol o1 = R2323 = 7 2m = ct1, Ro2o2 = Ro3o3 = R13 13 = -F G a2. The curvature tensor of M can be described by components Rabcd written as a symmetric matrix in a well known wayg
Rabcd
= diag
(a11a 2 1
(121 a 2 ,
a21
a1)1
(5)
where a1 and a2 are real functions of r only. Here we are considering the open sub region where a1 and a2 are nowhere equal and a1 # 0 and a2 # 0 (situations when a1 = a2 or a1 = 0 or a2 = 0 were discussed in ref. [l]). It follows from ref. [lo] that
hab = Cgab
+ D (rarb - t a t b ) ; $a;b = Egab + F (rarb - t a t b ) .
(6)
for some real functions C,D , F and E on M . Next one substitutes the first equation of (6) in ( 2 ) to get
Ccgab
+ Dcrarb+ Dra;,rb + Drbicra- Dctatb = 2gab$c + gac$b + gbc$a
-
Dta;ctb
+ Dtb;cta
(7)
Contracting the above equation with eapb, and then comparing both sides, we have qaea= 1Cla(pa = 0. Now contracting equation (7) with 8"Ob we get C,c= 214~.Once again contracting equation (7) with tatb and Tarb,we get D = D ( t )and $ = +(t),respectively. Hence on M we have D = D ( t )-, C = C ( t ) and $ = $ (t).Now consider the first equation of (6). Using (4) we get the following non-zero components
hoo
=
-D(t) -C ( t ) h22 = C ( t )r 2 , h33 = C ( t )r2 sin2 8.
(8)
Now we are interested in finding projective vector fields by using the relation
LXgab = hab
Valb = 0,1, 2,3.
403
Writing the above equation explicitly and using (4) and (8), we get
2m -x1+2 r2
(1--
x,'o- (1 -
(1 - $ ) - l
r2xi - (1 T 2 sin2
ex5
-
(1 -
2m>
xo l o -
=0
(10)
X>
=0
(11)
X>
=0
(12)
F) F)
r + r X 5 = -C (t) 2
sin2ex;
+ xZ,
Solutions of equations (9)-(18) are
+ (1 -
+ A4 ( r )+ ci,
?)
+
(9)
$)xpl
X1
X o = B6 (t,T )
(1-- 2r)C ( t ) + D ( t )
R6(t),
x2= c4 cos (p + c2 sin (p, x 3= cot 6 {c2 C O S Y - c4 sincp} + c3
=
o
404
provided that
5 [:( ':)-' J' { 1--
2 (1 -
C ( t ) (1 -
:)-+
+ D ( t )(1 -
F)'}
dr
F)'J' [. r 2
R6 ( t )= C ( t )-, where C I , C ~ , C ~ , E C ~ R and R6 ( t ) , A4 ( r ) , B6 ( t , r ) are some functions. Suppose
X = (77 ( t , r ), p ( t , r ) , c4coscp
+ c2 sincp, cotB{c2 coscp - c4 sincp} + cg) ,
where 77 ( t ,r ) = B6 ( t ,r )
+ A4 ( r )+ c1,
+ (1 - - ':)'
R6 ( t )
and one form = Xlt, , where X1 = X1 (t).The vector field X is then a projective vector field if it satisfies (2). Substituting the above information in (2) we get X1 = 0 which implies Ga = 0. Hence n o proper projective vector field will exist. It also follows from ref. [7] that the metric (4) admits no proper affine vector fields. This is because the admission of a proper affine vector field requires that the rank of the 6 x 6 Riemann matrix would be at most three. One can also check that the above metric does not admit proper homothetic vector fields. Hence projective vector fields admitted by (4) are the Killing vector fields which are
x0 = c1, x1= 0, x 2= c:!cos(p +cSsincp, x3= cot e {cgcOscp - cz sin cp} + c4
(22)
405
where c1, c2, CY, c 4 E R. Now consider the Reissner-Nordstrom metric in the usual coordinate system ( t ,r, 8, ‘p) (labeled by (xold ,x 2 ,z3) respectively) with line element 2m
+
g)
-1
dr2+r2 (do2
+ sin2 8 d p 2 ) (23) . ,
It follows from ref. [8] that the above space-time admits four independent Killing vector fields which are given in equation (22). The non-zero independent components of the Riemann tensor are
R2323
=
2mr - 3Q2 = 012. r4
The curvature tensor of M can be described by components Rab cd written as a 6 x 6 symmetric matrix in a well known wayg
Rab cd = diag (&lo, 011 where that
a10,all
hab
and
a12
= G g a b -tH
are real functions of
(rarb
(24)
ail,ail,011 Q.12) T
only. It follows from ref. [I11
- t a t b ) , $a;b = I g a b
+J
(rarb
-tatb) ,
(25)
for real functions G , H , I and J on M . Also G = G ( t ), H = H ( t ) and $ = ( t ) . Now consider the first equation of (25). Using (23) we get the following non-zero components
+
-1
= -H ( t )- G ( t )
~ O O
h22
= G ( t )r 2 ,
h33
+ $) + H ( t )
= G ( t )r2 sin2 0
Now we are interested in finding the projective vector field. If one proceeds further exactly on the same lines as we did in the Schwarzschild case, one can easily find that no proper projective vector field will exist. It follows from ref. [9] that the above space-time (23) does not admit proper affine vector fields. One can also check that the above space-time does not admit proper homothetic vector fields. Hence projective vector fields admitted by (23) are the Killing vector fields which are given in (22).
406
References 1. G. S. Hall, Class. Quantum Grau. 17,4637 (2000). 2. G. S. Hall, Symmetries and Curvature Structure in General Relativity, (World Scientific, 2004). 3. A. Barnes, Class. Quantum Gmu. 10, 1139 (1993). 4. G. S. Hall, Differential Geometry and its application, (Masaryk University, Brno Czech Republic, 1996). 5. G. S. Hall and D. P. Lonie, Class. Quantum Grau. 12, 1007 (1995). 6. G. S. Hall and C. B. G. McIntosh, Int. J . Theor. Phys. 22,469 (1983). 7. G. S. Hall, D. J. Low and J. R. Pulham, J. Math. Phys. 35,5930 (1994). 8. R. M. Wald, General Relativity, (The University of Chicago Press, 1984). 9. G. Shabbir, Class. Quantum Grau. 21,339 (2004). 10. G. Shabbir, to appear in Modern Physics Letters A, in 2006. 11. G. S. Hall, Class. Quantum Gmu. 17,3073 (2000).
MATTER SYMMETRIES OF NON-STATIC PLANE SYMMETRIC SPACETIMES M. SHARIF* and TARIQ ISMAEEL Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan * E-mail: [email protected] The matter collineations of plane symmetric spacetimes are studied according to the degenerate energy-momentum tensor. We have found many cases where the energy-momentum tensor is degenerate but the group of matter collineations is finite. Further we obtain different constraint equations on the energy-momentum tensor. Solving these constraints may provide some new exact solutions of Einstein’s field equations.
1. Introduction The General Theory of Relativity (GR), which is a field theory of gravitation and is described in terms of geometry, is highly non-linear. Because of this non-linearity, it becomes very difficult to solve the gravitational field equations unless certain symmetry restrictions are imposed on the spacetime metric. These symmetry restrictions are expressed in terms of isometries possessed by spacetimes. These isometries, which are also called Killing Vectors (KVs), give rise to conservation In GR, the Einstein tensor Gab plays a significant role, since it relates the geometry of spacetime to its source. However, GR does not prescribe the various forms of matter and takes over the energy-momentum tensor T a b from other branches of physics. Einstein’s field equations (EFEs) are given by 1 (a,b = 0 , 1 , 2 , 3 ) , (1) 2 where Gab are the components of the Einstein tensor, R a b those of the Ricci and T a b of the matter (energy-momentum) tensor. Also, R = gabRab is the Ricci scalar, IC is the gravitational constant and for simplicity we take A = 0. Gab
Rab - -&Jab = 6Tab,
407
408
Let ( M ,g) be a spacetime, i.e., M is a smooth four-dimensional, Hausdorff (C") manifold, and g a smooth Lorentz metric tensor of signature (+ - - -) defined on M . We shall use the usual component notation in local charts, and a covariant derivative with respect to the symmetric connection I? associated with the metric g will be denoted by a semicolon and a partial derivative by a comma. We define collineations as geometrical symmetries which are given by a relation of the form LcA = B, where L is the Lie derivative operator, Ea is the symmetry or collineation vector, A is any of the quantities g a b , I?&, R a b , RtCdand geometric objects constructed by them and B is a tensor with the same index symmetries as A. One can find all the well-known collineations by requiring the particular forms of the quantities A and B. For example A,b = g a b and Bat, = 2$g,b defines a conformal Killing vector (CKV), which specializes to a special conformal Killing vector (SCKV) when '$;& = 0, to a homothetic vector (HV) field when = constant and t o a Killing vector (KV) when = 0. If we take @ab = R a b and B a b = 2'$Rab the symmetry vector Ea is called a Ricci Inheritance collineation (RIC) and reduces to a Ricci collineation (RC) for B,b = 0. When A,b = T a b and B a b = 2 + T a b , where T a b is the energy-momentum tensor, the vector Ea is called a matter inheritance collineation (MIC) and it reduces to a matter collineation (MC) for B,b = 0. In the case of CKVs, the function is called the conformal factor and in the case of inheriting collineations the inheriting factor. We shall define an MC to be proper which is not a KV or a HV otherwise it is improper. The MC equation can be written as
+
+
LcTab
=o
H
LcGab = 0,
(2)
or in component form Tab,&'
+ Tact:b + TcbE:a
= 0.
(3)
There is a recent growing interest in the study of MCs3-'. Carot, et al.4 have discussed MCs from the point of view of the Lie algebra of vector fields generating them and, in particular, they discussed spacetimes with a degenerate Tat,.Hall, et al.5, in the discussion of RCs and MCs, have argued that the symmetries of the energy-momentum tensor may also provide some extra understanding of the the subject which has not been provided by Killing vectors, Ricci and curvature collineations. This paper studies the problem of calculating MCs for plane symmetric spacetimes when the energy-momentum tensor is degenerate only. A complete solution of the MC equations for the plane symmetric spacetimes will
409
be reported elsewhereg. The paper has been organised as follows. In the next section we write down the MC equations for plane symmetric spacetimes. In section three, we solve these MC equations when the energy-momentum tensor is degenerate only. We end with the conclusions. 2. Matter Collineation Equations
The most general plane symmetric metric is givenlo as
ds2 = e v ( t , z ) & - eA(t,z)&2 - ePL(t,z)(dy2+ & 2 ) .
(4)
The non-zero components of the energy-momentum tensor are TOO, Tol, T11, T 2 2 , T33 given in Appendix A. The MC equations for plane symmetric spacetime can be written as
where T3 = T2. It is to be noticed that we are using the notation T,, = T,. Further, we have written these equations under the restricyion To1 = 0. We shall solve these equations for the degenerate case only. 3. Solution of Matter Collineation Equations
In this section we solve MC equations (5)-(14) when the determinant of the energy-momentum is zero, i.e., det(T,b) = 0. This means that we would require at least one of T, = 0. First, we consider the trivial case, where T, = 0. In this case, Eqs.(5)-(14) are identically satisfied and thus every direction is a MC. By Eq. ( 2 ) these are Ricci flat spaces. The other possibilities can be classified in three main cases: (1) when only one of T,
# 0;
410
(2) when two of T, # 0; (3) when three of T, # 0.
We report only the case for which MCs are finite. This is the third case when three of T, # 0. In this case, there could be only two possibilities, i.e., either
(3a) (3b)
TO= 0,
Ti# 0,
Ti = 0,
Tj
(i = 1,2,3);
# 0 (j = 0,2,3).
We restrict ourselves to discuss the finite MCs of the first case only. Case (3a): In this case, Eq.(5) is identically satisfied and Eqs.(6)-(8) respectively give Ei = p ( ~y, ,z ) . The remaining equations will become
+
+ m ; 2 + T2J:l = 0, m ; 3 + T25?1 = 0, T2,0E0+ T2,1E1+ 2T2E:2 = 0,
T1,oE0 Tl,lE1 2TlE;l = 0,
E; + E $
+
= 0,
+
T2,0E0 Tz,lE1 2T2E$ = 0.
(15) (16)
(17) (18)
(19) (20)
From these equations, we have the following sixteen possibilities:
(i) TI = constant # 0 , T2 = constant # 0 , (ii) T1,o # 0, TIJ = 0 , T2 = constant # 0 , (iii) T1,o = 0, T ~ # J 0 , Tz = constant # 0 , (iv) 7'1 = constant # 0 , T2,o # 0, T2,1 = 0, (v) 7'1 = constant # 0, T2,o = 0, T z ,#~ 0, (vi) TI = constant # 0 , T2,o # 0, T2,1 # 0, (vii) T1,o # 0, T1,1 # 0 , TZ= constant # 0 , (viii) T1,o # 0, TIJ= 0, Tz,o# 0, T2,1 = 0, (ix) TI,O# 0, T1,1 = 0, T2,o = 0, Tz,i# 0, (x) Ti,o = 0, Ti,i# 0, T2,o # 0, T2,i = 0, (xi) T1,o = 0, T1,1 # 0, Tz,o = 0, T2,i # 0, (xii) TI,O# 0, T1,1 # 0, T2,o # 0, Tz,i= 0, (xiii) TI,O# 0, T1,1 = 0, T2,o # 0, Tz,i # 0, (xiv) TI,O = 0, T1,l # 0, T2,o # 0, T z , # ~ 0, (xv) T1,o # 0, T1,l # 0, T2,o = 0, T2,l # 0, (xvi) TI,O# 0, TIJ # 0, T2,o # 0, T2,i # 0. We list here only the finite cases. Case(3aiv): Solving MC equations simultaneously, we obtain the following
411
MCs
This gives six MCs out of which three are the usual KVs and three are the proper MCs. Case(3avi): Solving MC equations under the constraints of this case, we obtain the following solution
+ c2z + C3], E1 = c1y + C2-z + c3, E2 = -c1/ E Tid x + c4-z + c5,
EO
= --[c1y T2 1
T2,o
giving rise to six MCs. Case(3aviii): This gives the same results as the case (3aiv) and hence we obtain six MCs. Case( 3aix): Solving MC equations, after some algebraic manipulation, we obtain the following solution
(e = 0, ,$2 = c1z E3
(e = 0 , l),
+ c2, = -c1y + c3.
In this case MCs turn out to be the usual three KVs. Case(3ax): Proceeding in a similar way as above, it follows that MCs are
to E
1
= 0,
+ c2z + c3],
1
= --[ClY
47
J
c3 = -2 T2
fidz+cs.
(24)
412
Here we get five MCs out of which two are proper. Case(3axii): This case turns out to be similar to the case (3ax). Case(3axiii): Here after some algebra, we have the following constraint
where a is a separation constant. This gives rise to the following two possibilities:
(*)
a=o,
(**)
a # 0.
Case(3axiii*): For a = 0, MCs are
E0 = o , E1 = c1y + c2z + c3, E 2 = -TI z d z + cqz + c5,
/
It follows that we have six MCs. Case(3axiii**): For a # 0, MCs are T2 1 to= --eaz[cly
T2,o
E2
=
c3 =
-cl -C2
1
/
Tl -eaxdx T2
+ c2z + cg], + cqz + c.5,
Tl Eeaxdx - C4y
+ Cg
(26)
giving six MCs. Case(3axiv): Using the same procedure as above, after some algebra, we obtain the following MCs
413
Here again we obtain three proper MCs. Case(3axvi): In this case, we further have the following constraint
Q
=
1
--[Tl,OT2,1 - Tl,lT2,0] 2Tl
giving the following two options
(*)
cy
= 0,
(**)
Q
# 0.
Case(3axvi*): This gives exactly the same results as the case (3avi). Case(3axvi**): It gives the results similar to the case (3axiii**).
4. Discussion
In the classification of plane symmetric spacetimes according to the nature of the energy-momentum tensor, we find that when the energy-momentum tensor is degenerate, then there are many cases of MCs with infinite degrees of freedom. However, we have restricted our attention to finite MCs only. It is very interesting to note that we have found many cases where the energymomentum tensor is degenerate but the group of MCs is finite dimensional. We obtain three, five and six MCs out of which three are the usual KVs of the non-static plane symmetric spacetimes and rest are the proper MCs. In the cases (1)-(3), we summarize some results in the following: 1. In this case, the rank of
T a b being 1, it is found that all the possibilities yield infinite dimensional MCs; 2. In all subcases of this case, the rank of T a b is 2. 3. In all subcases of this case, the rank of Tab is 3. A point worth mentioning is that there are cases with finite dimensional groups of MCs even though the energy-momentum tensor is degenerate. We obtain three, five and six MCs.
We have obtained a number of constraint equations. If these constraint equations could be solved, then one can expect to find new interesting exact solutions. A complete classification of the degenerate and non-degenerate energy-momentum tensor will be reported somewhere elseg.
414
Appendix A The surviving components of the Ricci tensor are 1 Roo = -(46 2fi2 - i/i+ 2j; + i 2 - 2fii/) 4 1 -eV-’(2v’’ d 2+ 2v’p’ - Y’x’), 4 1 * I Rol = --(2fi’ fip’ - /id - Xp )7 2 1 R11 = -ex-”(2i i2 - i/i+ 2 i f i ) 4 1 - - (2v” d 2 4 4 ’ - V’X’ 2p’2 - 2p’XI), 4 1 R22 = -eP-’(2ji 2fi2 - i / f i ifi) 4 1 - -e~-’(2p’’ 2p12 - VIA’ p’v’), 4
+
+
+
+
+
+ + + +
R33
+
+
+
= R22.
(All
The Ricci scalar is given by 1 R = --e-’(2j; i2 - i/i- 2i/fi 2 f i i 2 1 -e-’(2~‘’ d2 - Y’X’ 2v’pI 2 - 2p’X’ 3p12 4 4 9 .
+
+
+ + + +
+ 3fi2 + 4ji)
+
(A21
Using Einstein field equations (l), the non-vanishing components of energymomentum tensor Tabare 1 . 1 T~~= -(fi2 + 2 f i ~ -) -eV-’(4p’’ 3pI2 - 2 p ’ ~ ’ ) , 4 4
+
To1 = Rol, 1 T11 = --eX-’(4ji
1 3fi2 - 2fiv) - - ( P ’ ~+ 2p’v’), 4 4 1 T 2 2 = --eP-”(2ji + 2j; + fi2 - fii/ + fii- u i + i2), 4 1 - e ~ - ~ ( 2 p ” 2v” pI2 - p ’ ~ ’ p’v’ - X’Y’ d 2 ) , 4 2 T33 = T22sin 0.
+
+
+
+
+
+
(A3)
Acknowledgment We would like to acknowledge the financial assistance of National Centre for Physics, Quaid-i-Azam University Islamabad to attend this conference.
415
References 1. A.Z. Petrov Einstein Spaces (Pergamon, Oxford University Press, 1969). 2. L. Hojman, L. Nunez, A. Patino and H. Rag0 J. Math. Phys. 27, 281 (1986). 3. J. Carot and J. d a Costa, Procs. of the 6th Canadian Conf. on General Relativity and Relativistic Astrophysics, (Fields Inst. Commun. 15, Amer. Math. SOC.WC Providence, RI, 1997) p. 179. 4. J. Carot, J. d a Costa and E.G.L.R. Vaz, J . Math. Phys. 35, 4832 (1994). 5. G.S. Hall, I. Roy, and E.G.L.R. Vaz Gen. Re1 and Grav. 28, 299 (1996). 6. I. Yavuz and U. Camci Gen. Re1 and Grav. 28, 691 (1996); U. Camci, I. Yavuz, H. Baysal, I.Tarhan and I.Yilmaz Int. J. Mod. Phys. D10, 751 (2001); U. Camci and A. Barnes Class. Quant. Grav. 19, 393 (2002). 7. M. Sharif Nuovo Cimento B116,673 (2001); U. Camci and M. Sharif Gen. Re1 and Grav. 35, 97 (2003); Class. Quantum Gravt. 20, 2169 (2003). 8. M. Sharif J. Math. Phys. 45, 1518 (2004); ibid 1532; Astrophys. Space Sci. 278, 447 (2001); M. Sharif and Sehar Aziz Gen. Re1 and Grav. 35, 1093 (2003). 9. M. Sharif and Tariq Ismaeel Ismaeel work in progress. 10. H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Hearlt Exact Solutions of Einstein’s Field Equations (Cambridge University Press, 2003).
SPACETIME FOLIATION AZAD A. SIDDIQUI Department of Basic Sciences, EME College, National University of Sciences and Technology, Rawalpindi, Pakistan.
and Department of Physics, Linkoping University Linkoping, Sweden E-mails: azadOceme.edu.pk, azadOifi.liu.se In this paper the concept of foliation is reviewed. The idea of spacetime foliation by hypersurfaces of zero intrinsic curvature that are orthogonal t o the world-lines of observers falling freely from infinity is also presented.
Keywords: Black Hole; Hypersurface; Foliation
1. Introduction Splitting of a space into a sequence of subspaces, such that every point in the space lies in one and only one of the subspaces, is called a foliation. Generally, a sequence of one lower dimensional subspaces called hypersurfaces are used for this purpose. In General Relativity (GR) foliation is often used to break the spacetime into ‘space’ and ‘time’. It might appear that foliation of a spacetime is undoing Einstein’s magnificent unification of ‘space’ and ‘time’ which led to the remarkable theories of Special and General Relativities. These theories in turn have provided great insights towards understanding the universe. Some physically reasonable solutions of the Einstein field equations’ are singular and represent black hole spacetimes. These spacetimes have special significance because of the horizon(s) in their geometry. To analyze the dynamics of such geometries, one often foliates the spacetime by a sequence of null or spacelike hypersurfaces. These foliating hypersurfaces are also used to generalize some basic concepts, like homogeneity and isotropy, in Cosmology.
416
417
Most of the models of the universe are idealized to be completely homogeneous and isotropic. Homogeneity and isotropy mean the universe looks the same to all observers in all directions at a given moment of time. Therefore, these basic concepts require a clear concept of time. Further, in any attempt at quantization in curved spacetime, using the usual canonical quantization procedure, the first problem that arises is the lack of a well defined time, so that the equal time commutation relations, which are the building blocks of the procedure, can be defined. There can be two approaches, local or global, for foliation of a spacetime, in order to obtain fresh insights into the consequences of GR, from a local or global viewpoint. Local refers to the individual observers and global to the spacetime as a whole. 2. Foliation The idea of a slicing or foliation (from the Latin word folia for leaf) dates back to the beginning of the theory of differential equations (i.e. the seventeenth century) where the trajectories of the solution space can be thought of as the leaves of the foliation. Towards the end of the nineteenth century, Poincar6 developed methods for the study of global, qualitative properties of solutions of dynamical systems in situations where the explicit solution methods had failed. He discovered that the study of the geometry of the space of trajectories of a dynamical system reveals complex phenomena. He emphasized the qualitative nature of the phenomena, thereby giving strong impetus to topological methods, which led to the subject of foliation. The foliation of an n-dimensional manifold, M , is a decomposition of A4 into submanifolds, all being of the same dimension, p . The submanifolds are the leaves of the foliation. The co-dimension, q, of a foliation is defined as q = n - p . A foliation of co-dimension one is called a foliation by hypersurfaces. The pioneers of foliation theory were Reeb2 and Ehresmann3, the former, in particular, coined the term foliation. The simplest and best understood cases of foliation are when p = q = 1, e.g. the two dimensional xy-plane, EX2, can be foliated by the straight lines, y = mx c, with c taken as the foliation parameter and any fixed m.One can also foliate the xy-plane by circles, x 2 y2 = a2 (0 < a < m), but in this case the origin is left out unless the degenerate circle, a = 0, is included. A foliation of a manifold is said to be complete if it covers the entire manifold by a sequence of non-intersecting submanifolds. For example, a disc of radius a can be completely foliated by circles but it can not be
+
+
r.
418
completely foliated by squares as there would be some portion of the disc left uncovered. Foliation of the spacetime by hypersurfaces (co-dimension one) may be obtained by timelike, null or spacelike hypersurfaces. Generally, the approach taken for obtaining a foliation of a spacetime by hypersurfaces, is to specify some geometrical property that this family of hypersurfaces must satisfy. A local or global time parameter, varying from one hypersurface to another, is then provided. However, there is no guarantee that a complete foliation can be so achieved. An example of the first approach is the requirement that the hypersurfaces look fiat to an observer, locally i.e. have zero intrinsic curvature. An example of the global approach is the requirement that this family of hypersufaces have zero or constant mean extrinsic curvature. 3. Foliation of a Black Hole Spacetime
Ttaditionally black holes are understood as embedded in asymptotically flat spacetimes. It is taken for granted that the spacetime is not compact. A classical black hole is a region from inside of which not even light can escape to infinity. Therefore, an infinity to escape to is required. There is a problem, if the spacetime did not tend to Minkowski space far away from the source, or more dramatically, if the universe was closed so that there was nothing sufficiently far away from the source. In a closed universe, as there is no infinity to escape to, the distinction between inside and outside a black hole is ambiguous, or alternatively, the distinction between the black hole singularity and the final cosmological singularity is not clear. It had been pointed out by Penrose4 that in a conformal sense it should be possible to regard the black hole singularity as part of the final singularity5, or in an open universe part of the compactification of the spacetime at future infinity. The picture he suggested was to view the black hole singularity as stalactites on the roof of a cave which represented the final singularity. Consequently, he argued, it should be possible to straighten outthe roof and have it appear smooth, by some appropriate conformal transformations. In other words, there should exist a foliation of the spacetime by a sequence of spacelike hypersurfaces which would approach the singularity smoothly without cutting it anywhere. Thus the entire spacetime would be foliated. The limit of some parameter going to some specific value should yield the entire singularity. Initial attempts focused on a foliation by maximal slicing6 i.e. by hypersurfaces of zero mean extrinsic curvature. Foliating even the most simple
419
of spacetimes for the purpose, namely the Schwarzschild spacetime, the hypersurfaces ran into a boundary, i.e. they did not pass through all the spacetime points7. This problem could mean either that Penrose's conjecture was incorrect, or that the maximal slicing procedure was inappropriate for the purpose. It was argued7" that York slicing would be more appropriate for the purpose. In this slicing the hypersurfaces are defined to have some given constants mean extrinsic curvature (hereafter called K-surfaces). Each K-surface has a different mean extrinsic curvature. As the mean extrinsic curvature is varied continuously so too is the time parameter. In other words, then, there is a 1-1correspondence between the usual time parameter and the constant value of the mean extrinsic curvature. For this reason one can define the York timeg as proportional to the mean extrinsic curvature. It would be natural, then, to expect the final and initial singularities to correspond to York time of f o o . Given a single K-surface, one can obtain a local K-surface foliation by solving an elliptic equation on the lapse function. This approach has been exploited by Estabrook et. al.1° for maximal surfaces (K = 0), and then by Eardley and Smarr7 for surfaces of K # 0. Alternatively, one can use a variational principle for minimizing the (3-) area of the hypersurface for a given (4-)volume of the world tube traced out by it. This approach was introduced by Goddard" and was then used by Brill et.al.12 and ~ t h e r s ' ~ ' ' ~ . It has been argued that such K-surfaces have special significance in a cosmological context'. To see this A.Qadir and J.A.Wheeler' first obtained a York slicing of the Schwarzschild lattice universe15. The slicing never ran into any problem, proceeding as expected. However, there was no process of formation of the black hole. What had been done was to cut the singularity out of the manifold and have the throat of the black hole joined to its counter part in a maximal extension16, using an Einstein-Rosen bridge. To ensure that the black hole actually form, and there be a singularity which finally is reached, the above model was modified. It was started at the phase of maximum expansion of the f i e d m a n n - p a r t of the lattice universe, with a dust shell of the Schwarzschild radius. Thus inside the shell there was Minkowski space, outside a Schwarzschild exterior geometry and at the boundary between cells a part that evolved with a Friedmann evolution. This model could be foliated from the phase of maximum expansion on without running into any boundary. Ignoring the question of what had happened before their start of the model, the implications of the model were discussed '. It was argued that there would be various consequences of York time being
420
taken as relevant for cosmological purposes. In particular, theories involving the time variation of the supposed universal constants more specifically the gravitational coupling should involve York time rather than the usual cosmological time parameter. This way problems at the singularities could be avoided. Of course, there was a serious objection to the second model used. How can one be sure that such a situation can arise? In other words, if one extrapolates back, can it be ensured that one will not violate some initial condition, or get the collapsing shell crossing the lattice cell boundary, or something similar? Another problem was the worry regarding the accuracy involved in the Schwarzschild lattice cell approximation. For this problem the earlier model was replaced by taking the same interior geometry, but rather than emerging into the lattice cell boundary, the outside consisted of a Friedmann universe with a 1/120 section of it removed and replaced by the Schwarzschild cell containing the collapsing shell. The previous results still held good. Thus the latter worry was not serious. To deal with the former problem a model was constructeds called the suture model. To construct this model two closed Friedmann universe models with different maximum sizes and minimum densities were taken. A section of the bigger model was cut out and replaced by a section of the smaller model. For simplicity both the sections were taken to be purely matter filled. As an initial condition, it was required that as one proceeds backwards to the big bang, the two regions merge smoothly together. To maintain continuity of the metric, the two regions were joined by a Schwarzschild region, called the suture. The minor (denser) section is an inhomogeneity in the initial condition in the Friedmann model. Due to the higher density the minor section evolves faster than the major section. Due to the different evolutions a gap opens up between the two sections which are now joined by a Schwarzschild suture. The minor section essentially plays the role of the black hole in this model. After a significant time, observers in the major section would see a black hole formed much like observers in an asymptotically flat spacetime would. Foliation of the suture model by spacelike hypersurfaces of constant mean extrinsic curvature was obtained numerically by A.Qadir and J.A.Wheeler17. The result of their foliation supported Penrose's conjecture. 4. Foliation by Flat Hypersurfaces
It has been argued" that special significance attaches to the frame of observers falling freely from infinity, starting at rest. In this frame the gravita-
42 1
tional force deduced for a Schwarzschild source would be just the Newtonian force. As such, for a more general source the relativistic correction to the Newtonian gravitational force could be computed in this frame. Thus one sees a repulsion due to charge in the RN-geometrylg and a rich structure of forces in a Kerr-Newmann spacetime2'. Such frames have been called $N-frames and have been identified21 as a special class of Fermi-Walker frames. These frames may also be useful to discuss Hawking radiation. The observer dependence of Hawking radiation is a source of worry for believing it to be a genuine physical effect. If it disappears in some frame and appears in another, according to some observers the black hole should never evaporate while according to others it would evaporate arbitrarily fast. How fast or slowly it disappears can be expected to be observer-dependent, but not whether it does so or not. There is no rigorous calculation demonstrating that Hawking radiation disappears in any frame. It can be naively argued that a freely falling observer sees a Minkowski spacetime around him and hence may not see Hawking radiation. This is not necessarily a valid argument, quantum theory being non local. The fact that the spacetime is locally Minkowski can not be used to deduce that there will be no effects of the spacetime curvature. More concretely, the wave functions to be used are defined over all space and not only at a point. As such, it is necessary to actually perform the calculation and check whether there will be Hawking radiation in a freely fall frame or not. To check this, using the canonical quantization procedure, it is required to have a complete foliation of the spacetime by the hypersurfaces corresponding to the $N-observers. Observers falling freely from infinity see Minkowski space about them. As such it might be expected that the hypersurfaces orthogonal to the world-lines of such observers would be flat, i.e. have zero intrinsic or Riemann curvature in their 3-geometry. We call them flat hypersurfaces. Thus the entire curvature of the spacetime would lie in the extrinsic curvature tensor. Of course, it is well nigh impossible to solve the system of nonlinear differential equations 3 R i j k l = 0. However, our expectation is found to hold true and we only need to obtain the free-fall geodesics and these hypersurfaces would be orthogonal to the world-lines of such observers. 5 . The Foliation Procedure
To obtain flat spacelike hypersurfaces that foliate the given spacetime, we have used the fact that the world-lines of observers falling freely from infinity, starting at rest, must be orthogonal to the flat foliating hypersurfaces. The procedure is to specify, in the given geometry, the world-lines of the
422
observers falling freely from infinity and require that the hypersurfaces are everywhere orthogonal to them. This enables one to write down the tangent vectors to the hypersurfaces themselves. Writing the unit tangent vector to the world-line of the freely falling observer as ta and the unit tangent vector to the flat hypersurface as T a ,we require that
tata= -TaTa = 1,
Tata= 0.
(1)
The geodesic equation (for the extremal path between two points) is X"
+rtCIC. bIC. c= 0,
(2)
where the dot represents the derivative with respect to the arc length parameter, s. The tangent vector to the hypersurface can be found by solving Eq.(2) with Eq.(l), for the given spacetime. The components of the tangent vector can be used to obtain an expression for the hypersurfaces. This procedure is used explicitly for the Schwarzschild and the RN spacetimes. The Schwarzschild spacetime is completely foliated by this procedure. For the RN spacetime (since the geodesics do not reach to the essential singularity because of the repulsion due to the chargelg) we can only obtain the hypersurfaces up to the limit the geodesics reach by this procedure. We are, nevertheless, able to obtain a complete foliation of the RN spacetime by spacelike flat hypersurfaces by analytically continuing them. It is also found that by using imaginary values of the constant of integration appearing in the equation of world-lines we obtain complete spacelike hypersurfaces. These hypersurfaces are not flat-but do provide a complete foliation of the RN spacetime. 6. Acknowledgements
The author wishes to thank the organisers of the meeting in Islamabad for their kindness, hospitality and financial support.
References 1.
2. 3. 4.
C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (W.H. Reeman and Sons 1973); HStephani, General Relativity (John Steward, 1982). G. Reeb, Actualites Sci. Indust. (1952). C. Ehresmann, Proc. Fifth Canadian Math. Congress, (1961). R. Penrose in Confrontation of Cosmological Theories with Observational Data, ed. M.S. Longair (D. Riedel, 1974).
423 5. 6.
7. 8.
9. 10. 11.
12. 13. 14. 15. 16. 17.
18.
19. 20. 21.
J.A. Wheeler in Proceedings of the Sixteenth Solvay Conference on Physics: Astrophysics and Gravitation, ed. R. Debever (Universite de Bruxelles, 1974). J.E. Marsden and F.J. Tipler, Phys. Reports 66, 109 (1980). L. Smarr and J.W. York Jr., Phys. Rev. D17, 2529 (1978); D.M. Eardley and L. Smarr, Phys. Rev. D19 , 2239 (1978). A. Qadir and J.A. Wheeler in &om SU(3) to Gravity: Yuval Neeman Festschrifi, eds, E.S. Gotsmann and G. Tauber (Cambridge University Press, 1985); A. Qadir and J.A. Wheeler in Spacetime Symmetries, Proceedings of the Workshop, College Park, Maryland, 1988, eds. Y.S.Kim and W.W.Zachary [Nucl. Phys. B (Proc. Supp.) 6 , 345 (1989)]. J. York, Phys. Rev. Letters 28 ,1656 (1971). F. Estabrook, H. Wahlquist, S. Christensen, B. DeWitt, L. Smarr and E. Tsiang, Phys. Rev. D7 , 2814 (1973). A. Goddard, Spacelike Surfaces of Constant Mean Curvature, Ph.D. thesis, Oxford University (1975); Comm. Math. Phys. 54 , 279 (1977); Math. Proc. Cambridge Philos. SOC.82 , 489 (1977); Gen. Rel. Grav. 8 , 525 (1977). D.R. Brill, J.M. Cavallo and J.A. Isenberg, J . Math. Phys. 21 , 2789 (1980). A. Pervez, A. Qadir and Azad A. Siddiqui, Phys. Rev. D51, 4598 (1995). A. Qadir and Azad A. Siddiqui, J. Math. Phys. 40 , 5883 (1999). R.M. Lindquist and J.A. Wheeler, Rev. Mod. Phys. 29, 432 (1957). S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime, (Cambridge University Press 1973). A. Qadir in Proceedings of the Fifth Marcel Grossmann Meeting, eds. D.G.Blair and M.Buckingham (World Scientific, Singapore, 1988) 593-624; A. Qadir and J.A. Wheeler, Black hole singularity as a part of big crunch singularity, Preprint of the Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. S.M. Mahajan, A. Qadir and P.M. Valanju, Nuovo Cimento B65 ,404 (1981); A. Qadir and J. Quamar in Proceedings of the Thitd Marcel Grossmann Meeting, ed. Hu Ning, (North Holland Publishing Co., 1983) 189. A. Qadir, Physics Letters A99, 419 (1983). A. Qadir, Europhysics Letters 2 , 426 (1986). A. Qadir and I. Zafarullah, Nuovo Cimento B l l l , 79 (1996).
SPINNING PARTICLE: ELECTROMAGNETIC AND GRAVITATIONAL INTERACTIONS* N. UNAL Physics Department, Akdeniz University, P. K. 510, Antalya, 07058, Turkey E-mail: [email protected]. tr We discuss the invariance of the spinning free particle Lagrangian under the global coordinate transformations for the classical model of the electron with internal degrees of freedom and obtain the conservation of the energy-momentum, total angular momentum and electric charge. The local transformations give the minimal and non-minimal gravitational and minimal electromagnetic interactions of the spinning particle in the Riemann-Cartan space from the generalized spin connections.
Keywords: electromagnetism, gravitation, spin
1. Introduction
Einstein proposed his gravitational theory in 1916. Soon after, Weyl tried to extend it to include electromagnetism. Later this idea was revitalized as a U ( l ) gauge invariance. In 1954, Yang and Mills extended it to SU(2). In 1956, Utiyama developed the gauge formulation of the gravitational interactions by the symmetry group SO(3,l) [l]and later Kibble extended this to the Poincart group, P [2]. In field theories, the matter fields are definite eigenstates of the mass, the electric (intrinsic) charge and the spin. In this connection, the minimal couplings of the matter field with gravitation and electromagnetism are discussed, separately. The spacetime symmetries are built by the requirement that the Lagrangian density for the free matter fields be invariant under the action of the corresponding symmetry group. Here, P , consists of translations and Lorentz rotations and SO(1, 3) frame rotations of matter fields, respectively [3-91. The global covariance of the spinless matter *This work is supported by Akdeniz University, Scientific Research Projects Unit.
424
425
field under P gives the conservation of the energy-momentum and the total angular momentum. The local extension of the gauge group, P , gives the minimal coupling of the matter field with gravitation: The spacetime part of the gauge group becomes the diffeomorphism group and the spinless matter field is invariant under the general coordinate transformations and the local SO(1, 3) frame rotations. In another approach, P is considered as the internal symmetry group of matter fields in Minkowski spacetime to obtain a complementary gauge formulation of gravitation [lo]. On the other hand, the inner symmetries require the invariance of the Lagrangian density for the matter fields under the action of some Lie group which is only represented in the space of fields and does not act on spacetime. Here it is invariance under U( 1) for electromagnetism. The Dirac equation is generalized to the curved spacetime by introducing the Fock-Ivanenko 2-vector or the spin connection [ l l ] . In this equation there are two kind of coupling with gravitation: The minimal coupling by introducing the vectors in the curved spacetime and the non-minimal coupling by the spin connection. The Dirac algebra relates the metric tensor of the spacetime to the anti-commutator of the spacetime dependent Dirac matrices. In their pioneering investigations, Schrodinger [12] and Bargmann [13] discussed a possible generalization of the spin connection and showed that it gives the spacetime curvature, the spin-2 gravitational field, and an Abelian spin-1 curvature. Since the spin-1 part of the connection is coupled to all spinors with an identical charge, Schrodinger did not identify it with electromagnetism. Recently, Crawford [14] investigated the coupling of the torsion with the spinning particle by evaluating the commutators of the covariant derivatives for the Dirac matrices, [V,, V,] y a and discussing the generalization of the spin connection. He showed that it is possible only by removing the constraint of the covariant constancy of the Dirac matrices. He came to the same conclusion for the spin-1 curvature as Schrodinger. The aim of this study is to derive the minimal and non-minimal coupling of electromagnetism and gravitation with the spinning particle in a complementary, unified approach. In this approach we perform only the spacetime transformations as the gauge transformations of the P and the internal coordinate transformations as the gauge transformations of the group U(2, 2). Due to the gauge group U(2, 2), we realize the Abelian and non-Abelian phase transformations represented by the generalized spin connection. For this purpose our motivation is the spinning particle model [15]. In this model, the spinning particle does not correspond to definite mass, charge and spin eigenstates, contrary to the field theories. In these theories the
426
mass and spin are identified by using the conserved dynamical variables of the particle without discussing the corresponding global gauge transformations, but the electric charge is introduced into the interaction Lagrangian by hand. The free-particle Lagrangian for a spinning particle is
where xa and p a are the coordinates and the momenta of the particle in the Minkowski spacetime, M4 with the metric qab = (1,-1, -1, -1). In Eq. (??) dot means the derivative with respect to s, the proper time of the particle, y a are constant Dirac matrices, z is the complex spinor with components zi for i = 1 , 2 , 3 , 4 , and the hermitian conjugate of z in 3+1 D Minkowski spacetime is defined as F = zty'. The dynamical variables zi are four complex internal coordinates and the internal configuration space of the particle is C4. The signatures of the bilinears in the quadratic form Zz are (+,+,-,-)and the transformations between the holomorphic coordinates conserve these signatures. We use Latin indices for local Lorentz frames and Greek indices for non-coordinate frames. In Eq. (??) p a may be considered as the four-Lagrange multiplier with the constraint
x" - z y a z = 0.
(2)
This constraint relates the external and internal dynamics of the particle. Since the Lagrangian in Eq. (??) is defined in phase space the Hamiltonian structure of the system is known or chosen at the beginning. 2. Global and Local Gauge Transformations
In M4 there are two kinds of coordinate transformations. These are the spacetime translations and external Lorentz rotations:
where P, and L c d are the generators of the translations and rotations, respectively. The Lagrangian in Eq. (??) is invariant under the translations and p a is conserved. In C4 there are two kinds of internal transformations which conserve the relation between the internal and external dynamics of the particle, (??). These are U(2, 2) phase transformations and they conserve the signatures
427
of the frequencies of the zitterbewegung oscillations (+,+,-,-):
where C,d is
In Eqs.(??) there are two kinds of phase transformations: The Abelian part involving exp (-i$l), and the non-Abelian part involving exp ( - i c c d C c d ) . The generators of the U ( 2 , 2 ) are 1 and Ccd and corresponds to the 0-vector and 2-vectors of the four dimensional Clifford or Dirac algebra, Ccd. The Lagrangian (??) is invariant under the external and internal Lorentz rotations and phase transformations and the conserved quantities are Zz and total angular momentum, Jab: Jab
1 = Lab+ -zCabz. 2
(5)
In this subsection we consider the local coordinate and phase transformations as gauge transformations. The external translations and Lorentz rotations are expressed by the non-holonomic transformations
dx"
--f
dx'
=
(6;
+ [', b + cPc, b 5')
dxb,
(6)
1
(9)
where we introduce the tetrads, eWb(x)as
The internal transformations are
where U is
i -i+ (x)1 - -ccd (x)cCd 4
Under these transformations
-
Zyaz -+ [Fyp(x)21 = e' a z y a ~ . Then the Lagrange multiplier term in the Lagrangian becomes pa (i"- ~y"z)4 p a e a p(2- Zepbybz),
(10)
428
where ea, is the inverse of the tetrads, eph.The metric tensor, gpV is defined as gpu
(x)= e a p (x)ehv (x)v a h .
The momentum, p , is defined in global coordinates as ~p
= eap (x)P a .
Then the Lagrange multiplier term is rewritten as p,
(i” - F y t ) ,
(11)
and it is form invariant under the transformations P and U(2, 2). The kinetic part of the Lagrangian is transformed as
2i Evaluating the derivatives and considering the Lagrange multiplier, the additional term becomes
where B, is
1
Bp
(x)= eap x) 4 , a (x)+ qfCd,a(x)c c d ] . (
[
(13)
Here i e a (x)E ~ (x)~ is the , ~non-holonomic connection, r, c d ( x ) .In Eq. (??), we identify ea (x)$ ~ , ~ as ( x the ) gauge potential of the internal phase transformations, the electromagnetic potential, A,(x) and ar, Cd(Z)&-d as the gauge potential of the internal Lorentz rotations, the Fock-Ivanenko 2-vectors, :
,
A,(x) = e a p (x>$ J , a ( x > ,
r, ).(
=
1 -rPcd (x)C c d . 4
Then B,(z) can be considered as the unified gauge potential of the internal coordinate transformations, U(2, 2). We rewrite the additional interaction terms as
z ~ , ( z ) z= F [ A , ( ~+) r
-
, ~ z~. ) ]
(14)
429
The Lagrangian of the spinning particle, interacting with electromagnetic and gravitational fields, is &-(A 1 z z - 2 5 +p,X’ - H, (15) 2i where ‘H is the Hamiltonian:
1
‘H = ( p ,
-
FB,z) FyPz = n,zy,z.
(16)
The ?-I in Eq. (??) looks like the Dirac Hamiltonian, but it corresponds to the classical Hamiltonian of the zitterbewegung system with internal degrees of freedom and continuous spin values. 3. Conclusions
We derived the electromagnetic and the gravitational interactions of spinning particle by using the local gauge transformations in the RiemannCartan spacetime and internal spacetime of the particle. In the presence of the gravitational and electromagnetic forces the electromagnetic current, j p is conserved. Since yp is covariantly constant the quantum Hamiltonian is obtained from the classical one in Eq. (??) by replacing the classical dynamical variables with the corresponding quantum operators for P,2, &, and without any ordering corrections.
Ti
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13. 14. 15.
R. Utiyama, Phys. Rev. 101, 1597 (1956). T. W. B. Kibble, J. Math. Phys. 2, 212 (1961). K. Hayashi and T. Nakano, Prog. Theor. Phys. 38, 491 (1967). K. Hayashi, Prog. Theor. Phys. 39, 464 (1968). K. Hayashi and A. Bregman, Ann. Phys. 75, 562 (1973). K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 64, 866 (1980). F. W. Hehl et al, Rev.Mod.Phys. 48, 393 (1976). Y. Neeman, in Lect. Notes in Math. 676, ed. by K. Bleuer et al., (Springer, Berlin, 1979). J. Hennig and J. Nitsch, Gen. Rel. Grav.13, 947 (1981). C. Wiesendanger, Class. Quant. Grav. 13, 68 (1996). V. A. Fock, Zeits. Phys. 57, 261 (1929); V. A. Fock and D. D. Ivanenko, Comptes Rendus des Seances d e L’Acadkmie des Sciences 188, 1470 (1929). E. Schrodinger, Sitz. Preuss Ak. D. Wiss. 25, 105 ( 1932). V. Bargmann, Sitz. Preuss. Ak. D. Wiss. 25, 346 (1932). J. P. Crawford, Class. Quant. Grav. 20, 2945 (2003). A. Proca, J. Phys. Radium 17, 5 (1956); A. 0. Barut and N. Zanghi, Phys. Rev. Lett. 52, 2009 (1984).
QUANTUM GRAVITY AND HAWKING RADIATION M. VARADARAJAN Theory Group, Raman Research Institute, Sir C. V. Raman Awe., Bangalore, Karnataka 560 080, India E-mail: [email protected] Recently, Ashtekar and Bojowald proposed a resolution of the Hawking Information Loss Puzzle based on their qualitative picture of the evaporation process. Here, we examine the puzzle in the solvable (and hence, mathematically precise) context of the Callen-Giddings-Harvey-Strominger (CGHS) toy model of black holes in 2 spacetime dimensions. The CGHS model is known to admit a non- perturbative quantization. We show that the resolution of the puzzle in the context of this quantization exhibits features of the Ashtekar-Bojowald paradigm and, therefore, provides a useful setting from which lessons for the 4 dimensional case may be learned. This article is a report of joint work with A. Ashtekar.
Keywords: black holes, information loss
1. Introduction It is widely accepted that the Hawking Information Loss Puzzle will find its resolution in a theory of quantum gravity. To date, no such (complete) theory is available. Therefore, it is useful to examine the puzzle in the context of toy models which admit black hole solutions as well as a complete quantization. In this article we report on some results obtained in the context of one such model, namely the CGHS model' of 2 dimensional black holes. We provide a broad-brush introduction to the Hawking Information Loss Puzzle in section 2 and to a recently proposed alternative view of the evaporation process in section 3. No attempt is made at a detailed description; details may be found in references [2] and [3] respectively. We describe our results for the CGHS model in section 4 and show that they share features with the proposal described in section 3; details may be found in reference [4]. We shall use obvious notation whenever possible.
430
43 1 2. The Information Loss Puzzle
By definition, a classical black hole cannot radiate to future null infinity. However, in his seminal work5, Hawking showed that black holes do radiate when leading order quantum effects are taken into account. Specifically, Hawking analysed the behaviour of quantum matter fields propagating on the classical spacetime geometry of a spherically symmetric black hole and showed that such a black hole spontaneously radiates particles of the quantum matter field at late times. The radiation is thermal at the ‘Hawking’ temperature TH where ~ T H ( F ) m p c 2 Here k is the Boltzmann constant, m p the Planck mass and M the black hole mass. Since energy conservation demands that the black hole mass must decrease due to the loss by radiation, the black hole is said to ‘evaporate’. Indeed, the relation TH N suggests the sequence: Hawking radiation + black hole mass loss -+higher Hawking temperature &more radiation. The following estimate shows that the evaporation process is very slow. The time scale for evaporation can be estimated as Tevap= with estimated through the Stefan- Boltzmann Law as U T ~ Rwhere ; Rs S M is the Schwarzschild radius of the black hole. The time scale over which the black holes settles down after it is perturbed can be estimated as the time it takes for light to travel across the black hole so that Tsettling Rslc. This yields TsettlinglTevapm $ / M 2 . Thus, as long as M >> m p , evaporation is a quasistatic process and the geometry can be modelled as that of a 1 parameter family of black holes of decreasing mass as depicted in Fig. (1).Evaporation continues till most of the mass (N M-mp) is lost as radiation, at which stage both the quasistatic approximation and the quantum field-on-curved-spacetime approximation break down. The collapsing matter admits an underlying quantum description and at early times this quantum state is pure. The final state after collapse is that of a ‘remnant’ of Planck size and lots of thermal radiation. The latter being a mixed state implies that the final state is pure only if the remnant has enough correlations with the radiation; this is ruled out by low energy physics arguments2 and we have the puzzle that information has been lost (i.e. quantum evolution is not unitary) during the evaporation process. Note that though quite robust, there is still a loophole in the above argument: although the evaporation process is very slow, the black hole lifetime is very large and small corrections to quasistaticity can have a cumulative effect such that there is a significant departure from the above picture after enough time has elapsed. Hence there is room for alternative pictures of the evaporation process and we describe one such alternative
-
N
&.
&
(&%)-’
%
N
N
N
432
MAlTER
Fig. 1.
The standard picture of black hole evaporation.
which was proposed by Ashtekar and Bojowald in Ref. [3]. 3. The Ashtekar Bojowald Paradigm
In the standard picture (see Fig. l),the final state is not just a remnant and radiation; there is also a (singular) boundary of spacetime. Ashtekar and Bojowald (AB) note that the notion of classical spacetime should break down not only at the end point of evaporation but all along the singularity. Using results from symmetry reduced models, they suggest that the classical singularity will be resolved also in the full theory of quantum gravity. The picture which AB put forward has the following features3: (a) Classical spacetime is not a viable concept near the singularity and is replaced by some, as yet unknown, quantum construct; (b) Quantum evolution (of gravity matter) is well defined through the classical singularity and hence spacetime does not end at the classical singularity but admits a quantum extension; (c) Quantum evolution is unitary and the ‘missing’ information is recovered from the correlations between the Hawking radiation and the quantum fields which re-emerge on the ‘other side’ of the classical singularity; (d) A significant part of the ADM mass evolves through the classical singularity and the area of a quasilocal construct, defined using the notion of trapped surfaces, called the dynamical horizon, evaporates6.
+
4. The CGHS model
The CGHS action’ depends on a 2 dimensional metric, g a b , a dilaton field, 4, and a scalar field, f . In appropriate units, the action is
433 SINGULARITY
LEFT PAST NULL INFINIT
Fig. 2.
RIGHT PAST
Penrose diagram for classical black hole formation in the CGHS model.
Here K is a ‘cosmological’constant. The model is exactly solvable. The general solution is as follows. Since spacetime is 2 dimensional, the metric is conformally flat so that gabdxadxb= *l where x* = x f t and 0-l is the conformal factor. Since the scalar field is conformally coupled, it satisfies the flat spacetime wave equation &8+f = 0 so that f is of the form f = f(+)(x+)+f(-)(x-). The functions f(+) and f(-) are called ‘left’ and ‘right’ movers. It turns out that (0 - 1) is given by double integrals of the stress energy of f . The dilaton can also be expressed in terms of f . In the interests of pedagogy we shall not discuss the dilaton any further in this article. Clearly, if f = 0, R = 1 and we have flat spacetime. The solution corresponding to matter collapse is depicted in Fig. 2 and is obtained when j(-) = 0 and f(+) is chosen to be some function of compact support which we denote by c(+). Note that the spacetime ends at the singularity as a result of which the spacetime manifold is a proper subset of the full (x+,x-) plane. A Hawking type analysis of a (right moving, conformally coupled) quantum test field on the background spacetime of Fig. 2 implies that the black hole emits Hawking radiation to right future null infinity (at a mass independent Hawking t e m p e r a t ~ r e )The ~ . model admits a non-perturbative quantizations-10 as follows. f^(x+,x-) is quantized in the standard flat spacetime Fock representation since its dynamics is that of a massless scalar field on the auxilliary flat background dx+dx-. Thus
434
where ii(*)(k) are the annihilation operators for the right and left moving modes. fi(x+,x-) can be obtained by substituting f for f in the classical expression for R and then normal ordering with respect to ii(*)(k), ii;*)(lc). Note that the natural arena for quantum theory is the entire (x+,x-) plane, since that is where f(z+,x-) ‘lives’ and where it admits the standard flat spacetime mode expansion (2). Note that the Hilbert space is the product of the left moving and right moving Fock spaces. Although the quantum states in this Hilbert space seem to be those of the scalar field, it is important to remember that they are quantum states of the full gravity- dilaton- scalar field system. Consider the state lc) = Ic(+))@ lo(-)) where lo(-)) is the right moving vacuum and Ic(+)) is the coherent state based on c(+)(z+). Thus ii(-)(k)lO(-))= 0 and ii(+)(k)lc(+)) = c(+)(lc)lc(+))where c(+)(k) are the mode coefficients of c(+)(x+). Then we have the following results: (i) For all (x+,x-) in Fig 2, (clfi(s+,x-)lc) = R(x+,z-) and (clf^(x+,x-)Ic)= c(+)(x+) so that the classical geometry is recovered in expectation value. The region of the (x+,x-) plane “above” the singularity (see Fig 3) offers a quantum extension of the spacetime. The expression for fi(x+,x-) as double integrals of f(x+,x-) is well defined in this extension. Thus the singularity is resolved in quantum theory.a In the extension, the expectation value of fi is negative so the metric suffers a ‘signature flip’ (-, +) 4 (+, -) across the singularity. (ii) Although fi is only a quadratic form and not a well defined operator, it can be suitably smeared to obtain a well defined operator. Thus f i ( a ) = J7dzcy(x)f2(x)is a well defined operator for suitable choices of smearing function a(.) and curve y (here x is a coordinate on the curve y). An analysis of the fluctuations of this operator indicates that the geometry fluctuates violently near the classical singularity so that classical geometry is not a viable concept near the singular region. (iii) The quantum theory of the model is unitary. The state lo(-)) restricted to the right null infinity of Fig. 2 is a mixed state. Its particle content as viewed by freely falling (in the geometry of Fig. 2) observers at infinity aThis is not deep; it is just a consequence of the classical equations admitting analytic continuation beyond the singularity which in turn is a consequence of the conformal coupling of f .
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"
," '',
Fig. 3.
QUANTUM EXTENSION OFCLASSICALSPACETIME
The quantum extension of the classical spacetime of Fig. 2
is thermally distributed at the Hawking temperature. The fact that lo(-,) is actually pure is apparent only in the full quantum extension of Fig. 3. The missing information is encoded in correlations with operators in this quantum extended region. Note that lo(-)) is not a 'test field state'; it is (the right moving part of) the underlying non-perturbative state for the dilaton- gravity- matter system. (i)- (iii) clearly exhibit features of (a) - (c) of the AB paradigm. The key open question is how to understand the backreaction of the Hawking radiation on the geometry from the perspective of the non- perturbative quantization. Thus, we would like to define and compute detailed O(h) (and higher) corrections to the classical equations from the underlying quantum theory. This constitutes work in progress and involves the non-trivial issue of interpretation in quantum gravity.
References 1. C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, Phys.Rev.
D45, 1005 (1992). 2. J. Preskill, in Proceedings of the International Symposium on Black Holes, Membranes, Wormholes, and Superstrings, Woodlands, T X , 1992, (World Scientific, Houston, TX, 1992). 3. A, Ashtekar and M. Bojowald, Class. Quant. Grav 22, 3349 (2005). 4. A. Ashtekar and M. Varadarajan, in preparation 2006. 5. S. W. Hawking, Comm. Math. Phys. textbf43, 199 (1975) . 6. A. Ashtekar and B. Krishnan, Phys. Rev. D68, 104030 (2003). 7. S. B. Giddings and W. M. Nelson, Phys. Rev. D46,2486 (1992). 8. K. Kuchaf, J. Romano and M. Varadarajan, Phys. Rev. D55, 795 (1997). 9. A. Mikovic, Phys. Lett. B355, 85 (1995). 10. M. Varadarajan, Phys. Rev. D57, 3463 (1998).
MEASURING PARAMETERS OF SUPERMASSIVE BLACK HOLES WITH SPACE MISSIONS A.F. ZAKHAROV National Astronomical Observatories of Chinese Academy of Sciences, Beijing 100012, China Institute of Theoretical and Experimental Physics, 25, B. Cheremuahkinskaya st., Moscow, 11 7259, Russia Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia; E-mail: zakharov0itep.m To describe black hole in astrophysics typically astronomers use Newtonian approaches for the gravitational field because usually one analyzes processes acting far enough (in Schwarzschild radius units) from black hole horizons. Here we discuss phenomena where we have t o use general relativistic approaches to explain present and future observational data like Fe K , line profiles and shapes of shadows around black holes. Different X-ray missions such as ASCA, XMM-Newton, Chandra etc. discovered features of Fe K , lines and other Xray lines as well. Attempts to fit spectral line shapes lead t o conclusions that sometimes the profiles line shapes should correspond t o radiating regions which are located in the innermost parts of accretion disks where contributions of general relativistic phenomena are extremely important. As an illustration we consider a radiating annulus model t o clarify claims given recently by Miiller & Camenzind (2004). We discuss properties of highly inclined disks and analyze the possibility t o evaluate magnetic fields near black hole horizons. We mention also that shadows could give us another case when one could evaluate black hole parameters (namely, spins, charges and inclination angles) analyzing sizes and shapes shadows around black holes. Keywords: Black hole physics; the Galactic Center; Tests of General Relativity
1. Introduction Here we discuss samples where we really need general relativistic approaches in the strong gravitational field limit to explain observational data generating by radiation arising in black hole vicinities and typically one could get the data with space missions such as ASCA, RXTE, XMM-Newton,
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Chandra etc. Several years ago it was predicted that profiles of lines emitted by AGNs and X-ray binary systems could have asymmetric double-peaked, double horned or triangular shape according to classification done by Muller and Camenzind [l]. A comprehensive review summarizes the detailed discussion of theoretical aspects of possible scenarios for generation of broad iron lines in AGNs [2] (an influence of microlensing on Fe K , line shapes and spectra was discussed in [3] but optical depths for the phenomena were calculated in [4-61). A formation of shadows (mirages) is another sample when general relativistic effects are extremely important and in principle they could be detected with forthcoming interferometrical facilities [7-151 (perspective studies of microlensing with Radioastron facilities were discussed recently [16]). Observations of shadows could give a real chance to observe ”faces” of black holes of black holes and confirm general relativitistic predictions in the framework of a strong gravitational field approach and obtain new constraints on alternative theories of gravity. 2. Toy Model Lessons Recently Muller and Camenzind [ 11 presented results of their calculations and classified different types of spectral line shapes and described their origin. In particular, the authors claimed that usually “... triangular form follows from low inclination angles...”, “...double peaked shape is a consequence of the space-time that is sufficiently flat. This is theoretically reproduced by shifting the inner edge to the disk outwards ... A relatively flat space-time is already reached around 25 rg...”We tested their hypothesis about an origin of doubled peaked and double horned line shapes. Using a radiating annulus model for numerical simulations we showed that double peaked spectral lines arise for almost a n y locations of narrow emission rings (annuli) (except closest orbits as we see below) although Muller and Camenzind [l]suggested that such profiles arise for relatively flat spacetimes and typical radii for emission region about 25 r g .We note here that in the framework of the model we do not use any assumptions about an emissivity law, but only that the radiating region is a narrow circular ring (annulus). But general statements (which will be described below) could be generalized on a wide disk case without any problem. We used an approach which was discussed in details in other papers [17-371. The approach was used in particular to simulate spectral line shapes. This approach is based on results of qualitative analysis [38, 391. Presenting their classification of different types of spectra line shapes
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Muller and Camenzind [l]noted that double peaked shapes arise usually for emission regions located far enough from black holes. Earlier, we calculated spectral line shapes for annuli for selected radii and distant observer position angles and found an essential fraction of spectral line gallery correspond to double peaked profiles [20].To check the Muller and Camenzind [l]hypothesis about an origin of double peaked profiles we calculated a complete set of spectral line shapes for emitting annuli. Let us discuss results of our calculations for rapidly rotating black holes (for a = 0.998 one could find a detailed description of the calculations in [40-421). We summarize results of the calculations. As was shown in the framework of the simple model the double peaked spectral line shape arises almost for all parameters T and a except the case when radii are very small T E (0.7,2) and inclination angles are in the band 0 E [45",90'1 (for these parameters the spectral line shape has triangular structure). The phenomenon could be easy understood, since for this case the essential fraction of all photons emitted in the opposite direction with respect to the emitting segment of annulus is captured by the black hole. Therefore the red peak is strongly damped. For other radii and angles spectral line profiles have double peaked structure. If we assume that there is a weak dependence of emissivity function on the radius, then the number of photons characterizes the relative intensity of the line (roughly speaking for T = 0.7 an intensity (in counts) is 10 times lower than an intensity for T = 2). Therefore in observations for small radii one should detect only a narrow blue peak but another part of the spectra is non-distinguishable from the background. Note also that for a fixed radius there is a strong monotone dependence of intensity on inclination angle (maximal intensity corresponds to photon motion near the equatorial plane and only a small fraction of photons reach a distant observer near the polar axis). That is a natural consequence of the photon boost due to circular motion of emitting fragments of the annulus in the equatorial plane and the influence of spin of the rotating black hole. In the framework of the simple model one could understand that sometimes the Fe K , line has only one narrow peak like in observations of the Seyfert galaxy MCG-6-30-15 by the XMM-Newton satellite [43]. If the radiating (or illuminating) region is a narrow annulus evolving along quasi-circular orbits, then initially two peak structure of the spectral line profile transforms in one peaked (triangular) form. Moreover, an absolute intensity in the line is increased for smaller radii since a significant fraction of emitted photons are captured by the black hole during the evolution of the emitting region towards the black hole. Hence in observations we only could detect a narrow
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blue peak and its height will be essentially lower than its height was before for larger radii. Other parts of the triangular spectral line shape could be non-distinguishable from the background. A relatively low intensity for a triangular spectral line shape could give a narrow single peak structure in observations.
3. Signatures of Accretion Discs with High Inclination
At inclination angles 19 > 80°,new observational features of GR could arise. Matt et al. [44] discovered such phenomenon for a Schwarzschild black hole, moreover the authors predicted that their results could be applicable to a Kerr black hole over the range of parameters exploited. The authors mentioned that this problem was not analyzed in detail for a Kerr metric case and it would be necessary to investigate this case. In the detailed consideration [23] we did not use a specific model of surface emissivity of accretion (we only assume that the emitting region is narrow enough). Therefore, we confirmed their hypothesis for the Kerr metric case and for a Schwarzschild black hole using other assumptions about surface emissivity of accretion disks. In principle, such a phenomenon could be observed in microquasars and X-ray binary systems where there are neutron stars and black holes with stellar masses. We confirmed also the conclusion that extra peaks are generated by photons which are emitted by the far side of the disk, therefore we have a manifestation of gravitational lensing in the strong gravitational field approach for GR [23]. Some possibilities to observe the above features of spectral line profiles were considered [44]. The authors argued that there are non-negligible chances to observe such a phenomenon in some AGNs and X-ray binary systems. Thus, such properties of spectral line shapes are robust enough with respect to wide variations of rotational parameters of black holes and the surface emissivity of accretion disks as was predicted [44]. Their conjecture was confirmed not only for the Kerr black hole case but also for other dependencies of surface emissivity of the accretion disk. Positions and heights of these extra peaks drastically depend on both the radial coordinate of the emitting region (annuli) and the inclination angle. It was found that these extra peaks arise due to gravitational lensing effects in the strong gravitational field, namely they are formed by photons with some number of revolutions around the black hole. This conclusion is based only on relativistic calculations without any assumption about physical parameters of the accretion disc like X-ray surface emissivity etc. A detailed description of the analysis was given in [23].
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4. Magnetic Fields in AGNs and Microquasars Magnetic fields play a key role in dynamics of accretion discs and jet formation. To obtain an estimation of the magnetic field we simulate the formation of the line profile for different values of magnetic field. As a result we find the minimal B value at which the distortion of the line profile becomes significant. Here we use an approach, which is based on numerical simulations of trajectories of the photons emitted by a hot ring moving along circular geodesics near the black hole, described in [18-201. The influence of the accretion disc model on the profile of spectral line was discussed [31]. Let us discuss the possible influence of high magnetic fields on real observational data (see details in [25]).We will try to estimate magnetic fields when one could find the typical features of line splitting from the analysis of the spectral line shape. Further we will choose some values of magnetic field and simulate the spectral line shapes from observational data for these values, assuming that these observational data correspond to an object with no significant magnetic fields. We will try to find signatures of the triple blue peak analyzing the simulated data when magnetic fields are rather high. Assuming that there are no essential magnetic fields (compared to lo1' G) for some chosen object (for example, for MCG 6-30-15) we could simulate the spectral line shapes for the same objects but with essential magnetic fields. From results of simulations one can see that classical Zeeman splitting in three components, which can be revealed experimentally today, changes qualitatively the line profiles only for rather high magnetic field. Something like this structure can be detected, e.g. for H = 1.2.1011 GI but the reliable recognition of three peaks here is hardly possible [25].It is known that neutron stars (pulsars) could have huge magnetic fields. So, it means that the effect discussed above could appear in binary neutron star systems. The quantitative description of such systems, however, needs more detailed computations. A detailed discussion of the magnetic field influence on spectral line shapes was discussed for flat accretion flows [25, 261 and for non-flat accretion flows [29].
Acknowledgments I would like to thank the organizers of the 12th Regional Conference on Mathematical Physics (Islamabad, Pakistan), especially Profs. Asghar Qadir, Riazuddin, Faheem Hussain for hospitality and their kind attention to this contribution. Also I am grateful to the National Natural Science Foundation of China (NNSFC) (Grant # 10233050) and National Key Ba-
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sic Research Foundation of China (Grant # TG 2000078404) for a partial financial support of the work.
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Participants Ahmed, Sarfraz Government College University, Lahore, Pakistan
Abbas, Gohar Government College University, Lahore, Pakistan Acharya, Bobby The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Afzal, Tafazul Government College University, Lahore, Pakistan
Ahmed, Zahid Punjab University, Lahore, Pakistan Ahmed, Zahoor National Centre for Mathematics, Government College University, Lahore, Pakistan
Ahmad, Niaz Quaid-i-Azam University, Islamabad, Pakistan
Akhtar, Naseem Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Ahmad, Sarfraz Quaid-i-Azam University, Islamabad, Pakistan
Al-Ajmi, Mudhahir Sultan Qaboos University, Muscat, Oman
Ahmed, Ali Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Ali, Sajid Centre for Advanced Mathematics and Physics, National University of Science and Technology, Rawalpindi, Pakistan
Ahmed, Akhlaq Qua id- i-Aza m University, Islamabad, Pakistan
Ah, Shaukat Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Ahmed, Ishtiaq National Centre for Physics, Islamabad, Pakistan
Alishahiha, Mohsin IPM School of Physics, Tehran, I r a n
Ahmed, Kamaluddin COMSATS Institute of Information Technology, Islamabad, Pakistan
Ansari, Hamid National Centre for Physics, Islamabad, Pakistan
Ahmed, Mofiz Uddin Bangladesh Open University, Gazipur, Bangladesh
Ardalan, Farhad IPM School of Physics, Tehran, I r a n
Ahmed, Muhammad National Centre for Physics, Islamabad, Pakistan
Arshad, Muhammad National Centre for Physics, Islamabad, Pakistan
Ahmed, Mushtaq Government College University, Lahore, Pakistan
445
446 Asghar, M. Irfan National Centre for Physics, Islamabad, Pakistan
Dhar, Avinash Tata institute of Fundamental Research Mumbai, India
Ashfaq, Muhammad Quaid-i-Azam University, Islamabad, Pakistan
Duff, Michael Imperial College, London, U. K.
Aslam, M. Jamil National Centre for Physics, Islamabad, Pakistan
Duru, Ismail Hakki Turkish Academy of Science, Istanbul Teknik Universitesi, Macka-Istanbu I, Turkey
Ayub, Muhammad Government College University, Lahore, Pakistan Aziz, Sehar Punjab University, Lahore, Pakistan Balachandran, A. P. Syracuse University, Syracuse, N.Y., U. S . A. Bhat, Naseer Iqbal University of Kashmir, Srinagar, Kashmir
Eagleton, T. S. The Canon Foundation for Scientific Research, Oxford, U. K. Ehsan, Zahida Government College University, Lahore, Pakistan Faridi, Ayub Centre for High Energy Physics, Punjab University, Lahore, Pakistan
Bilal, Amer Quaid-i-Azam University, Islamabad, Pakistan
Fayyazuddin, Ansar Baruch College, The City University of New York, New York, N.Y., U. S. A.
Camci, Ugur Canakkale Onsekiz Mart University, Canakkale, Turkey
Fayyazuddi n National Centre for Physics, Islamabad, Pakistan
Dadich, N. K. Inter-University Centre for Astronomy and Astrophysics, Pune, India
Feroz, Tooba Centre for Advanced Mathematics and Physics, National University o f Science & Technology, Rawalpindi, Pakistan
Dar, Amanullah Quaid-i-Azam University, Islamabad, Pakistan Dayi, Omer F. Istanbul Technical University, Maslak-Istan bu I, Turkey Deeba, Farah Government College University, Lahore, Pakistan
Firdous, Hina Quaid-i-Azam University, Islamabad, Pakistan Ghoshal, Debashis Ha rish-Cha nd ra Research Institute, Allahabad, India
447 Gilani, Amjad H. Shah National Centre for Physics, Islamabad, Pakistan Hall, Graham University of Aberdeen, Aberdeen, Scotland, U. K. Hannan, Abdul Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan Anwar UI Haq, M. University of Sargodha, Sargodha, Pakistan Haque, Qamar Pakistan Institute o f Nuclear Science and Technology, Islamabad, Pakistan Haseeb, Mahnaz COMSATS Institute of Information Technology, Islamabad, Pakistan Hoodbhoy, Pervez Quaid-i-Azam University, Islamabad, Pakistan Hoorani, Hafeez National Centre for Physics, Islamabad, Pakistan Husain, Tasneem Zahra Ha rvard University, Cambridge, Mass., U. S. A. Husain, Viqar University of New Brunswick, New Brunswick, Canada Huseynaliev, Yashar Bahauddin Zakaria University, Multan. Pakistan Hussain, Faheem National Centre for Physics and COMSATS Institute of Information Technology, Islamabad, Pakistan
Hussain, Ibrar Centre for Advanced Mathematics and Physics, National University o f Science & Technology, Rawalpindi, Pakistan Hussain, Manzar Quaid-i-Azam University, Islamabad, Pakistan Hussain, Safdar University of Sargodha, Sargodha, Pakistan Hussain, Sajjad Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan Idrees, Shaneela Centre for High Energy Physics, Punjab University, Lahore, Pakistan Imran, Muhammad Quaid-i-Azam University, Islamabad, Pakistan Iqbal, Kaleem Pakistan Institute o f Nuclear Science and Technology, Islamabad, Pakistan Iqbal, Muhammad Government College University, Lahore, Pakistan Ismail, Tariq Punja b University, Lahore, Pakistan Israr, Zeba Government College University, Lahore, Pakistan Jamil, Mubashar Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan
448 laved, Tariq Quaid-i-Azam University, Islamabad, Pakistan
Koca, M. Sultan Qaboos University, Muscat, Oman
layaraman, T. Institute of Mathematical Sciences, Chennai, India
Korner, 3 . G. Institut fur Physik, Univ. Mainz, Mainz, Germany
Kadeer, Alimjan Institut fur Physik, Univ. Mainz, Mainz, Germany
Kumar, Alok Institute of Physics, Bhubaneswar, India
Kaladze, T. I.Vekua Institute of Applied Mathematics of Tbilisi State University, Tbilisi, Georgia
Leites, D. A. S MPIMiS, Leipzig, Germany
Kashif, Abdul Rahman Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan
Mahajan, S. Institute for Fusion Studies, University of Texas a t Austin, Texas, U. S. A. Mahboob Quaid-i-Azam University, Islamabad, Pakistan
Khan, Waqas Mahmood Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Mahmood, Ahmer Qua id- i-Aza m University, Islamabad, Pakistan
Khan, Majid Quaid-i-Azam University, Islamabad, Pakistan
Mahmood, Shahzad Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Khan, Mariam Saleh National Centre for Physics, Islamabad, Pakistan Khan, Shahid Government College University, Lahore, Pakistan
Mahmood, Waqas National Centre for Physics, Islamabad, Pakistan Malik, Asim Ali Government College University, Lahore, Pakistan
Khawaja, Atta UI Latif Centre for High Energy Physics, Punjab University, Lahore, Pakistan
Malik, Assad Abbas Quaid-i-Azam University, Islamabad, Pakistan
Kh ursh id, Ta i moor National Centre for Physics, Islamabad, Pakistan
Mann, Amer Quaid-i-Azam University, Islamabad, Pakistan
Kiran, Zubia Government College University, Lahore, Pakistan
449
Masood, Bilal Centre for High Energy Physics, Punja b University, Lahore, Pakistan Mir, Azeem COMSATS Institute of Information Technology, Islamabad Pakistan Mirza, Arshad Majeed Quaid-i-Azam University, Islamabad, Pakistan Mukhtar, Qaisar Quaid- i-Aza m University, Islamabad, Pakistan Murtaza, Ghulam Government College University, Lahore, Pakistan Jameel Un Nabi Ghulam Ishaq Khan Institute, Topi, Swabi, Pakistan
Pasupathy, J. Bangalore, India Qadir, Asghar Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan Qaisar, Suleman Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan Qureshi, M. Jamil Punjab University, Lahore, Pakistan Rafique, M. Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpind, Pakistan i
Naqvi, Asad University of Wales, Swansea, U. K.
Hafeez Ur Rahman Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Narlikar, J.V. Inter-University Centre for Astronomy and Astrophysics, Pune, India
Muneeb Ur Rahman Ghulam Ishaq Khan Institute, Topi, Swabi, Pakistan
Nasim, Maqsood UI Hassan Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan Nouman, M. COMSATS Institute of Information Technology, Islamabad, Pakistan Panda, Sudhakar Harish-Chandra Research Institute, Allahabad, India Paracha, Muhammad Ali National Centre for Physics, Islamabad, Pakistan
Ramzan, Muhammad Ghulam Ishaq Khan Institute, Topi, Swabi, Pakistan Rashid, Muneer A. Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan Raza, Syed Shabbar National Centre for Physics, Islamabad, Pakistan Rehman, Ayesha Government College University, Lahore, Pakistan
450 Riazuddin National Centre for Physics, Islamabad, Pakistan
Sarfaraz, Muhammad Quaid-i-Azam University, Islamabad, Pakistan
Rubab, Nazish Government College University, Lahore, Pakistan
Sarwar, M. Adnan Quaid-i-Azam University, Islamabad, Pakistan
Sabeeh, Kashif Quaid-i-Azam University, Islamabad, Pakistan
Sazhenkov, S. Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan
Fauzia Saddiq National Centre for Physics, Islamabad, Pakistan Saifullah, Khalid Quaid-i-Azam University, Islamabad, Pakistan Sajid Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan Sajjad, M. Ghulam Ishaq Khan Institute, Topi, Swabi, Pakistan Sajjad, Muhammad Quaid-i-Azam University, Islamabad, Pakistan Salahuddin, Muhammad Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Sen, Ashoke Harish-Chandra Research Institute, Allahabad, India Shabbir, Ghulam Ghulam Ishaq Khan Institute, Topi, Swabi, Pakistan Shafiq, M. Quaid-i-Azam University, Islamabad, Pakistan Sharif, Muhammad Quaid-i-Azam University, Islamabad, Pakistan Sharif, Muhammad Punjab University, Lahore, Pakistan Shah, Hassan Amir Government College University, Lahore, Pakistan
Saleem, Hamid Pakistan Institute of Nuclear Science and Technology, Islamabad, Pakistan
Shaikh, Umber Punjab University, Lahore, Pakistan
Saleem, Kamran Government College University, Lahore, Pakistan
S heik h - l a bbari, M. M. IPM School of Physics, Tehran, I r a n
Salimullah, M. Jahangirnagar University, Dhaka, Bangladesh
Siddique, Asif National Centre for Physics, Islamabad, Pakistan
451 Siddiqui, Azad Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan Siddiqui, Hira Government College University, Lahore, Pakistan Tabassam, Hajira National Centre for Physics, Islamabad, Pakistan Tahir, Farida COMSATS Institute of Information Technology, Islamabad, Pakistan Tarhan, Ismail Canakkale Onsekiz Mart University, Ca na kka le, Turkey Tehseen, Naghmana Punjab University, Lahore, Pakistan Thompson, George The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Hooft, G.'t Hooft Institute of Theoretical Physics, Universi tei t Utrec ht, Utrecht, Holland Tiwana, Mazhar Quaid-i-Azam University, Islamabad, Pakistan Tsintsadze, Nodar Government College University, Lahore, Pakistan and Institute of Physics, Georgian Academy of Sciences, Tbilisi, Georgia
Unal, G. Istanbul Technical University, Ayazaga Campus, Maslak, Istanbul, Turkey Unal, Nuri Akdeniz University, Antalya, Turkey Usman, Karnran Quaid-i-Azam University, Islamabad, Pakistan Varadarajan, Madhavan Raman Research Institute, Bangalore, India Virdag, Fariha National Centre for Physics, Islamabad, Pakistan Wadia, Spenta Tata Institute of Fundamental Research, Mumbai, India Yan, Yihua National Astronomical Observatories of CAS, Changyang District, Beijing, China Yasir, Muhammad Quaid-i-Azam University, Islamabad, Pakistan Shar-e-Yazdaan Centre for Advanced Mathematics and Physics, National University of Science & Technology, Rawalpindi, Pakistan Yousafzai, Saima Quaid-i-Azam University, Islamabad, Pakistan Zaheer, Sadia Government College University, Lahore, Pakistan Zakharov, A. F. Alikhanov Institute For Theoretical and Experimental Physics, Moscow, Russia
