Geometrical Aspects of Quantum Fields: Proceedings of the First Workshop State University of Londrina, Brazil Held on 17-22 April 2000

104

leads to the real action

s

=-iid2x (^i-CT+y^^+8(i - ^* w). (3.40)

4

The twisted N A Toda Models

The twisted affine Kac-Moody algebras are constructed from a finite dimensional algebra possessing a nontrivial symmetry of their Dynkin diagrams (folding). Such symmetry can be extended to the algebra by an outer automorphism a 15 , as a(Ea) = r,aEa(a)

(4.41)

where r/a = ± 1 . For the simple roots, r\ai = 1. The signs can be consistently assign to all generators since nonsimple roots can be written as sum of two roots other roots. The no torsion theorem require a Bn- "tail like" structure which is fulfilled only by the A\^ and £>„+! (see appendix N of ref. 1 5 ). In both cases the automorphism is of order 2 (i.e. a2 = 1). Let us denote by a the roots of the untwisted algebra Q. For the A\^ case, the automorphism is denned by
a(a2) = a2n-i

• • • ,tr(an-i)

= an

(4.42)

whilst for the D\+x, the automorphism acts only in the "fish tail" of the Dynkin diagram of Dn+i, i.e. a(Eai)=Eai,

•••,a(Ean_l)=Ean_1,

a{Ean)

= Ea„+1 .

(4.43)

The automorphism a decomposes the algebra Q = Geven [J Godd- The twisted affine algebra is constructed from Q assigning an affine index m 6 Z to the generators in GeVen while m € Z + \ to those in Q0dd (see appendix N of 1 5 ). The simple root step operators for A2ll are E0i = Eg

+ E
i = 1, • • • ,n

E0O= E(}]ai_..._a2n

(4.44)

corresponding to the simple and highest roots Pi = o ( Q i + Q 2n-i+i) i = l,---,n, respectively.

V = «i H

Ha 2 „ = 2(/3i-l

/?„) (4.45)

105

For £>n+i> simple root step operators are E0,=E^,

i = l,...,n-l,

E0O = El}2l-...-an.1-an+l

E0n=E^

+

E^+l,

~ ^ - . . . - ^ - ^

(4.46)

corresponding to the simple and highest roots Pi = at i = l , - - - , n - 1,

Pn = - ( Q „ + a n + i ) ,

^ = a x + • • • a n _ i + - ( a „ + a„+i) = A + • • • /?„

(4.47)

where have denoted by ft the roots of the twisted (folded) algebra. The torsionless affine NA Toda models are defined by 2™

o\

If

Q = 2(2n-l)d + J2 —4—'

( 4 - 48 )

and Q = (2n - 2)d + £

^

j

^

(4.49)

for A 2 n and £>„+! respectively, where A; are the fundamental weights of the untwisted algebra Q, i.e. —i-£J- = 8{j. Both models are specified by the constant grade ± 1 operators e± 71-2

e

± = ]T) c±iE±0i

+ c±(n-i)-E±(/3„_i+/3„) + c±„-ET/3o

(4.50)

where A are the simple roots of the twisted affine algebra specified in (4.45) and in (4.47). According to the grading generators (4.48) and (4.49), the zero grade subalgebra is in both cases Go = SL(2) ® t / ( l ) n _ 1 ® C/(l)g ® f ( l ) j generated by {E±pn, hi, • • •, hn, c, d}. Hence the zero grade subgroup is parametrized as in (2.30) where we have taken -q — 0, responsible for breaking the conformal invariance. The factor group is given in (2.13), where QQ is generated by y - # = ( ^ - ^ ) - t f and Hi are the fundamental weights of the twisted algebra i.e. - ^ r 1 = 6ij In order to decouple the ipt, i = 1, • • •, n — 1 we chose

106

an orthonormal basis for the Cartan subalgebra, i.e. $(H) = Hupi + rjh + vc where ni = (ai + ---a2n-i+i)-H,

YH

= nn,

TrCHMj) = 2 % i, j = 1, • • -n (4.51)

and Hi = (an-i+1+---

+ an+1)-H,

Y-H

= nn,

TrCHiHj) =6tj,

i,j =

l,---n (4.52)

for Ay2ll and - D ^ respectively. The Lagrangean density is obtained from (2.16) leading to —

YI

1

< ^ - V>

(4-53)

2^^-VD»U

(^4)

* > = TTTT- + \ E

9

and C

2

SxdV' + i n ~

»-ir T^X

where n-2

V2' = E

1

M 2 e - * " + v ! i + 1 + x|cn| 2 e 2 v i + I c n . i l ^ - ^ - ^ H - V x )

(4-55)

and n-2

VD(2, = T

1

| C i | 2 e - ^ + ^ + > + - | c „ | 2 e ^ + I c ^ f e - ^ - ' a + 2^x) • (4.56)

j=i

The models described by (3.37), (4.53) and (4.54) coincide with those proposed by Fateev in 6 . 5

Zero Curvature

The equations of motion for the NA Toda models are known to be of the form 16

d(B-ldB)

+ [e-,B-1e~+B}=0,

d{dBB~l)

- [e+.BelB-1] = 0 .

(5.57)

The subsidiary constraint JY-H = Tr{B-xdBY • H) = JY-H = Tr(dBB~lY • H) — 0 can be consistenly imposed since [Y • H, e±] = 0 as can be obtained

107

from (5.57) by taking the trace with Y.H. nonlocal field R yields,

Solving those equations for the

9R = ( ^ ) ^ e * < - > , BR = ( ^ ) * ^ e * ( - > . (5.58) Y* A Y2, A The equations of motion for the fields ip,x and tpi,i = 1, • • • ,n — 1 obtained from (5.57) imposing the constraints (5.58) coincide precisely with the EulerLagrange equations derived from (4.53) and (4.54). Alternatively, (5.57) admits a zero curvature representation dA — &A + [A, A] = 0 where A = el + B^dB,

A = -B'h+B .

(5.59)

Whenever the constraints (5.58) are incorporated into A and A in (5.59), equations (5.57) yields the zero curvature representation of the NA singular Toda models. Such argument is valid for all NA Toda models, in particular for the torsionless class of models discussed in the previous two sections. Using the explicit parametrization of B given in (2.30), the corresponding e± specified in (4.50), (4.45) and (4.47) together with (5.58) where Y is given in (4.51) and (4.52), we find, in a systematic manner, the following form for A and A

AAi, = g c^l + ^1„_, + 1 ) + <*-!(£
i=l

+ %e**VPL+&l„),

(5-60)

and n-2

-AAp

= £ Cie-^^

(Eg) + E&_t+l)

+ C„_ie •*- {&an+an_1

+

C„_lXe^-^-(<

)

+ Ean

cne^E{X-

+ 1+an+2)

_1-<)+2)

+c^xe-^-HE^^ 4. 1 -

+

- E^+i+an+2)

./.2 v „-«>,-i-JJi/p(0)

~ •^aI,+a„ + i+a„ +2 ) '

(5.61)

108 n-2

= Y.CiE(X

ADi:u

+ c B - 1 ( ^ - . m _ l +^-°i._I-..+I)

i=l c

+

n(-B(a1 + -..+Qn_1+a„ + 1)

+ £ ^

^(ai + -+a„+Q n + i ) )

+ ^ e * * ( £ * l + E<1+1) ,

(5.62)

t=l

and n-2 m

D

~ 2-~i

l

<*i +C„_ie

l\ta.-i

+ £ /

««+iK-i'

+ 2cn_1^-^-1(<)+l+a»_1 +<)_l+an)

+ cn+le*HE%+...+an_i+an+l)

- E%i+...+ari+an+i))

.

(5.63)

For the untwisted afnne 5 „ model of the previous section the zero curvature representation is obtained from n-2

ABp

=^CiE^+Cn-lEiX-a,-1+CnE^2{a2

+ drPe-^E^

+ £ dwHi + %ei«E<»

+

...+an)

,

(5.64)

«=i

n-2

__A ... — ST^ r.P-i+i pio) I r „vi+v»2 zjU)

+9Yp

v

' " - 1 + 5fl

t=i

+ c„_ 1 (l + 2 ^ ) e - v ' - 1 £ ; i 0 i 1 + O n - 2c„_1e-^-1-^^(l + i M ^ L - M a . •

(5-65)

The zero curvature representation of such subclass of torsionless NA Toda models shows that they are in fact classically integrable field theories. The construction of the previous sections provides a sistematic afnne Lie algebraic structure underlying those models.

109 6

Conclusions

We have constructed a class of affine NA Toda models from the gauged twoloop WZNW models in which left and right symmetries are incorporated by a suitable choice of grading operator Q. Such framework is specified by grade ± 1 constant generators e± and the pair (Q, e±) determines the model in terms of a zero grade subgroup Go- We have shown that for non abelian Go, it is possible to reduce even further the phase space by constraining to zero the currents commuting with e± , (J £ Go)) to the fields lying in the coset Go/Go only. Moreover, we have found a Lie algebraic condition which defines a class of T-selfdual torsionless models, for the case Go = ^(1)- The action for those models were sistematicaly constructed and shown to coincide with the models proposed by Fateev 6 , describing the strong coupling limit of specific 2-d models representing sine-Gordon interacting with Toda-like models. Their weak coupling limit appears to be the Thirring model coupled to certain affine Toda theories 6 . Following the same line of arguments of the previous sections, one can construct more general models, say, Go/Go — — c/(i)' ' ^O/GQ = sW)®sW)WW"-\ gQ/go = ^ ( W ) - 2 , etc. Those models represent more general NA affine Toda models obtained by considering specific gradations Qa,b,-- = hatbt...d+ YM^tab-- a • However the important problem of the classification of all integrable models obtained as gauged two loop GWZNW models remains open. The connections constructed above provide soliton solution by using dressing transformation formalism. Specific examples of soliton behaviour will be reported in a separate publication. Acknowledgments We are grateful to FAPESP and CNPq for financial support. References 1. L.D. Faddeev and L. Takhtajan, Hamiltonian methods in the Theory of Solitons, (Springer, Berlin, 1987) 2. A. N. Leznov, M. V. Saveliev, Group Theoretical Methods for Integration of Nonlinear Dynamical Systems, (Birkhauser Verlag, Berlin, Progress in Physics, Vol. 15 1992)

110

3. H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Phys. Lett B 254, 372 (1991). 4. L.A. Ferreira, J.L. Miramontes and J.S. Guillen, Nucl. Phys.B 449, 631 (1995); C. R. Fernandez-Pousa, M. V. Gallas , T. J. Hollowood and J.L. Miramontes Nucl.Phys. B 484, 609 (1997). 5. F. Lund, Ann. of Phys. 415, 251 (1978); F. Lund and T. Regge, Phys. Rev. D 14, 1524 (1976). 6. V.A. Fateev, Nucl. Phys. B 479, 594 (1996). 7. A. Bilal, Nucl. Phys. B 422, 258 (1994). 8. J.F. Gomes, E.P. Gueuvoghlanian, G.M. Sotkov and A.H. Zimerman, IFT-Unesp preprint (in preparation). 9. J.F. Gomes, F.E.M. da Silveira, G.M. Sotkov and A.H. Zimerman, "Singular Non-Abelian Toda Theories", hepth 9810057, also in Nonassociative Algebras and its Applications, Lee. Notes in Mathematics, Ed. R. Costa, et. al., Marcel Dekker, p. 125-136 (2000); J.F. Gomes, E. P. Guevoughlanian ,F.E.M. da Silveira, G.M. Sotkov and A.H. Zimerman, Singular Conformal, and Conformal Affine Non-Abelian Toda Theories Ed. A.N. Sassakian, to appear in M.V. Saveliev Memorial Volume, Dubna (2000). 10. J.F. Gomes, G.M. Sotkov and A.H. Zimerman, SL(2, R)q Symmetries of Non-Abelian Toda theories hepth 9803122, Phys. Lett. B 435, 49 (1998). 11. J.F. Gomes, G.M. Sotkov and A.H. Zimerman, Parafermionic reductions of WZNW model, hepth 9803234, Ann. of Phys. 274, 289-362 (1999). 12. E. Witten, Phys. Rev. Lett. 38, 121 (1978). 13. J. Balog, L. Feher, L. O'Raifeartaigh, P. Forgas and A. Wipf, Ann. of Phys. 203, 76 (1990). 14. J.-L. Gervais and M. V. Saveliev, Phys. Lett. B 286, 271 (1992). 15. J.F. Cornwell, Group Theory in Physics , Vol. 3 (Academic Press 1989). 16. A. N. Leznov and M. V. Saveliev, Commun. Math. Phys. 89, 59 (1983).

111

D U F F I N - K E M M E R - P E T I A U EQUATION IN R I E M A N N I A N SPACE-TIMES J.T. LUNARDI*, B.M. PIMENTEL, AND R.G. TEIXEIRA Institute) de Fisica Tedrica - Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 - Sao Paulo, S.P., Brazil; * On leave from Departamento de Matemdtica e Estatistica. Setor de Ciencias Exatas e Naturais. Universidade Estadual de Ponta Grossa. Ponta Grossa, PR - Brazil E-mail: [email protected], [email protected], [email protected] In this work we analyze the generalization of Duffin-Kemmer-Petiau equation to the case of Riemannian space-times and show that the usual results for KleinGordon and Proca equations in Riemannian space-times can be fully recovered when one selects, respectively, the spin 0 and 1 sectors of Duffin-Kemmer-Petiau theory.

1

Introduction

The Duffin-Kemmer-Petiau (DKP) equation is a convenient relativistic wave equation to describe spin 0 and 1 bosons with the advantage over standard relativistic equations, like Klein-Gordon (KG) and Proca ones, of being of first order in derivatives. As a matter of fact, this equation was developed specifically to fulfill this characteristic, and provide an equation for bosons similar to Dirac spin 1/2 equation. The first one to propose what is now known as the 16 x 16 DKP algebra was Petiau x , a de Broglie's student, who took as starting point the former's work on first order wave equations on 16 x 16 matrices that were products of different Dirac matrices spaces. Latter it was showed that this algebra could be decomposed into 5, 10 and 1 degrees irreducible representations, the latter one being trivial 2 . Anyway, Petiau's work remained unknown to the majority of scientific community so that Kemmer, working independently, wrote Proca's equation as a set of coupled first order equations as well as the equivalent spin 0 case 3 . Although these set of equations could be written in 10 x 10 (spin 1) and 5 x 5 (spin 0) matrix forms, it was not clear which algebraic relations these matrices should obey. Based on this work, Duffin was able to put the equations sets in a first order /? matrix formulation presenting 3 of the 4 commutation relations present in the DKP algebra 4 . This result provided the motivation to Kemmer to complete formalism and present the complete theory of a relativistic wave equation for spin 0 and 1 bosons 5 . These and other facts related with the historical development of DKP theory,

112

as well as a detailed list of references on the subject, can be found in Krajcik and Nieto paper on historical development of Bhabha first order relativistic equations 6 . More recently there have been an increasing interest in DKP theory. Specifically, it has been applied to QCD (large and short distances) by Gribov 7 , to covariant Hamiltonian dinamics by Kanatchikov 8 and has been used in its generally relativistic version by Red'kov to study, using a specific representation of the 10 degrees DKP algebra, spin 1 particles in the abelian monopole field 9 . But, although DKP equation is known to be completely equivalent to KG and Proca in the free field case, doubts arise, specially with respect to KG equation, when minimal interaction with eletromagnetic field comes into play. This equivalence has been proved recently at classical level 1 0 , l i and at classical and quantum levels too by Fainberg& Pimentel 12 ; but leaves open the question about whether other processes or interactions could distinguish DKP from KG and Proca equations. So, our intention in this work is to analyze the generalization to Riemannian space-times of DKP equation and its equivalence to the Riemannian versions of KG and Proca equations. This demonstration will be performed by showing that one obtains the generalized KG and Proca equations when the spin 0 and 1 sectors of the generalized DKP equation are selected. In order to make this work more self contained we will dedicate some space to basic results on DKP theory and to the construction of its Riemannian generalization using the tetrad formalism. So, we start by presenting in section 2 the basic results on DKP equation on Minkowski space-time. We shall not enter in full details of the theory but simply quote the most important results and properties, specially about the projectors of physical spin 0 and 1 components of DKP field, necessary to the understanding of this work. For further details we suggest the reader to original works 1 ' 4 , 5 or classic textbooks 13 - 14 . In section 3 we will present some basic results on the tetrad formalism and perform the passage from Minkowskian to Riemannian space-times. In section 4 the equivalence between DKP equation and KG and Proca equations in Riemannian space-times will be demonstrated using the projectors of physical spin 0 and 1 sectors of DKP theory. Finally, we present our conclusions and comments in section 5. Throughout this work we will adopt the signature (H ) to the metric tensors as well as Einstein implicit summation rule, except otherwise stated. Moreover, Latin letters will be used when labelling indexes concerned to the Minkowski space-time or to the Minkowskian manifolds tangent to the Riemannian manifold, while Greek letters will label indexes referred to the Riemannian manifold. Both Latin and

113

Greek indexes will run from 0 to 3, except when we clearly state the opposite. 2

D K P equation in Minkowski space-time

The DKP equation is given by (iPada -m)tl> = 0,

(1)

where the matrices /3a obey the algebraic relations Pa(3b(3c + Pcl3bf3a = /3ar]bc + 0cr)ba,

(2)

being rjab the metric tensor of Minkowski space-time. This equation is very similar to Dirac's equation but the algebraic properties of /?a matrices, which have no inverses, make it more difficult to deal with. From the algebraic relation above we can obtain (no summation on repeated indexes)

(n3 = vaaPa,

(3)

so that we can define the matrices r?a = 2 ( / r ) 2 - 7 T ,

(4)

(r?a)2 = l, f , V - i j V = 0 ,

(5)

that satisfy

vapb + pbr]a

= 0 (ffl

_£ h) ^

r)aa/3a =7j a /3 a = p " y \

(6)

(7)

With these results we can write the Lagrangian density for DKP field as

£ = l-Wa VaV - m^,

(8)

where ip 1S defined as ^ = ip^r)0 .

(9)

From this Lagrangian we can obtain DKP equation through a variational principle. Moreover, we will choose /?° to be hermitian and /?* (i = 1,2,3) antihermitian so that the equation for tp can also be easily obtained by applying hermitian conjugation to equation (1). Besides that, under a Lorentz transformation x'a = h.af,xb we have r/, - •
(10)

114

U-lpaU which gives, for (ujab = -coba)13,

infinitesimal

= Aabpb,

transformations,

U = l + ±uabSab,

(11) ab

A

=

rj

ab

Sab = l0a,/3b}.

+ u>ab

(12)

If one uses two sets of Dirac matrices 7" and j ' a acting on different indexes of a 16 component ip wave function it can be verified that the matrices Pa = \ hai'

+ iia)

(13)

satisfy the algebraic relation (2), but these matrices form a reducible representation since it can be shown 5 ' 1 3 that this algebra has 3 inequivalent irreducible representations: a trivial 1 degree (/3a = 0) without physical significance; a 5 degree one, corresponding to a 5 component ip that describes a spin 0 boson; and a 10 degree one, corresponding to a 10 component %p describing a spin 1 boson. 2.1

The spin 0 sector of DKP theory

The spin 0 sector can be selected from a general representation of /3 matrices through the operators P = -(f3°)2(P)2(P2)2(P3)2,

(14)

2

which satisfies P = P, and Pa = P0a. It can be shown a b

PP

ab

13

(15)

that

= Pr] , PS

ab

= SabP = 0, PaSbc = (r)abPc - r)acPb) ,

(16)

and, as a consequence, under infinitesimal Lorentz transformations (12) we have PUxjj = Pip,

(17)

so that Pip transforms as a (pseudo)scalar. Similarly PaUrp = Paip + uabPbip,

(18)

a

showing that P xp transforms like a (pseudo)vector. Applying these operators to DKP equation (1) we have da ( P » = T-^Pip,

(19)

115

and Pfy = -db m

(Pip),

(20)

which combined provide dada (Pip) + m 2 (Pi/>) = D (PV) + m2

(PI(J)

= 0.

(21)

These results show that all elements of the column matrix Pip are scalar fields of mass m obeying KG equation while the elements of Paip are ^ times the derivative with respect to xa of the corresponding elements of Pip. Moreover, we can choose a 5 degree irreducible representation of the 0a matrices in such a way that

^ •'(!) ^t^1) •'••*=ft1 )•

m

so that equation (20) results in V'a = —daipi , m allowing us to make ip^ = \/rrnp and obtain ^ [ ^ ° n , D

(23)

l m V

V

= 0,

(24)

V vW ) making evident that the DKP equation describes a scalar particle. 2.2

The spin 1 sector of DKP theory

In order to select the spin 1 sector of DKP equation from a general representation of j3 matrices we can use the operators i?^^

1

)

2

^

2

2

)

^

3

)

2

^

0

-^

0

],

(25)

and Rab

_ Rapb

From the definitions it can be shown following properties Rab

=

( 2 g) 13

that these operators have the

_Rba^

Ra0bl3c = Rabpc = T]bcRa - r]acRb,

( 2? )

(28)

116 Rasbc

_ ^abjfc _ ^cRb^

gbcRa

=

0 ;

(29)

and Rabscd

=

rfcjtad _ vacRbd _ ^bdgae + ^dtfc

(gp)

Then, under infinitesimal Lorentz transformations (12), we have RaUip = Raip + ujabRbi> ,

(31)

and RabU

+ uacRcbip,

(32)

showing that Raxj) transforms like a (pseudo)vector while Rabijj transforms like a rank 2 (pseudo)tensor. The application of these operators to DKP equation results in db (Rabx(j) = ™Raip,

(33)

1

and

fl«ty = —ll/ 0 6 ,

(34)

m where Uab = daRbtp - dbRaxl) ,

(35)

a

is the strength tensor of the massive vector field R xj). Combined, these results provide db (--Uab) = ~Ra^j \ m J i dbUba + m2Ra%l> = 0,

(36) (37)

or equivalently (D + m 2 ) Raip = 0; daRaip = 0. a

(38)

So, all elements of the column matrix R tp are components vector fields of mass m obeying Proca equation; being the elements of Rabip equal to ~ times the field strength tensor of the vector field of which the corresponding elements of Raip are components. So, similarly to the spin 0 case, this procedure selects the spin 1 content of DKP theory, making explicitly clear that it describes a massive vectorial particle. Similarly to the spin 0 sector of DKP theory, we can find a 10 degree irreducible representation of the /3 a matrices in such a way that

117

fa

(39)

09x1

^H.^J.iW.(* 1 ),iW=(* 1 ], <«>

^=U.J ,an * = U™J , * , * = U.Jfa A

D

„/.

( fa \

n

.1.

(

fa

(41)

Denning V>„ = v ^ a ,

(42)

where Z?a is a vector field, we get from equation (35) that

t,

,

8

-=^(* V. 1 * ")=^Ur,)'

(43

»

where Kab = (daBb - 8bBa).

(44)

Consequently equation (34) will result in

R

^=~M^)'

<45)

which, together with equations (40 and (41), determines the components ^4 to tpg in ift. So equation (37) can now be written as da (Kab) + m2 (Bb) = 0, (46) making explicit that the DKP equation describes a massive vector field. 3

Passage to Riemannian space-times

Before constructing the DKP equation in Riemannian 1ZA space-times we will construct the tensor quantities on 1iA using the tensors defined on a Minkowski manifold tangent to each point of 1Z4, and for this purpose we will use the standard formalism of "tetrads". Here we will just mention the necessary fundamental results and point out the reader to reference 15 .

118

A tetrad is constituted by a set of four vector fields eMa (x) that satisfy, at each point x of 7l4, the relations T,ab=ella{x)evb(x)g'"'{x),

(47)

= e^ (x)evb {x)r]ab,

(48)

lab = e M a (a;) e"b (x) g^ {x),

(49)

g»v{x)=e»a{x)evb{x)Vab,

(50)

e"0W=rWV/(x),

(51)

g^{x) and

where

the Latin indexes being raised and lowered by the Minkowski metric r]ab and the Greek ones by the metric g^v of the manifold 1Z4. The components Bab in M4 of a tensor B^" denned on TZ4 are given by Bab = e / e / B ' " ' , Bab = e\evbB^,

(52)

B*v = e»aevbBab,

(53)

or inversely B^

= e^e^Bab,

a

and it is easy to see that A^B^ = AaB . The Lorentz covariant derivative D^ is defined as D^Ba=dflBa+uj^b(x)Bb,

(54)

such that DllBa transforms like a vector under local Lorentz transformations xa = Aab (x) xb on the M4 tangent manifold, where the connection uliab (x) on M4 satisfies the transformation rule = A ° c < « (A- 1 )'" - (5MA)° c (A-1)0".

J* a

Requiring D^ (B Aa)

a

= d^ (B Aa), D^Ba

a

since B Aa b

= dfiBa-Ljfi aBb.

(55)

is a scalar, one gets (56)

The covariant derivative VM of an object Bv defined in the Riemannian manifold is given, as usual, as VMB„=aMBv-r^aBa,

(57)

119

where r M „ a is the connection on the Tl4 manifold. Then the total covariant derivative VM of a quantity Bva, with Lorentzian and Riemannian indexes, will be given by = D^B,,0, - T^uaBaa,

(58)

V M B" Q = D»Bva + r„a"Baa.

(59)

V^JBJ,"

or

wMaf>

a

The relation between connections and T^v ment that V M e,/ a = 0, which implies that UV°6 = eaaevbTllva

can be found by require-

- evbd^eva.

ua

(60)

Moreover, the metricity condition V^g = 0 will imply that ujfl = -w^ 6 ". Besides this, it is easy to see that V^A1* = VaAa, where V a = e M a V M , and that e'aDvB* = V^B" = VM (e\Ba) = eW^B". 3.1

ab

Generalized DKP equation

The procedure to generalize DKP equation is very similar to the case of Dirac one 15 . First we will generalize the matrices 0a defined on flat Minkowski manifold M4 obeying equation (2) to matrices /?M defined on Riemannian manifold Tl4 by P = e\l3a,

(61)

that will satisfy

PFF* + $a$v^ = /?V" + 0 V .

(62)

a

as it can be shown using the properties of /3 and equation (50). Moreover, under a local infinitesimal Lorentz transformation on M4, the multicomponent field ip will transform as given by equations (10) and (12) so that its variation will be H = \uabSabiP

,

(63)

and the variation of the tetrad vectors will be <5eMa = w a 6 e / .

(64)

If we consider two nearby points x\ and x2 with the local tetrads eM° {x{) and eMQ (x2) then ip (xi) and ip (x2) are the field ip referred to these tetrads, respectively. Performing a parallel displacement from x\ to x2 on the tetrad eMa (xi) we get a new one denoted by e'M° (x2) and the field ip at x2 with

120

respect to this new tetrad will be ip'(x2). Then, the covariant differential of the field can be defined as Dip = dxaVaip = dx^V^ip = ip' (x2) - V

fri),

(65)

or Dip = iP (x2) - ib (m) - & {x2) - ip' (x2)},

(66)

where we separated the translation term ip {x2)—ip (xi) from the local Lorentz transformation term ip'(x2) — ip (x2). Explicitly we have xP (x2) - ip (xi) = dxadaip = dx^d^ip,

(67)

and ip{xi)-rP'{x2)

= -\uabSa\

(68)

so that Dip = dxadaip + ^ujabSabip.

(69)

From the expression (56) for the Lorentz covariant derivative we see that the variation of the tetrad vector eM° under Lorentz transformations on the M4 manifold is <JeM° =uvabellbdxv,

(70)

so, comparing with equation (64) we can identify wab = ujvabdx" ,

(71)

and rewrite (69) as 1 Dip = dx» ( d» + $ultabSab

) ip.

(72)

From this expression we can obtain the Lorentz covariant derivative D^ of field ip and, since ip has no Riemannian index, this derivative will be equal to the total covariant derivative VM of ip; so we have that

VMV = D^iP = fa + ^absA

ip.

(73)

Applying the hermitian conjugation to this expression and using the definition of ip we get V^

= dl3-1-utiab^Sab.

(74)

121 Now we can write the generalized expression for the ip field Lagrangian (V>/?MVM - VnWrj))

£ =

- rm/np

(75)

that can be written explicitly as l

£ = v ^ - Lp» 1

(d^+i^o65°v

- ( dMV - ^abipSab

)/3»iP)-

mW

(76)

So we have dC dip

l

+ w„„ 6 0"S o fy - u,/ 6 /?V) - ™V>

- (nr/,

'-9

(77)

and dv

dC = - 2J [3, (V=ff/5") ^ + A / ^ ' M d (3^) fir

where we used the result

j

(78)

(79)

15

(80) valid for a Riemannian manifold. Combining these results we get the generalized DKP equation of motion for Riemannian space-times i/?MVM -mip = 0. 4

(81)

The equivalence with KG and Proca equations

Now we will analyze the equivalence between DKP and KG and Proca theories in Riemannian space-times. In order to do this we will generalize the operators defined in Section 2 to select the spin 0 and 1 sectors and apply them to the generalized DKP equation. Then we can compare the obtained results with KG and Proca equations in Riemannian space-times, in a way analogous to what was done in the Minkowskian free field case.

122

4-1

The spin 0 case: Equivalence with KG

From the Pa operators denned in Section 2 we can construct the generalized projectors P M as P " = e»aPa = e M a P/?° = PP",

(82)

where P is given by equation (14) in terms of the matrices (3a. Using equations (16) and (50) it is easy to verify that P»pv = pg^t

PS»V = S^P

= 0, P"Sav

= (g'"xPv - g^vPa).

(83)

As each component of Pip was shown to be a scalar under Lorentz transformation on the Minkowski tangent manifold M4 they will also be scalars under general coordinate transformations on the Riemannian manifold H4. Similarly, each component of Paip was shown to be a vector under Lorentz transformations so that the components of P^ip will also be vectors under general coordinate transformations 15 . So, one should expect that the total covariant derivative VM of Pip and P^ip should be that of a scalar and a vector, respectively. As a matter of fact, we have that VM {Pip) = D„ (PV) = d„ {Pip) + lu>„abSabPip = aM {PiP) , 2

(84)

and VM {P"iP) = e" c V„ {PciP) = e\Dil

{Pcify

d» (Pci>) + \^abSabPciP

+ w/fcPfy

V„ {P"ip) = e\ [9M {Pcip) + uj^tPt'ip] , since SabPc = SabPpc

(85) (86)

= 0, so that the use of equation (60) results in

V„ (P"V) = % (i*V) + V P V

(87)

These results show that we can calculate the total covariant derivatives VM of Pip and P^ip by applying the derivative to each of their components as if they were, respectively, a scalar and a vector on the Riemannian manifold 1Z4 and neglect its matrix character. Similarly, Pip and Paip can also be treated as scalar and vector, respectively, when calculating the Lorentz covariant derivative £>M. Moreover, we can also see that PV M V = % (p^)

+ \^abPSabiP

= d„ {PiP) ,

(88)

123

and P"V M V = e%PcV^

PVV^

= ev

d» (PC*P) + - w M a 6 P c 5 a V

= e\ [0„ (PciP) + w M c 6 P 6 V] = VM (P"ip)

(89)

(90)

so that it becomes obvious that PV^ip = VM (Pip) and PVV ^ip = VM ( P " ^ ) . Now, applying the operators P M and P to the generalized DKP equation (81), we get P»ih = — V (P
(91)

and VM ( P ^ ) = -

(P^)

(92)

Combining equations (91) and (92) we obtain the generalized KG equation VMVM (Pip) + m 2 (Pip) = V a V a (P) + m2'(Pip) = 0.

(93)

These results make clear that when we select the spin 0 sector of the generalized DKP equation (81), describing a scalar particle on a Riemannian manifold, we get a complete equivalence with the generalized KG equation. Once more we can make this equivalence more explicit using the specific choice of the matrices /?a that satisfies condition (22). Then we get as result PiP =

°J*X ) , P"V = e^PaiP = e»

IP =

( ^C^)

4x1

Ipa

, V„W + mV = 0,

(94)

(95)

and, finally, we note that it is possible to make a change in the representation of j3 matrices in such a way that the form of DKP field ip becomes iP -»• 1PR = ( v ^ Z ^

(96)

124

4-2

The spin 1 case: Equivalence with Proca

Similarly to the spin 0 case we can also generalize the operators Ra and Rab defined in Section 2 as R" = e»cRc, R?v = eticevdRcd,

(97)

which can be seen to satisfy the relations W

= -R^;

R"P"PP = R^p" R^S"" = g^R"

(98)

= gVf'R>1 - g»pRv, - g^R",

S^R"

(99)

= 0,

(100)

+ g»P R™.

(101)

and R^Scc/3

=

g>«*Rrf

_

gl*<*Rv0

_ gVPRfa

Analogously to the case of Pip and P^ip, we would expect the total covariant derivatives V^ of R^ip and R^ip to be, respectively, those of a vector and a tensor since Raip is a vector and Rabip a tensor on the Minkowski tangent manifold M4. This turns out to be true since VM (R"il>) = e"cVM {Rc4>) =

c e\d» (R iP) + ^atS^R^

+ u^cbRbrP (102)

V„ (iTt/>) = e% [aM {Rci>) + UncbRbip] ,

(103)

VM (Ravi>) = e V d V M {Rcdi>) ,

(104)

and

VM (Ra»iP) =

cd a6 c c bd cb eacevd 3M (R iP) + i w M a 6 5 i ? V + LJ„ bR ip + cj/bR i>

,

(105) VM (Ravi>) = eace"d [0M {Rcd*l>) + ^cbRbdxjj

+ UpdbRcbil>] ,

(106)

where we used SabRcd = SabRc/3d and SabRc = 0, so that using equation (60) in equations (103) and (106) results in VM (J?"V) = e" c % (R^)

+ T^Rfy]

,

(107)

125

and VM (RauiP) = eacevd

[9M {Rcd^) + T^R^rP

+ Y^Ra^}

.

(108)

So we can neglect the matrix character of R**ip and R^ip when calculating their total covariant derivative and proceed by applying the derivative to each of their components as if they were usual vector and tensor, respectively, on the Riemannian manifold 7£4. Moreover we can also show that iTV M V = e"cRcV^

=

e\Rc % W +

i r v M v = e"c aM (i?cv) +

1

o^abSa"lP

(109)

-^abRcsay

(110)

R'V^rp = eve [dy. (Rcip) + w^iity] ,

(111)

and RavV^

= eacevdRcaV^ RO'V^

= eacevd

= eace"d dp {RCdi>) + [0„ {RcdxP) + uj/bRcb^

\iOyabRCdSab^

+ uj^bRhd^}

V

,

(112) (113)

au

so that it becomes clear that R VMV = VM (R"ip) and R V^ = VM {Ravip). M Now we can use the operators R^ and i? " on the generalized DKP equation (81), obtaining m

(114)

i

and (115)

m

where now we have a covariant strength tensor U"v = V i ? > -

VRTTI).

(116)

Combining equations (114) and (115) we get the generalized Proca equation VA ([/ A ") + m2R"i/> = 0.

(117)

This makes clear that the spin 1 sector of the generalized DKP equation (81), describing a massive vectorial particle on a Riemannian manifold, is completely equivalent to the generalized Proca equation.

126

5

Conclusions and comments

In this work we analyzed the generalization of DKP theory to Riemannian space-times. We followed the standard procedure to perform the generalization by constructing the Riemmanian quantities from the flat space-time ones through the use of tetrad formalism. We have obtained a first order relativists wave equation that describes spin 0 and 1 particles coupled to the gravitational field. Then we used the operators that select the spin 0 and 1 sectors of the theory and obtained, respectively, the generalized KG and Proca equations for spin 0 and 1 bosons in Riemannian space-times, proving the equivalence between the theories. It should be also mentioned that the form (95) for DKP field when the matrices /? satisfy condition (94) is, in principle, analogous to the result obtained when we consider the interaction with eletromagnetic field: the "gradient" components of free field case (the first four components) axe replaced by the covariant derivatives 12 . This analogy is not clearly manifested in the case of the spin 0 sector because the covariant derivatives in a curved space-time will, of course, simply reduce to the usual derivatives due to the scalar character of ip. But if we perform for the spin 1 sector of DKP field in Riemannian space-times the same construction made in equations (39) to (45) in Minkowski space-time through a specific choice of a 10 degree representation of the /3 matrices, we will find that the usual derivatives will be replaced by covariant ones in those equations, changing the forms of the components ipi to ^9 of ip field. This procedure shows, together with the result of 12 , that care must be taken in account when using specific representations of the (3 matrices to obtain the components of DKP field ip: the forms of the components obtained in the free field case may not be usable to interacting fields, requiring a new construction. The perspectives for future developments are diverse and some of them are currently under our study. We could mention the application of the DKP theory to the electromagnetic field in Riemannian space-times. Moreover, it is interesting to look for the construction of a generalization similar to the one proposed by Dirac to its electron equation 16 . Subsequently, it comes the generalizations to Riemann-Cartan space-times 15 and to teleparallel description of gravity 17>18-19 as a step to compare the question of coupling of DKP fields to torsion in both theories, similarly to what has been done to usual spin 0 and 1 formalisms 2 0 ' 2 1 . Independently, it will be analyzed the quantic processes in DKP theory with gravitation viewed as an external field, as it has already been done with DKP field in an external electromagnetic field n .

127

Acknowlegdments J.T.L. and B.M.P. would like to thank CAPES's PICDT program and CNPq, respectively, for partial support. R.G.T. thanks CAPES for full support. The authors also wish to thank Professor V. Ya. Fainberg by his critical reading of our manuscript. References 1. G. Petiau, University of Paris thesis (1936). Published in Acad. Roy. de Belg., Classe Sci., Mem in 8° 16, No. 2 (1936). 2. J. Geheniau, Acad. Roy. de Belg., Classe Sci., Mem in 8° 18 (1938), No. 1 (1938). 3. N. Kemmer, Proc. Roy. Soc. A 166, 127 (1938). 4. R. J. Duffin, Phys. Rev. 54, 1114 (1938). 5. N. Kemmer, Proc. Roy. Soc. A 173, 91 (1939). 6. R. A. Krajcik and M. M. Nieto, Am. J. Phys. 45, 818 (1977). 7. V. Gribov, Eur. Phys. J. C 10, 71 (1999). Also available as hepph/9807224. 8. I. V. Kanatchikov, hep-th/9911175. 9. V. M. Red'kov, quant-ph/9812007. 10. M. Nowakowski, Phys. Lett. A 244, 329 (1998). 11. V. Ya. Fainberg and B. M. Pimentel, hep-th/9911219. To appear in Theoretical and Mathematical Physics. 12. J. T. Lunardi, B. M.Pimentel, R. G. Teixeira and J. S. Valverde, hepth/9911254. To appear in Physics Letters A. 13. H. Umezawa, Quantum Field Theory (North-Holland, 1956). 14. A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience, 1965). 15. V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, 1985). 16. P. A. M. Dirac, Max Planck Festschrift, page 339, (Veb Deutscher Verlag der Wissenschafter, 1958). 17. K. Hayashi and T. Shirafuji, Phys. Rev. D19, 3524 (1979). 18. V. C. de Andrade and J. G. Pereira, Phys. Rev. D 56, 4689 (1998). 19. J. W. Maluf and J. F. da Rocha-Neto, gr-qc/0002059. 20. V. C. de Andrade and J. G. Pereira, Gen. Rel. Grav. 30, 263 (1998). 21. V. C. de Andrade and J. G. Pereira, Int. J. Mod. Phys. D8, 141 (1999).

128

W E A K SCALE COMPACTIFICATION A N D C O N S T R A I N T S O N N O N - N E W T O N I A N GRAVITY IN SUBMILLIMETER RANGE

V. M. M O S T E P A N E N K O Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, Urea 22290-180, Rio de Janeiro, RJ — Brazil; On leave from A.Friedmann Laboratory for Theoretical Physics, St.Petersburg, Russia. E-mail: [email protected] M. N O V E L L O Centro Brasileiro

de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 22290-180, Rio de Janeiro, RJ — Brazil E-mail: novelloOlafex.cbpf.br

150,

Urea

We discuss the recent ideas that the gravitational and gauge interactions may become united at the weak scale of about 10 3 GeV. They lead to the existence of n > 2 compact (but large comparing the Planck length) extra spatial dimensions. The consequence of this ideas is the rise of Yukawa-type corrections to the Newtonian gravity at the distances much larger than the compactification dimension, i.e., in a submillimeter range. The application of the Casimir effect for obtaining new constraints on non-Newtonian gravity is covered. The conclusion is made that the Casimir effect suggests the new important possibilities for the experimental test of the geometrical structure of space-time at small distances.

1

Introduction

The role of geometrical ideas and methods in Quantum Field Theory is universally known. The superstring approach which suggests the existence of six extra spatial dimensions with some compactification scale appears to be particularly promising as it avoids the problem of divergencies. Here we discuss the constraints on non-Newtonian gravity which is predicted by the different variants of string theory, supersymmetry and supergravity. As it is shown below, the laboratory tests of these predictions are possible in the nearest future, especially if to speak about the theories with the weak scale unification energy. It is common knowledge that the gravitational interaction holds a unique position with respect to the other fundamental interactions described by the Standard Model. Up to now, there is no renormalizable Quantum Gravity and the obstacles placed on the way to such a theory seem to be insurmountable. Gravitational physics suffers also from a poor experimental test. Even

129

the classical Newtonian gravitational law which is reliably proved at large distances is lacking experimental confirmation at the distances of order 1 mm or less. At the same time it is generally believed to be correct up to the Planck distances yjGfi/c3 ~ 10~ 35 m. Needless to say that it is a far-ranging extrapolation over 33 orders of magnitude. Corrections to Newtonian gravitational law at small distances are predicted by unified gauge theories, supersymmetry, supergravity, and string theory. They are mediated by light and massless elementary particles like scalar axion, graviphoton, dilaton, arion, and others. The exchange of these particles between atoms leads to an interatomic interaction described by Yukawa- or power-type effective potentials. Constraints for their parameters (interaction constants a, Xn, and interaction range A in case of a massive particle) is the subject for considerable study (see the monograph 1 and references therein). The gravitational experiments of Cavendish- and Eotvos-type lead to rather strong constraints over a distanse range 1 0 _ 2 m < A < 10 6 km 2 . At submillimeter range the existence of corrections to Newtonian gravity is not excluded experimentally which are in excess of it by many orders. The only constraints on non-Newtonian gravity at a submillimeter range follow from the measurements of the van der Waals and Casimir force (see, e.g., 3 ' 4 - 5 ). Until recently, they were not enough restrictive in spite of the quick progress in experimental technique and increased accuracy of force measurements. In 6 the Casimir force between the metallic surfaces of a disk and a spherical lens was measured by the use of torsion pendulum. The obtained experimental results and the extent of their agreement with theory were used in 7 to obtain stronger constraints on the corrections to Newtonian gravity in a submillimeter range. The strengthening up to 30 times comparing the previously known constraints was obtained within the range 2.2 x 10" 7 m < A < 1.6 x 10~4 m (see also 8 ) . In 9 ' 1 0 ' n the results of the Casimir force measurements between a metallized disk and a sphere attached to a cantilever of an atomic force microscope were reported and accurately confronted with a theory. They were used in 12,13 to constrain the nonNewtonian gravity. The strengthening of constraints up to 560 times comparing the former Casimir force measurements between dielectrics was obtained in the interaction range 5.9 x 10~ 9 m < A < 1.15 x 10~ 7 m. In the present report we discuss the possible violation of Newtonian gravitational law at small distances and constraints on it following from the Casimir effect. In Sec. 2 the recent impressive ideas are considered that the gravitational and gauge interactions become united at the weak scale (see, e.g., 1 4 ). These ideas in the context of string theory inevitably lead to the existence of Yukawa-type corrections to the Newtonian gravitational law at moderate dis-

130

tances in addition to the well known arguments in support of such corrections presented above. Extra spatial dimensions cause even more drastic change of gravitational law at small distances. In Sec. 3 the constraints on the parameters of Yukawa-type interactions are discussed which were obtained recently from the Casimir force measurements of papers 9'10>11 between gold and aliminum surfaces. In Sec. 4 the new constraints on the Yukawa interaction are presented following from the latest Casimir force measurement between gold surfaces by means of an atomic force microscope (these experimental results can be found in 1 5 ). In Sec. 5 (conclusions and discussion) the importance of the new Casimir force measurements for the elementary particle physics, astrophysics and cosmology is underlined. Also the prospects for further strengthening of constraints on non-Newtonian gravity in the submillimeter range are outlined. Below we use units in which h = c= 1. 2

Corrections to Newtonian Gravity in the Theories with a Weak Unification Scale

As was told in the introduction, the corrections to Newtonian gravitational law are predicted by the unified gauge theories, supersymmetry, supergravity, and string theory. The effective potential of gravitational interaction between two atoms with account of such corrections can be represented in the form V(ri3) = - ^ ^ ( l

+ a o

e-A)

>

(1)

where M\^, are the masses of the atoms, r\i is the distance between them, G is Newtonian gravitational constant, aa is a dimensionless interaction constant, A is the interaction range. In the case that the Yukawa-type interaction is mediated by a light particle of mass m the interaction range is described by the Compton wave length of this particle, so that A = 1/m. According to recent ideas the gravitational and gauge interactions may become united at the weak scale F ~ 1 TeV= 103 GeV, and the weakness of gravity at macroscopic distances is explained by the existence of n > 2 compact (but rather large) extra spatial dimensions 14 . 16 . 17 ' 18 . The consequence of these ideas is that the gravitational interaction is described by Eq. (1) for the distances much larger than the size of characteristic compactification dimension 19>20>21. One can arrive to potential (1) as follows. Let n extra dimensions be compactified by making a periodic identification with a period R. If one mass M\ is placed at the origin and the test mass M 2 is at the distance ri2 C R from Mi the force law of N = (4 + n)-

131

dimensional space-time is FN(r12)

= - G

^ ± ,

N

(2)

which provides the continuity of force lines in (N — l)-dimensional space. If ri2 3> R one obtains the usual Newtonian force MXM2 F(r12) = -G—2—, (3) r

12

where the Gauss law can be used to find the connection between the Ndimensional and the usual Newtonian gravitational constants 19

^L

G=^^

(4)

If there is no extra dimensions (n = 0, N = 4) one finds from (4) G — GN as it should be. Taking into account the connection between the gravitational constants G, GN and the respective Planckean energy scales M

PI

— n"1 "

1V1

PI,N

—

„

N-3 • GNTT~2~

\°)

Eq. (4) turns out to be equivalent to Mh = M»:*RN-*.

(6)

This result gives the possibility to estimate the allowed values of the compactification dimension R and the required number of extra dimensions. Putting JV-dimensional Planckean scale be equal to the weak scale, i.e. MpitN — F, one obtains from (6) 14 1 fMpA7^

R=-[—^) F \ F J

30 1

30

17

~10«^4- ~10^-17cm F

(7)

(we remind that MPI « 2.4 x 10 18 GeV). Evidently, one extra dimension is impermissable because for N = 5 it follows R ~ 10 13 cm which is in contradiction with the confirmed validity of Newtonian gravitational law at the scales of solar system. But already n = 2 (TV = 6) leads to R ~ 10~ 2 cm which is permitted by the results of gravitational measurements. The above considerations show that the gravitational law may vary from (2) at small, submillimeter distances to the usual form (3) at relatively large distances. The corrections to Eqs. (2), (3) can be found by considering the Newtonian limit of iV-dimensional Einstein gravity 19>20>21. At the distances ri2 <S R the corrections to Eq. (2) are of power type. The corrections at

132

the distances r i 2 ^> R hold the greatest interest because they can be tested experimentally. It is significant that they are of Yukawa-type, so that remote from the compactification scale the gravitational potential is given by Eq. (1). Coefficient aa of Eq. (1) depends on compactification geometry and on the number of extra dimensions. By way of example, if n extra dimensions have the topology of n-torus CCQ — 2n, and if they have the topology of n-sphere o>> = n + l 2 1 . The above models with n = 6 can be formulated within type I or type IIB string theories 16 ' 18 (in the case of M theory n = 7). In so doing, gravitons are described by closed strings and propagate in TV-dimensional bulk. The particles of the Standard Model are described by open strings living on (3+1)dimensional wall. This wall should have a thickness of order F _ 1 ~ 10~ 17 cm in the extra dimensions. Gravity becomes unified with the gauge interactions of Standard Model at the weak scale F. The usual Newtonian gravitational constant G loses its status of a fundamental constant. It is a multidimensional gravitational constant GN which acquires a meaning of the fundamental one. Note that the separated character of gravitons which may propagate freely in extra dimensions, while all ordinary particles cannot do so, is in some analogy with the field theory of gravity 22 where the gravity to gravity interaction is quite distinct from the interaction of matter to gravity. The possibility of serious variations of the gravitational law in submillimeter range makes much-needed the performance of new experiments. As was stressed, e.g., in 3 ' 5 , the Casimir effect may well become a new method for experimental verification of fundamental physical theories. In 23 the measurements of the Casimir force between plane plates were considered in order to restrict the extra dimensions and string-inspired forces. In the next two sections the recent experiments on measuring the Casimir force between a disk and a spherical lens (sphere) are used for the same purpose and new more strong constraints are obtained.

3

What Constraints are Known up to Date?

It has been known that the Casimir and van der Waals force measurements between dielectrics (see, e.g, 24 ) lead to the strongest constraints on the constants of Yukawa-type interaction given by the second term of Eq. (1) with a range of action 10~ 9 m < A < 1 0 - 4 m 3 ' 4 . In 6 the Casimir force between two metallized surfaces of a flat disk and a spherical lens was measured with the use of torsion pendulum. The outer metallic layer of gold, covering the test bodies, had the thickness of 0.5 /zm. The absolute error of the force measure-

133

ments in 6 was A F = 1 0 ~ n N for the distances a between the disk and the lens in the range 1 (im < a < 6 /im. In the limits of this error the theoretical expression for the Casimir force

^(0)(a) = - ^ 4

(8)

|Fth(a)-F(°>(a)|
(9)

360 a J was confirmed (where R is lens curvature radius). No corrections to Eq. (8) due to surface roughness, finite conductivity of the boundary metal or nonzero temperature was recorded. These corrections, however, may not lie in the limits of the absolute error A F . By way of example, at a w 1 /im the roughness correction may be around 12% of F^ or even larger 7 , and the finite conductivity correction for the gold surfaces at 1 /im separation is 10% of F^ 25 . (Remind that A F is around 3% of F<°) at a = 1/xm.) As to the temperature correction, it achieves 174% of F^ 0 ' at the separation a = 6/tm, where, however, A F is around 700% of F ' 0 ' . By this reason the constraints for Yukawa-type interaction following from the experiment 6 were found from the inequality 7

where F t /, is the theoretical force value, including F^°\ all the corrections to it mentioned above, and also the hypothetical Yukawa-type interaction calculated in 7 at experimental configuration (remind that the sign of finite conductivity correction is opposite to the signs of other corrections). The obtained results are usually shown by the curves in (aa, A)-plane where the regions above the curves are prohibited and below the curves are permitted. The strengthening of constraints calculated by Eq. (9) in 7 comparing the constraints obtained earlier (see, e.g., 3 ) by the results of Casimir force measurements between dielectrics is up to a factor 30 in the interaction range 2.2 x 10~ 7 m < A < 1.6 x 10~ 4 m (a bit different result was obtained later in 8 where the corrections to the ideal Casimir force (8) were not taken into account). In 9 ' 10 the results of the Casimir force measurements between a flat disk and a sphere by means of an atomic force microscope were presented in comparison with the theory taking into account the finite conductivity and roughness corrections. Temperature corrections are not essential in the interact i o n range 0.1/im < a < 0.9/xm of 9 , 1 °. The test bodies were covered by the aluminum layer of 300 nm thickness and Au/Pd layer of the thickness 20 nm (the latter is transparent for electromagnetic oscillations of characteristic frequency). The absolute error of the force measurements in 9 ' 10 was A F = 2 x 1 0 - 1 2 N. In the limits of this error the theoretical expression for

134

the Casimir force with corrections to it due to both surface roughness and finite conductivity was confirmed. The theoretical expression for the Yukawatype interaction in experimental configuration of 9 ' 10 was obtained in 12 . The constraints on the parameters UG, A calculated in 12 from the inequality \FYu(a)\

< AF,

(10) 9

turned out to be the best ones in the interaction range 5.9 x 10~ m < A < 10~ 7 m. They are stronger up to 140 times than the previously known ones from the Casimir force measurements between dielectrics (note that here all the corrections were included into the force under measuring; by this reason the constraints on \ao\ rather than on aa were obtained). In n the improved precision measurement of the Casimir force was performed by means of an atomic force microscope. The experimental improvements which include vibration isolation, lower systematic errors, and independent measurement of surface separation gave the possibility to decrease the absolute error of force measurement by a factor 2. Also the smoother Al coating with thickness 250 nm was used and thinner external Au/Pd layer of the thickness 7.9 nm. The Yukawa-type hypothetical force in the configuration of 11 was calculated in 13 where the stronger constraints on aa, A were also obtained using the inequality (10). These constraints turned out to be up to four times stronger than the constraints 12 obtained from the previous experiment 9 10 ' within a bit wider interaction range 5.9 x 1 0 _ 9 m < A < 1.15 x 10~ 7 m. The total strengthening of constraints on the corrections to Newtonian gravitational law from the measurements of the Casimir force by means of atomic force microscope has reached 560 times within the A-interval mentioned above. 4

Constraints from the Recent Measurement of the Casimir Force Between Gold Coated Lens and Disk

Recently one more measurement of the Casimir force was performed using the atomic force microscope 15 . The test bodies (sphere and a disk) were coated by gold instead of aluminum which removes some difficulties connected with the additional thin Au/Pd layers used in the previous measurements 9 ' 1 0 > n to prevent the oxidation processes on Al surfaces. The used polystyrene sphere coated by gold layer was of diameter 2R = 191.3 /mi and a sapphire disk had a diameter 2L = 1 cm, and a thickness D — 1 mm. The thickness of the gold coating on both test bodies was A = 86.6 nm. This can be considered as an infinitely thick in relation to the Casimir force measurements. The root mean square roughness amplitude of the gold surfaces was decreased until 1 nm which makes roughness corrections negligibly small. The measurements

135

were performed at smaller separations, i.e. 62 nm < a < 350 nm. The absolute error of force measurements was, however, AF = 3.5 x 1 0 - 1 2 N, i.e., a bit larger than in the previous experiments. The reason is the thinner gold coating used in 15 which led to poor termal conductivity of the cantilever. At smaller separations of about 65 nm this error is less than 1% of the measured Casimir force. Now let us calculate the gravitational force acting in experimental configuration due to the potential (1). The Newtonian contribution is found to be negligible. Actually, due to the inequality fl«L each atom of the sphere can be considered as if it would be placed above the center of the disk. Then the vertical component of the Newtonian gravitational force acting between the sphere atom of a mass Mi situated at a height I <S L and the disk is L

f„,(l)=I

GMxp2-K

l+D

rdr

J

-2irGMlPD\l-^^

.

J V^Tz

I (11) where p is the disk density, and only the first order terms in D/L and l/L are retained. The Newtonian gravitational force acting between the disk and the sphere is obtained from (11) by integration over the sphere volume 0

FN,Z « -^Gpp'DR3

(l - g - - | ) ,

(12)

where p' is the density of the sphere material. Even with sphere and disk made of the vacuo-distilled gold as a whole with p = p' = 18.88 x 103 kg/m 3 one arrives from (12) to the negligibly small value of FN
[ Pl - ( P l - p)e~^x]

[Pl - (Pl - p')e~^x

.

(13) According to 15 the theoretical value of the Casimir force was confirmed within the limits of AF = 3.5 x 10~ 12 N and no hypothetical force was observed. In such a situation, the constraints on aG can be obtained from the

136

inequality (10). The strongest constraints follow for the smallest possible values of a « 65 nm 27 . The Casimir force measurement between the gold surfaces by means of an atomic force microscope gives the possibility to strengthen the previuosly known constraints up to 19 times within a range 4.3 x 10~ 9 m < A < 1.5 x 10 _ 7 m. The largest strengthening takes place for A =(5-10) nm. Comparing the constraints obtained from the Casimir and van der Waals force measurements between dielectrics the strengthening up to 4500 was achieved by the Casimir force measurement 15 between gold surfaces using the atomic force microscope. 5

Conclusions and Discussion

We have discussed above the new geometrical ideas on the possible existence of relatively large extra spatial dimensions. The consequence of these ideas is the deviation from Newtonian gravitational law in submillimeter range. It was indicated that there are numerous evidences that the gravitational interactions at small distances undergo deviations from the Newtonian law. These deviations can be described by the Yukawa-type potential. They were predicted in the theoretical schemes with the quantum gravity scale both of order 1018 GeV and 103 GeV. In the latter case the problem of experimental search for such deviations takes on great significance. The existence of large extra dimensions can radically alter many concepts of space-time, elementary particle physics, astrophysics and cosmology. To cite an example, the highest temperature at which the Universe was born turns out to be ~ 103 GeV instead of 1018 GeV. Although this does not touch the era of big-bang nucleosynthesis which begins at lower temperatures of about 1 MeV the theory of the very early Universe including inflation may be significantly changed 19 . The Casimir and van der Waals force measurements is the main source of constraints on the Yukawa-type interactions at small distances. In the present paper we have reviewed the latest advances deduced in such a manner in the submillimeter interaction range. The new constraints were obtained also following from the recent Casimir force measurements between gold surfaces by means of an atomic force microscope 15 . They are found to be up to 19 times stronger than the previously reported 13 and up to 4500 stronger than the well-known constraints obtained from the Casimir and van der Waals force measurements between dielectrics. Although the interaction range where the new constraints are valid was extended, there is yet a gap up to the results obtained from the Casimir force measurement by the use of torsion pendulum 6 .

137

As it is clear from the above, much work should be done in order to test experimentally predictions of the models with weak-scale compactification characterized by the value ao ~ 10 (see Sec. 2). As was shown in 7 the constraints following from the experiment 6 can be improved up to four orders of magnitude in the range around A = 10~ 4 m. This is just right to attain the desirable values of aa. As to the experiments using the atomic force microscope Mo.n.is^ where the larger strengthening of the previously known constraints is already achieved (up to 4500 times), there remains almost fifteen orders more needed to achieve the value aa ~ 10 in the interaction range A < 10~ 7 m. Thus, it is desirable here not only to increase the strength of constraints but also to move the interaction range under examination to larger A (e.g., by the increase of a sphere radius and the space separation to the disk). In conclusion it may be said that the laboratory experiments on the measurement of the Casimir force offer new possibilities for the test of the geometrical structure of space-time at small distances. Acknowledgments The authors are grateful to G. L. Klimchitskaya, D. E. Krause, and U. Mohideen for helpful discussions. V.M.M. is grateful to the Centro Brasileiro de Pesquisas Fisicas where the work was performed for kind hospitality. He acknowledges the financial support from FAPERJ. M.N. was partly supported by CNPq. References 1. E. Fischbach and C. L. Talmadge, The Search for Non-Newtonian Gravity (Springer-Verlag, New York, 1998). 2. G. L. Smith, C. D. Hoyle, J. H. Gundlach, E. G. Adelberger, B. R. Heckel, and H. E. Swanson, Phys. Rev. D 6 1 , 022001 (1999). 3. V. M. Mostepanenko and I. Yu. Sokolov, Phys. Rev. D 47, 2882 (1993). 4. M. Bordag, V. M. Mostepanenko, and I. Yu. Sokolov, Phys. Lett. A 187, 35 (1994). 5. V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and Its Applications (Clarendon, Oxford, 1997). 6. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); 81, 5475(E) (1998). 7. M. Bordag, B. Geyer, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. D 58, 075003 (1998). 8. J. C. Long, H. W. Chan, and J. C. Price, Nucl. Phys. B 539, 23 (1999). 9. U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998).

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10. G. L. Klimchitskaya, A. Roy, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. A 60, 3487 (1999). 11. A. Roy, C.-Y. Lin, and U. Mohideen, Phys. Rev. D 60, 111101(R) (1999). 12. M. Bordag, B. Geyer, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. D 60, 055004 (1999). 13. M. Bordag, B. Geyer, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. D 62, 011701(R) (2000). 14. N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998). 15. B. W. Harris, F. Chen, and U. Mohideen, quant-ph/0005088; Phys. Rev. A, 2000, to appear. 16. G. Shiu and S.-H. Henry Tye, Phys. Rev. D 58, 106007 (1998). 17. I. Antoniadis, S. Dimopoulos, and G. Dvali, Nucl. Phys. B 516, 70 (1998). 18. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 436, 257 (1998). 19. N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Rev. D 59, 086004 (1999). 20. E. G. Floratos and G. K. Leontaris, Phys. Lett. B 465, 95 (1999). 21. A. Kehagias and S. Sfetsos, Phys. Lett. B 472, 39 (2000). 22. M. Novello, V. A. De Lorenci, and L. R. de Freitas, Ann. Phys. (N.Y.) 254, 83 (1997). 23. D. E. Krause and E. Fischbach, hep-ph/9912276. To appear in Testing General Relativity in Space: Gyroscopes, Clocks, and Interferometers, edited by C. Lammerzahl, C.W.F. Everitt, F.W. Hehl (Springer-Verlag, 2000). 24. B. V. Derjaguin, 1.1. Abrikosova, and E. M. Lifshitz, Quart. Rev. Chem. Soc. 10, 295 (1956). 25. G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. A 61, 062107 (2000). 26. V. P. Mitrofanov and O. I. Ponomareva, Sov. Phys. JETP (USA) 67, 1963 (1988). 27. V. M. Mostepanenko and M. Novello, hep-ph/0008035.

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FINITE A C T I O N , H O L O G R A P H I C C O N F O R M A L A N O M A L Y A N D Q U A N T U M B R A N E - W O R L D S IN D5 G A U G E D SUPERGRAVITY S. NOJIRI Department of Mathematics and Physics, National Defence Academy, Yokosuka 239, JAPAN E-mail: [email protected]

Hashirimizu

O. OBREGON Instituto de Fisica de la Universidad de Guanajuato, Apdo.Postal E-l\3, Leon, Gto., MEXICO E-mail: [email protected]

37150

S. D. ODINTSOV Tomsk State Pedagogical University, 634041 Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato, Apdo.Postal E-143, 37150 Leon, Gto., MEXICO E-mail: [email protected] S. OGUSHI Department of Physics, Ochanomizu University, Otsuka, Bunkyou-ku JAPAN JSPS Research Fellow E-mail: [email protected]

Tokyo 112,

We report our recent results concerning d5 gauged supergravity (dilatonic gravity) considered on AdS background. The finite action on such background as well as d4 holographic conformal anomaly (via AdS/CFT correspondence) are found. In such formalism the bulk potential is kept to be arbitrary, dilaton dependent function. Holographic R.G in such theory is briefly discussed. d5 AdS brane-world Universe induced by quantum effects of brane CFT is constructed. Such brane is spherical, hyperbolic or flat one. Hence, the possibility of quantum creation of inflationary brane-world Universe is shown.

140

1

Introduction

AdS/CFT correspondence 1 may be realized in a sufficiently simple form as d5 gauged supergravity/boundary gauge theory correspondence. The reason is very simple: different versions of five-dimensional gauged SG (for example, N = 8 gauged SG 2 which contains 42 scalars and non-trivial scalar potential) could be obtained as compactification (reduction) of ten-dimensional IIB SG. Then, in practice it is enough to consider 5d gauged SG classical solutions (say, AdS-like backgrounds) in AdS/CFT set-up instead of the investigation of much more involved, non-linear equations of IIB SG. Moreover, such solutions describe RG flows in boundary gauge theory (for a very recent discussion of such flows see 3>4>44.5>6>7>8 a n d refs. therein). To simplify the situation in extended SG one can consider the symmetric (special) RG flows where scalars lie in one-dimensional submanifold of total space. Then, such theory is effectively described as d5 dilatonic gravity with non-trivial dilatonic potential. Nevertheless, it is still extremely difficult to make the explicit identification of deformed SG solution with the dual (non-conformal exactly) gauge theory. As a rule 4,T , only indirect arguments may be suggested in such identification". From another side, the fundamental holographic principle 9 in AdS/CFT form enriches the classical gravity itself (and here also classical gauged SG). Indeed, instead of the standard subtraction of reference background 1 0 ' n in making the gravitational action finite and the quasilocal stress tensor welldefined one introduces more elegant, local surface counterterm prescription 12 . Within it one adds the coordinate invariant functional of the intrinsic boundary geometry to gravitational action. Clearly, that does not modify the equations of motion. Moreover, this procedure has nice interpretation in terms of dual QFT as standard regularization. The specific choice of surface counterterm cancels the divergences of bulk gravitational action. As a byproduct, it also defines the conformal anomaly of boundary QFT. Local surface counterterm prescription has been successfully applied to construction of finite action and quasilocal stress tensor on asymptotically "Such dual theory in massless case is, of course, classically conformally invariant and it has well-defined conformal anomaly. However, among the interacting theories only A/ = 4 SYM is known to be exactly conformally invariant. Its conformal anomaly is not renormalized. For other, d4 QFTs there is breaking of conformal invariance due to radiative corrections which give contribution also to conformal anomaly. Hence, one can call such theories as non-conformal ones or not exactly conformally invariant. The conformal anomalies for such theories are explicitly unknown. Only for few simple theories (like scalar QED or gauge theory without fermions) the calculation of radiative corrections to conformal anomaly has been done up to two or three loops. It is a challenge to find exact conformal anomaly. Presumbly, only SG description may help to resolve this problem.

141

AdS space in Einstein gravity 1216 and in higher derivative gravity 17 . Moreover, the generalization to asymptotically flat spaces is possible as it was first mentioned in ref.18. Surface counterterm has been found for domain-wall black holes in gauged SG in diverse dimensions 19 . However, actually only the case of asymptotically constant dilaton has been investigated there. In the current report we present our recent results on the construction of finite action, consistent gravitational stress tensor and dilaton-dependent Weyl anomaly for boundary QFT (from bulk side) in five-dimensional gauged supergravity with single scalar (dilaton) on asymptotically AdS background. Note that dilaton is not constant and the potential is chosen to be arbitrary. The implications of results for the study of RG flows in boundary QFT are presented, in particular, the candidate c-function is suggested. The comparison with holografic RG is done as well. As an extension, the brane-world solutions in dilatonic gravity are discussed (with quantum corrections).Indeed, after the discovery that gravity on the brane may be localized 35 there was renewed interest in the studies of higher-dimensional (brane-world) theories. In particular, numerous works 36 (and refs. therein) have been devoted to the investigation of cosmology (inflation) of brane-worlds. In refs. 38,37,42 it has been suggested the inflationary brane-world scenario realized due to quantum effects of brane matter. Such scenario is based on large N quantum CFT living on the brane 38 - 37 . Actually, that corresponds to implementing of RS compactification within the context of renormalization group flow in AdS/CFT set-up. Note that working within large N approximation justifies such approach to brane-world quantum cosmology as then quantum matter loops contribution is essential. In the last section we report on the role of quantum matter living on the brane in the study of brane-world cosmology in 5d AdS dilatonic gravity with non-trivial dilatonic potential (bosonic sector of the corresponding gauged supergravity). We are mainly interested in the situation when the boundary of 5d AdS space represents a 4d constant curvature space whose creation (as is shown) is possible only due to quantum effects of brane matter. Thus, the possibility of dilatonic brane-world inflation induced by quantum effects is proved. In different versions of such scenario discussed here the dynamical determination of dilaton occurs as well. This finishes the discussion of our results in the study of AdS/CFT aspects of d5 gauged supergravity (bosonic sector).

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2

Holografic Weyl anomaly for gauged supergravity with general dilaton potential

In the present section the derivation of dilaton-dependent Weyl anomaly from gauged SG will be given. This is based on 23>48. We start from the bulk action of d + 1-dimensional dilatonic gravity with the potential $

~

!6TTG

[

dd+1xJ-d \R + IW(W) 2 + F()A4>+ $(
JMd+1

(1) Here M
d I2 . . 2 = —p~ dpdp + V , <7ijda;Ma;J , 4

§ij = p~lgij

•

(2)

»=i

Here / is related with A2 by 4A2 = d(d — I)/I2. If gij = %•, the boundary of AdS lies at p = 0. We follow to method of calculation of conformal anomaly as it was done in refs. 21 ' 22 where dilatonic gravity with constant dilaton potential has been considered. The action (1) diverges in general since it contains the infinite volume integration on Ma+i • The action is regularized by introducing the infrared cutoff e and replacing / dd+1x -»• / d d x Je dp, JMd ddx(- • • ) - > / ddx (• • • j

We 2

also expand g^ and <j> with respect to p: g^ = g(o)ij + P9(i)%j + P 9(2)ij + ' • •> (j) = (0) + p4>(i) + p2<j>(2) + • • •• Then the action is also expanded as a power series on e. The subtraction of the terms proportional to the inverse power of e does not break the invariance under the scale transformation Sg^u = 26agfil/ and 5e = 2<5
143

term under the scale transformation is finite when e —> 0 and should be canceled by the variation of the finite term (which does not depend on e) in the action since the original action (1) is invariant under the scale transformation. Therefore the In e term S\n gives the Weyl anomaly T of the action renormalized by the subtraction of the terms which diverge when e —• 0 (d = 4)

Sin = -±Jdix^T.

(3)

The conformal anomaly can be also obtained from the surface counterterms, which is discussed in Section 3. For d = 4, by solving g{1)ij, g{2)ij, 0 ( 1 ) and 0(2) with respect to g ^ , (/>(o) and by using the equations of motion, we obtain the following expression for the anomaly: T = --?—

87rG

{hxR2 + h2RijRij

+ h3Ri]drfdjtp

+ h4Rgijdi
+h5-jLdi(V=99ijdj) + W ' f t W ) 2 +h7 ( - L c ^ v ^ S ^ ) ) +hsgkldk(j>dl) (4) '-9

/

V-9

Here /ii = [3 {(24 - 10$)$' 6 + (62208 + 22464$ + 2196$ 2 + 7 2 $ 3 + $ 4 ) $ " ( $ " + 8 V)2 + 2$' 4 {(108 + 162$ + 7 $ 2 ) $ " +72 ( - 8 + 14$ + $ 2 ) V } - 2$' 2 {(6912 + 2736$ + 192$ 2 + $ 3 ) $ " 2 +4(11232 + 6156$ + 552$ 2 + 13$ 3 )$"V + 32 ( - 2592 + 468$ + 96$ 2 +5$3)V2}

- 3 ( - 2 4 + $)(6 + $ ) 2 $ ' 3 ( $ ' " + 8V')\]

I

[l6(6 + $ ) 2 { - 2 $ ' 2 + (24 + $ ) $ " } {-2$' 2 + (18 + $ ) ( $ " + %V)f 3 {(12 - 5 $ ) $ ' 2 + (288 + 72 $ + $ 2 ) $ " }

^ =~

, L„2 +. (24 ,„, +. ^ 8„„ ( 6 +. $^2 ) 2 {-2$' $ ) $„" }

'

(5)

and V{4>) = X(<j)) - Y'(4>). The explicit forms of h3, • • • h$ are given in 48 . This expression which should describe dual d4 QFT of QCD type, with broken SUSY looks really complicated. The interesting remark is that Weyl anomaly is not integrable in general. In other words, it is impossible to construct the anomaly induced action. This is not strange, as it is usual situation for conformal anomaly when radiative corrections are taken into account.

144

In case of the dilaton gravity in 21 corresponding to $ = 0 (or more generally in case that the axion is included 24 as in 2 2 ) , we have the following expression: /3 T=

8^G

r

J

j

M

dix

V-9(o)

[g-R(0)ij#(o) - ^4^(0)

2

+4 |

r —

d

i ( ^ - 5 ( 0 ) ^ 0 ) ^ ^ ( 0 ) ) \ + 3 (5(o)^V(o)5^(o))

.(6)

Here ip can be regarded as dilaton. In the limit of $ —> 0, if one chooses V = — 2 and makes AdS/CFT identification of SG parameters one finds that the standard result (conformal anomaly of Af = 4 super YM theory covariantly coupled with Af = 4 conformal supergravity 2 5 ) in (6) is reproduced 21>41. We should also note that the expression (4) cannot be rewritten as a sum of the Gauss-Bonnet invariant G and the square of the Weyl tensor F, which are given as G = R2 - 4RijRij + RijkiRijkl, F = \R2 - 2RijRii + RijklR^kl. This is the signal that the conformal symmetry is broken already in classical theory. When

) = ae 6 *, (|a| C 1), we find _ ai

1

1 a2

~ ~a2 ~ T&G ' 8 ' 36 '

1 a3

a2

(

5

b2

\

~ ~&G ' 24 ' I," 162 + 576yJ '

(?)

Here V should be arbitrary but constant. We should note $(0) 7^ 0. One can absorb the difference into the redefinition of / since we need not to assume $(0) = 0 in deriving the form of hi and hi- Hence, this simple example suggests the way of comparison between SG side and QFT descriptions of non-conformal boundary theory. Let us discuss the properties of conformal anomaly. In order that the region near the boundary at p = 0 is asymptotically AdS, we need to require $ - • 0 and $ ' ->• 0 when p -> 0. One can also confirm that hi -> ^ and

145

hi -¥ - § in the limit of $ ->• 0 and $ ' -> 0 even if $ " ^ 0 and $'" ^ 0. In the AdS/CFT correspondence, hi and h2 are related with the central charge c of the conformal field theory (or its analog for non-conformal theory). Since we have two functions hi and h2, there are two ways to define the candidate c-function when the conformal field theory is deformed: 24-rrhi ci = — £ — .

c

8irh2 2 =

Q-

•

... (8)

If we put V() - 4A2 + $(0), then / = ( v^y) 2 • One should note that it is chosen / = 1 in (8). We can restore I by changing h —> l3h and k —» Z3A; and $' -» /*', $ " -> Z 2 $" and *'" -> / 3 $ ' " in (4). Then in the limit of $ -» 0, 3

one gets ci, c2 —> ^ ( y£pr J , which agrees with the proposal of the previous work 28 in the limit. The c-function ci or c2 in (8) is, of course, more general definition. It is interesting to study the behaviour of candidate c-function for explicit values of dilatonic potential at different limits. It also could be interesting to see what is the analogue of our dilaton-dependent c-function in non-commutative YM theory (without dilaton, see 2 9 ) . The definitions of the c-functions in (8), are, however, not always good ones since our results are too wide. They quickly become non-monotonic and even singular in explicit examples. They presumbly measure the deviations from SG description and should not be taken seriously. As pointed in 3 3 , it might be necessary to impose the condition $' = 0 on the conformal boundary. Such condition follows from the equations of motion of d5 gauged SG. Anyway as $' = 0 on the boundary in the solution which has the asymptotic AdS region, we can add any function which proportional to the power of $ ' = 0 to the previous expressions of the c-functions in (8). As a trial, if we put $ ' = 0, we obtain 3TT 288 + 72$ + $ 2 l = 2 = ' G (6 + $) 2 (24 + $) (9) instead of (8). We should note that there disappear the higher derivative terms like $ " or 4>'". That will be our final proposal for acceptable c-function in terms of dilatonic potential. The given c-functions in (9) reproduce the known result for the central charge on the boundary. Since ^ j —> 0 in the asymptotically AdS region even if the region is UV or IR, the given c-functions in (9) have fixed points in the asymptotic AdS region jfy = ^ ^ ^ j -> 0, C

2TT 62208 + 22464$ + 2196* 2 + 72$ 3 + $ 4 TP, to , ^o,2nA *\,-.n 3G (6 + $) (24 +, $)(18 +. <^3>) '>

c

where U = p~? is the radius coordinate in AdS or the energy scale of the boundary field theory.

146

We can now check the monotonity in the c-functions. For this purpose, we consider some examples in 6 and 7 , where V = — 2. In the classical solutions for the both cases, <j> is the monotonically decreasing function of the energy scale U — p~i and = 0 at the UV limit corresponding to the boundary. Then in order to know the energy scale dependences of Ci and c-i, we only need to investigate the dependences of c\ and c2 in (9). The potentials in 6 and 7 , and also $ have a minimum $ = 0 at \, we only need to check the monotonities of c\ and c2 with respect to $ when $ > 0. From (9), we find ^ i l , ^p^1 < 0. Therefore the c-functions c\ and c2 are monotonically decreasing functions of 4> or increasings function of the energy scale U as the c-function in 4 ' 7 . We should also note that the c-functions c\ and c2 are positive definite for non-negative $ . In 2 8 , another c-function has been proposed in terms of the metric as follows:

CGPPZ =

(S) 3 '

(10)

where the metric is given by ds2 = dz2 + e2Adxtldx^L. The c-function (10) is positive and has a fixed point in the asymptotically AdS region again and the c-function is also monotonically increasing function of the energy scale. The c-functions (9) proposed in 23,48 are given in terms of the dilaton potential, not in terms of metric, but it might be interesting that the c-functions in (9) have the similar properties (positivity, monotonity and fixed point in the asymptotically AdS region). These properties could be understood from the equations of motion. We can also consider other examples of c-function for different choices of dilatonic potential. In 30 , several examples of the potentials in gauged supergravity are given. They appeared as a result of sphere reduction in Mtheory or string theory, down to three or five dimensions. We find, however, that the proposed c-functions have not acceptable behaviour for the potentials in 30 . The problem seems to be that the solutions in above models have not asymptotic AdS region in UV but in IR. On the same time the conformal anomaly in (4) is evaluated as UV effect. If we assume that $ in the expression of c-functions c\ and c^ vanishes at IR AdS region, $ becomes negative. When $ is negative, the properties of the c-functions c\ and c-i become bad, they are not monotonic nor positive, and furthermore they have a singularity in the region given by the solutions in 30 . Thus, for such type of potential other proposal for c-function which is not related with conformal nomaly should be

147

made. Hence, we discussed the holografic Weyl anomaly from SG side and typical behaviour of candidate c-functions. However, it is not completely clear which role should play dilaton in above expressions as holographic RG coupling constant in dual QFT. It could be induced mass, quantum fields or coupling constants (most probably, gauge coupling), but the explicit rule with what it should be identified is absent. The big number of usual RG parameters in dual QFT suggests also that there should be considered gauged SG with few scalars. 3

Surface Counterterms and Finite Action

Let us turn now to discussion of of surface counterterms which are also connected with holografic Weyl anomaly. As well-known, we need to add the surface terms to the bulk action in order to have the well-defined variational principle. Under the variation over the metric G"*" and the scalar field , the variation of the action (1) SS = SSMd+i + ^ M j is given by

+ (X{cj>) - ¥'{)) (V0) 2 + $() - Y'{4>))

d^tf}

2

+^{(x'()-Y'{<j>))dvcj> SSMd = Y^Q j ddxyf^n^

[&> (Giv6G^)

(11) - Dv ( « ? " " ) + ¥{)&> (5
Here g^ is the metric induced from GM1/ and nM is the unit vector normal to Mi. The surface term SSM* of the variation contains nM<9M and n^dfj, (8o JM ddx\[^-g \5&" {•••}+ 8(f) {•••}] after using the partial integration. If we put {• • •} = 0 for {• • •}, one could obtain the boundary condition corresponding to Neumann boundary condition. We can, of course, select Dirichlet boundary condition by choosing S&" = & = 0, which is natural for AdS/CFT correspondence. The Neumann type condition becomes, however, necessary later when we consider the black

148

hole mass etc. by using surface terms. If the variation of the action on the boundary contains n^d^ or n^dn {5<j)), however, we cannot partially integrate it on the boundary since nM expresses the direction perpendicular to the boundary. Therefore the "minimum" of the action is ambiguous. Such a problem was well studied in 10 for the Einstein gravity and the boundary term was added to the action. It cancels the term containing n^d^ We need to cancel also the term containing n^d^ (Sep). Then one finds the boundary term 21 S{b1] = - g ^ G /

ddx^g[D^

+ Y{<j>)ntid») .

(12)

We also need to add surface counterterm 5£ ' which cancels the divergence coming from the infinite volume of the bulk space, say AdS. In order to investigate the divergence, we choose the metric in the form (2). In the parametrization (2), nM and the curvature R are given by

«"={'¥,<>,..

^-n^nkln'

n', - *£&&'.'.

-

£.&&

(13) Here R is the scalar curvature defined by g^ in (2). Expanding gij and <j> with respect to p, we find the following expression for 5 + 5,( i ) .

S+

^

I, =

iSGft/^'^ +P\ - JZ^R(°)

~2d

•9(0)

- J2«(o)0(iW - JZ^

! * ^

( X ^ ( ° ) ) (V(o)0
(14)

\ + [m

(15)

+r(0(o))A^ O ) + *'(0 ( o))0(i) )>+0 Then for d = 2 :(2)

16TTG J

ddx^-g

and for d = 3,4,

!p)--L-[a'x\y/ZEl™^2

+ d-2

+

JL-R- d{d21- 2);*(0) (16)

149

Note that the last term in above expression does not look typical from the AdS/CFT point of view. The reason is that it does not depend from only the boundary values of the fields. Its presence may indicate to breaking of AdS/CFT conjecture in the situations when SUGRA scalars significally deviate from constants or are not asymptotic constants. Thus we got the boundary counterterm action for gauged SG. Using these local surface counterterms as part of complete action one can show explicitly that bosonic sector of gauged SG in dimensions under discussion gives finite action in asymptotically AdS space. The corresponding example will be given in below. Let us turn now to the discussion of deep connection between surface counterterms and holographic conformal anomaly. It is enough to mention only d = 4. In order to control the logarithmically divergent terms in the bulk action S, we choose d — 4 = e < 0. Then S + Sb = 7Sin + finite terms. Here S\n is given in (3). We also find g^j&j-Sin = - f £ l n + O (e 2 ). Here £ l n is the Lagrangian density corresponding to 5i n : S\n = Jdd+1C\n. obtain the following expression of the trace anomaly:

r=lim

Then we

JfL^+5^^ 1

«-"- y/=m

2

6g?0)

which is identical with the result found in (3). We should note that the last term in (16) does not lead to any ambiguity in the calculation of conformal anomaly since <7(o) does not depend on p. If we use the equations of motion, we finally obtain the expression (4). Hence, we found the finite gravitational action (for asymptotically AdS spaces) in 5 dimensions by adding the local surface counterterm. This action correctly reproduces holographic trace anomaly for dual (gauge) theory. In principle, one can also generalize all results for higher dimensions, say, d6, etc. With the growth of dimension, the technical problems become more and more complicated as the number of structures in boundary term is increasing. Let us consider the black hole or "throat" type solution for the equations of the motion when d = 4. The surface term (16) may be used for calculation of the finite black hole mass and/or other thermodynamical quantities. For simplicity, we choose X(<j>) = a (constant), Y(
ds2 = -e2"dt2

+ e2°dr2 + r2 £

(dx1)2

(18)

150

and p, a and (p depend only on r. We now define new variables U and VbyU — &P+", V = r V - 0 ' . When $(0) = $'(0) = <j> = 0, a solution corresponding 4

to the throat limit of D3-brane is given by U = 1, V = Vo = JT — fJ- In the following, we use large r expansion and consider the perturbation around this solution. Then we obtain, when r is large or c is small, one gets U = 1 + c2u ,

u = u0 + ^-r~213 6

,

" = w . . »=».- 6( /_V_ 2) r ' M+ ^ <19> Here uo and VQ are constants of the integration. Here we choose VQ = UQ = 0. The horizon which is defined by V = 0 lies at r = r

^

/ 2

4 + C

^

24(^-4)/-2)

<M>

'

Hawking temperature is l_dV_ 4w r2 dr

1 l\,_3 i < 4/

4

4*\ '

2/i(^-6)(2/3-3),i_0

2 /J,4 4 - C L-1—

M +C

'-^S.

i/2

6(/?-4)(/?-2)'

i_el

P//4

M

2 ^

J'

(21) We now evaluate the free energy of the black hole within the standard prescription 31>32. The free energy F can be obtained by substituting the classical solution into the action S: F = TS. Here T is the Hawking temperature. Since we have 0 = | ($() + )f) + R + a (Vcj>)2 by using the equations of motion, we find the following expression of the action (1) after Wick-rotating it to the Euclid signature 16TTG

\j

d5VG ($()+

1 2V(3, r°° , ,„/,,., 12 JC°drr3uU(cl>) + ~) 167TG ' 3 T Jrh

.

(22)

Here V(3) is the volume of the 3d space ( / d5x • • • = /?V(3) / drr 3 • • •) and /? is the period of time, which can be regarded as the inverse of the temperature T (^). The expression (22) contains the divergence. We regularize the divergence by replacing f°° dr —> J r m a x dr and subtract the contribution from a zero temperature solution, where we choose \i = c = 0, and the solution corresponds to the vacuum or pure AdS:

s°=^HW''^::rV>^

<»

151

The factor /G"{r-r^x^-c-o)

ig c n o s e n s o

that the proper length of the circles

which correspond to the period ^ in the Euclid time at r m a x coincides with each other in the two solutions. Then we find the following expression for the free energy F = l i m ^ ^ o o T (S - S0), F =

V((3)

'V

2TTG12T2

2 i1-- a2 - / (0-1) A» /M 12/9(/9 - 4)(/9 - 2)

(24)

+

Here we assume /9 > 2 or the expression S - So still contains the divergences and we cannot get finite results. However, the inequality /9 > 2 is not always satisfied in the gauged supergravity models. In that case the expression in (24) would not be valid. One can express the free energy F in (24) in terms of the temperature T instead of /J,: V,( 3 )

-TTT4/6

16TTG

+

c2l*-i0T4-Wji

2/33 - 15/92 + 2 2 0 - 4 60O9-4)(/9-2)

+

(25) Then the entropy S = - 4£ and the energy (mass) E = F + TS is given by Vi(3) 16-KG

E =

V(t 3 ) IGTTG

4TTT316 +

M-WTt-Wji

20 3 - 15/92 + 2 2 / 9 - 4 3/9(/9 - 4)

+ ••

STrT^ + c 2 / 8 - 4 ^ ^ 4 ) 1 " ^

{2(3 - 3)(2/93 - 15/92 + 220 - 4) 6/9(/9-4)(/9-2)

+ ••

(26)

We now evaluate the mass using the surface term of the action in (16), i.e. within local surface counterterm method. The surface energy momentum tensor TV, is now defined by6 6St2) = 16TTG b

S does not contribute due to the equation of motion in the bulk. The variation of S + Sj gives a contribution proportional to the extrinsic curvature 6ij at the boundary: 8 [S + Sj ' ) = ^~^, (9ij — 8gij)5g'^. The contribution is finite even in the limit of r —> oo. Then the finite part does not depend on the parameters characterizing the black hole. Therefore after subtracting the contribution from the reference metric, which could be that of AdS, the contribution from the variation of S + S[ vanishes.

152

+^(^))} + ^n^{v

C

?^J'5i^W}

•

(27)

Note that the energy-momentum tensor is still not well-defined due to the term containing nM9M. If we assume Sg1* ~ O(pai) for large p when we choose the coordinate system (2), then nM3M (Sg**-) ~ jSg** (a\ + dp) (•)• Or if Sg1* ~ O (ra2) for large r when we choose the coordinate system (18), then nM9M (6gij-) ~ 5gijea ( ^ + 0 r ) (•)• As we consider the black hole-like object in this section, one chooses the coordinate system (18). Then mass E of the black hole like object is given by E=

(28)

fdt-ixy/ZNSTttiu*)'

Here we assume the metric of the reference spacetime (e.g. AdS) has the form of ds2 = f(r)dr2 - N2(r)dt2 + £ ? 7 ^ a^dx^x^ and 5TU is the difference of the (t, t) component of the energy-momentum tensor in the spacetime with black hole like object from that in the reference spacetime, which we choose to be AdS, and ul is the t component of the unit time-like vector normal to the hypersurface given by t =constant. By using the solution in (19), the (t, t) component of the energy-momentum tensor in (27) has the following form: l3V L Tu = IGTTGI3 1 - -T r4 + 0-6 6(/9-4)G!3-2) 3r

l2fic2 ( 1 1 2 ' r ? \12 6/3(/?-6) ( 3 - / 3 ) ( l + a2) 12 J +'"J "

(29)

If we assume the mass is finite, /3 should satisfy the inequality /? > 2, as in the case of the free energy in (24) since y/aN (u*) = lr2 for the reference AdS space. Then the ^-dependent term in (29) does not contribute to the mass and one gets E — ~^§- and using (21) _ 3*%)7rr* r _ tj

~

16TTG

i

1

* y - 6 )(2/? - 3) l Cfl1

[nl

>

(/3-4)(/3-2) /

'

[M)

which does not agree with the result in (26). This might express the ambiguity in the choice of the regularization to make the finite action. A possible origin of it might be following. We assumed <j) can be expanded in the (integer) power series of p when deriving the surface terms in (16). However, this assumption seems to conflict with the classical solution, where the fractional power seems to appear since r 2 ~ | . In any case, in QFT there is no problem in regularization dependence of the results. In many cases (see example in ref.17) the explicit choice of free parameters of regularization leads to coincidence of

153

the answers which look different in different regularizations. As usually happens in QFT the renormalization is more universal as the same answers for beta-functions may be obtained while using different regularizations. That suggests that holographic renormalization group should be developed and the predictions of above calculations should be tested in it. As in the case of the c-function, we might drop the terms containing $ ' in the expression of S^ ' in (16) but the result of the mass E in (30) does not change. 4

Comparison with other counterterm schemes and holografic RG

In this section we compare the surface counterterms and the trace anomaly obtained here with those in ref.34 (flat 4d case) and give generalization for 4d curved space. We start with the following action:

S= ^ ~ 8^G /

J dhx\[^6 (R - IguG^d^d^ d4x

^9(D^

- V(0))

+ Lct) •

(31)

Here we choose V((f>) = — 1 | and nM is given by n** = (1,0, • • •, 0), where the first component corresponds to r-component. As an extension of 39,34 , one takes the following metric: ds2 = dr2 + e2A{r)gijdxidxi

.

(32)

Here glj is the metric of the Einstein manifold, where Ricci tensor Rij given by glj satisfies the following condition: Rij = k9ij ,

(33)

where A; is a constant. The equations of motion from varying (31) with respect to the metric lead to the following form instead of (8),(9) in 3 4 . d2A 1 dtf dJ k 2A 2 dr 6y'J dr dr 3 J dA\2 1 1 dJ k 2 + 9lJ IT =n+7T,9iJ~-~ dr) ~ I 2$ dr dr + ' 3^ -

iV - - —

2A 2 A

.

(35)

154

If -^r is not zero, we can treat A' = ^A and 4>" E ^ a s functions of

(36)

If we assume the solution of (36) in the following form: ^ - = f(4>K ,A)gIJ we obtain f {", A)gu

dA' dA' 1 K~ 2„2IJ rjdA'dA' 1 J ~ -^f{4> ,A) g &4) dcj) dcp1 d(f>J

k 2A »~2A 36

||y, (37)

which can be solved with respect to f(4>K,A): f(cf>K,A) = -3±

, 9 -

2ke~2A

(38)

a A1' nr.jdA> y 7)471)47

Then we find dr

=

2ke-2A

-3±4/9-

nKL

JJ

dA'

dA'

(39)

dAL

In ± sign in (39), the — sign reproduces the result in 34 when k = 0. We should note that (j)1 can be regarded as a coupling constant associated with the operator 0 j , J d4xcj)IOi, from the AdS/CFT correspondence. Since In A can be also regarded as a logarithm of the scale, the ^-function could be given by

j = dtf_ = l_d£_ P

~ dA

2ke~2A \ jjd ^KT. 9A1 dA' j 9

-3±4/9-

A' dr

84>K

(In

A') 1

dip

84>L

(40)

First, we recall the surface terms:

Ss.t = - ^

d4x^g- (D^

I

+ Lc.t) (41)

and varying these terms with respect to the boundary metric gij one gets 1 SS -g SgV 1

surface term

f

i

+V ki kl

1

SSs.t " ,

" 1x

x1

"

9i

-- >

T

Lct +

^

L c t

\

JgWf

(42)

155

One can take Lc.t as in

34

, Let = T

1-

(43)

T^R

12

where Rg — gtJRgij = g13 Rij = 4ke~2A(r\ We denote 4 dimensional curvatures given by g^ and its derivatives with respect to xl by the suffix g. Then the variation of Rg with respect to the boundary metric g%3 is given by 8Lc.t _ 6gij

Here as

_[D

(44)

•. = - i * e - " M f t , + . . .

AHgij

~

expresses total derivative terms. And the equation (42) is rewritten 1 SS r^g 6gv 1 =

~8^rG

+ surface term

B

^9ij9

1 6S,.t r=g Sg'i

jw,

9kl,r +

I ^9ij,r

ke-W-

9ij

-!*(!-£•«.-"<«

(45)

Since one can regard g^ as metric of the 4 dimensional spacetime where the field theory lives, we could define trace anomaly by T =

2 2eiA

-a 6S „ 6S

-a3—-

'-9 ~

Sgv

2

..iSSs.t

+ V-9=z9' Sgij surface term 2e4A _ sss.t

(46)

f=g-y' 8gV

surface term

Then a4A

T = -f—

i -6A' + lke~2A^

- ^

(47)

}•

We now compare the above result with the one given in this report. The solution of the Einstein equations where the boundary has constant curvature is given in 4 4 . When the scalar fields vanish, the solution is given by

ds2 = f(y)dy2 +yJ2

kji^Wdx3

, / =

—^ .

(48)

Here the boundary lies at y -> oo. If we change the coordinate by z = f fy , , , the metric in (48) can be rewritten in the form of (32), where i + 3— 2vy]i->V

156

y(z).

Then the anomaly T in (47) is y2

T=

kl2

6

-4^{-yV

1+

Ik

6 1

+

(49)

¥ ^-7/-

On the boundary, where y -t oo, T has the finite value:

On the other hand, if we use the previous expression (6) of the trace anomaly T with constant
which is identical with (50). Thus, holografic RG consideration gives the same conformal anomaly as in second section. For simplicity, one can consider the case that the boundary is flat and the metric g^ in (2) on the boundary is given by gij = F(p)t]ij. We also assume the dilaton

(p). This is exactly the case of ref.34. Then the conformal anomaly (4) vanishes on such background. Let us demonstrate that our discussion is consistent with results of ref.34. 34 In , the following counterterms scheme is proposed

?(2) -

1

f *- r ^ f 6 » W ,

i

instead of (16). Here u is obtained in terms of this paper as follows: u()2 = 1 + Y2 $(). Then based on the counter terms in (52), the following expression of the trace anomaly is given in 34 :

r

=dbi ( - 2B - tt) -

(53)

Here B = pdpA. The above trace anomaly was evaluated for fixed but finite p. If the boundary is asymptotically AdS, F goes to a constant F —• F0 (F0: a constant). Then, we find the behaviors of A and B as A —»• | In ^k, £? —> — | . Then from (39), we find becomes a constant. Since we have the following equation of the motion 48 0

I2 (=, , , 12\ w + +

= -v r

v)

3 . 3 ,„ „ 2 + {dpF)

7

**

6

n

dpF

„

~P~p ~2

1 {dp
'

(54)

157

one gets

=\/1 + S $ W "* 1 •

u

(55)

Since B —> — | , this tells that the trace anomaly (53) vanishes on the boundary. Thus, we demonstrated that trace anomaly of 34 vanishes in the UV limit what is expected also from AdS/CFT correspondence. We should note that the trace anomaly (4) is evaluated on the boundary, i.e., in the UV limit. We evaluated the anomaly by expandind the action in the power series of the infrared cutoff e and subtracting the divergent terms in the limit of e —> 0. If we evaluate the anomaly for finite p as in 34 , the terms with positive power of e in the expansion do not vanish and we would obtain non-vanishing trace anomaly in general. Thus, the trace anomaly obtained in this paper does not not have any contradiction with that in 34 ,i.e. with holografic RG. 5

Dilatonic brane-world inflation induced by quantum effects: Constant bulk potential

In this section we consider brane-world solutions in d5 dilatonic gravity following ref.49 when brane CFT is present. We start with Euclidean signature for the action S which is the sum of the Einstein-Hilbert action SEH with kinetic term for dilaton , the Gibbons-Hawking surface term 5GH , the surface counter term S\ and the trace anomaly induced action Wc: S =

SEH

+

SQH

+ 2Si + W,

(56) 12N

5EH =

16^G /

5GH =

8^G /

d 5 x

^^

(RW ~ \V»yfl(t> +

d4x

^WV"""'

4 » = - 8irGl : " fd x^g^,

W

= b f d4x^FA

(58) (59)

+ b' J <£xv^ \A [2D2 + £M„VMV„

-\R&+ \{V»R)V» A+fG-^DR^A] 3 3V c

For the introduction to anomaly induced effective action in curved space-time (with torsion), see section 5.5 in 4 0 .

158

~h {b" + \{b + b>)} \^x^ +C f d*xA^

[* - 6°A - 6(VMA)(VM)]

a2 + 2ivvMv„ - ^R& + i(v"i?)v„

<£ .

(60)

Here the quantities in the 5 dimensional bulk spacetime are specified by the suffices (5) and those in the boundary 4 dimensional spacetime are specified by (4). The factor 2 in front of Si in (56) is coming from that we have two bulk regions which are connected with each other by the brane. In (58), nM is the unit vector normal to the boundary. In (60), one chooses the 4 dimensional boundary metric as ff(*W

= Q2A

( 61 )

9W

and we specify the quantities given by g^v by using ". G (G) and F (F) are the Gauss-Bonnet invariant and the square of the Weyl tensor. In the brane effective action (60), we consider the case corresponding to M = 4 SU(N) Yang-Mills theory, where 41 , b = -b' = § = f j ^ ? . The dilaton field <j> which appears from the coupling with extended conformal supergravity is in general complex but we consider the case in which only the real part of <j> is non-zero. Adopting AdS/CFT correspondence one can argue that in symmetric phase the quantum brane matter appears due to maximally SUSY Yang-Mills theory as above. Note that there is a kinetic term for the dilaton in the classical bulk action but also there is dilatonic contribution to the anomaly induced effective action W. Here, it appears the difference with the correspondent construction in ref.42 where there was no dilaton. In the bulk, the solution of the equations of motion is given in 4 4 , as follows

ds2 = f(y)dy2 + y £

g^x^dx'dx^

,

f =

= / dy

2

-y ,

VA (l + sfa + # ) d(d - 1)

+2

/ ^ " ^ 2

2>J =

\J V A (l + ^

+# ) '

(62)

Here A2 = j£ and §ij is the metric of the Einstein manifold, which is defined by nj = kgij, where r^ is the Ricci tensor constructed with (jij and k is a constant. We should note that there is a curvature singularity at y = 0 44 . The solution with non-trivial dilaton would presumbly correspond to the deformation of the vacuum (which is associated with the dimension 4 operator, say trF 2 ) in the dual maximally SUSY Yang-Mills theory.

159

If one defines a new coordinate z by z=

d{d l)

dy

2

\

~

(63)

VA (l + afc + ^]

and solves y with respect to z, we obtain the warp factor e2A(z'k*> = y(z). Here one assumes the metric of 5 dimensional space time as follows: ds2 = dz2 + e2A{s'cr)gfU/dx'idx,/

,

g^dx^dx"

= I2 (da2 + dn23) .

(64)

Here d£l\ corresponds to the metric of 3 dimensional unit sphere. Then we find A(z, a) = A(z, k = 3) — In cosh a, for unit sphere (k = 3) (65) A(z, a) = A(z, k = 0) + a, for flat Euclidean space (k = 0) (66) A(z,a) = A(z,k = —3) — lnsinher, for unit hyperboloid (k = —3) . (67) We now identify A and g in (64) with those in (61). Then we find F = G = 0, R= fi etc. According to the assumption in (64), the actions in (57), (58), (59), and (60) have the following forms:

SEH = { ^

J dzda {(~&d2A - 20(dzA)2) eiA + (-6%A - 6(daA)2

+6)e2A

Son = - ^

W = V3fda

_ | e "(0,fl a - ^2\dA)2

+^ }

,

(68)

J dae4AdzA,

(69)

[b'A (2%A - 802aA) - 2(6 + b') (l - 8% A -

+CA4>{dU-4d2atp)]

{daA)2f

.

(71) 2

Here V3 is the volume or area of the unit 3 sphere: V3 = 2n . On the brane at the boundary, one gets the following equations Q

- ^ G ( ^ - } ) ^

A

+

b'(AdiA-16d2aA)

-4(6 + 6') (%A + 2dlA - 6(d„A)2dlA)

+ 2(7 (% - 4d2cf>) , (72)

160

from the variation over A and 0 = -^4Adz4>

+ C{A {dt<j> - 4 ^ 0 ) + d$(A)} ,

(73)

from the variation over <j>. We should note that the contributions from 5EH and Son are twice from the naive values since we have two bulk regions which are connected with each other by the brane. The equations (72) and (73) do not depend on k, that is, they are correct for any of the sphere, hyperboloid, or flat Euclidean space. The k dependence appears when the bulk solutions are substituted. Substituting the bulk solution given by (62), (63) and (65), (66) or (67) into (72) and (73), one obtains 1 +

O =-

1 2 8b — 32/o ++ 2^A4 ^ )y o+ '>

^ + 6C0o.

(74) (75)

Here we assume the brane lies at y = j/o and the dilaton takes a constant value there 0:

(76)

+> = *kc-

Note that eq.(74) does not depend on b and C. Eq.(75) determines the value of 4>Q. That might be interesting since the vacuum expectation value of the dilaton cannot be determined perturbatively in string theory. Of course, (76) contains the parameter c, which indicates the non-triviality of the dilaton. The parameter c, however, can be determined from (74). Hence, in such scenario one gets a dynamical mechanism to determine of dilaton on the boundary (in our observable world). The effective tension of the domain wall is given by 3 n ,

^

3

/

= ^GdyA=^Glf+3y-o

kl2

Pc2 +

Wo-

(7?)

One should note that the radial (z) component of the geodesic equation in the metric (64) is given by ^ ^ 4- dzAe2A time and we can normalize e2A (7^7)

( ^ r ) = 0 . Here r is the proper

= 1 and obtain ^ £ + dzA = 0. Since

the cosmological constant on the brane is given by j | ^ ,
161

As in 3 , defining the radius R of the brane as R? = yo, we can rewrite (74) as

1+

^ + ii|- 1 ) fi4 + 8t'-

(78)

Especially when the dilaton vanishes (c = 0) and the brane is the unit sphere (k = 3), the equation (78) reproduces the result of ref.37 for TV = 4 SU(N) super Yang-Mills theory in case of the large N limit where b' -> - jr^rj • R3

I

73-V

1+

GN2

R? R* +

(?9)

^ = -F 8^--

Let us define a function F(R, c) as

^ ! s/ i t ffi t ffl- i r

(8o)

It appears in the r.h.s. in (78). First we consider the k > 0 case. Since d {In F(R,c)) dR

_1( f ~ R\y

~W_ l2c2 ZR2 + 24R*

\ j

1

/ / \V

kl2 l2c2 \ ZR? + 24i?8 j

2 1 2 k2l4 2l2c2\ (A kl ki2AA F r kP ki2 W^\ ^ " V 1 (f8kl + 4 1+ + V 3 ^ 2i^ "(w+lF-lWr)-

(81)

F(R, c) has a minimum at R = R0, where RQ is defined by 0 = §^5- + ^ — 2

j^-- When k > 0, there is only one solution for RQ. Therefore F(R, c) in the case of k > 0 (sphere case) is a monotonically increasing function of R when R > Ro and a decreasing function when i? < RQ. Since F(R,c) is clearly a monotonically increasing function of c, we find for k > 0 and 6' < 0 case that i? decreases when c increases if i? > Ro, that is, the non-trivial dilaton makes the radius smaller. Then, since 1/.R corresponds to the rate of the inflation of the universe, when we Wick-rotate the sphere into the inflationary universe, the large dilaton supports the rapid universe expansion. Hence, we showed that quantum CFT living on the domain wall leads to the creation of inflationary dilatonic 4d de Sitter-brane Universe realized within 5d AdS bulk spaced Of course, such ever expanding inflationary brane-world is understood d

Such brane-world quantum inflation for the case of constant dilaton has been presented in refs. 3 8 , 3 7 , 4 2 . In the usual 4d world the anomaly induced inflation has been suggested in ref.43 (no dilaton) and in ref.46 when a non-constant dilaton is present.

162

in a sense of the analytical continuation of 4d sphere to Lorentzian signature. It would be interesting to understand the relation between such inflationary brane-world and inflation in D-branes, for example, of Hagedorn type 47 . Since one finds F(R0,c) = j£§, Eq.(78) has a solution if ^ < -86'. That puts again some bounds to the dilaton value. When \c\ is small, one obtains i?4, ~ jjjip, F{Ro,c) ~ J ^ Q ^ - Therefore Eq.(78) is satisfied for small 2

\c\. On the other hand, when c is large, we get RQ ~ f^, F(Ro,c) ~ (fc|cD? . 43

KG

Therefore Eq.(78) is not always satisfied and we have no solution for R in (78) for very large \c\. Then the existence of the inflationary Universe gives a restriction on the value of c, which characterizes the behavior of the dilaton. We now consider the k < 0 case. When c = 0, there is no solution for R in (78). Let us define another function G(R,c) as follows: /2 2

h.]2

Since G(R, c) appears in the root of F(R, c) in (80), G(R, c) must be positive. Then ^ g £ l = _ ^ _ |W^ G(R,c) has a minimum 1 + *£ ( —f#)* when R6 = — f j . Therefore if c2 > ^ - , F(R,c) is real for any positive value of R. Since F(0,c) = ^ J j ^ , and when i? -»• oo, F(i?,c) -^ | ^ < 0, there is a solution R in (78) if J^U- > -86'. If we Wick-rotate the solution corresponding to hyperboloid, we obtain a 4 dimensional AdS space, whose metric is given by rfs

Ls4

=

dz2

+ e ^ {-M2 + dx2 + dy2) .

(83)

Then there is such kind of solution due to the quantum effect if the parameter c characterizing the behavior of the dilaton is large enough. Thus we demonstrated that due to the dilaton presence there is the possibility of quantum creation of a 4d hyperbolic wall Universe. Again, some bounds to the dilaton appear. It is remarkable that hyperbolic brane-world occurs even for usual matter content due to the dilaton. One can compare with the case in ref.42 where a hyperbolic 4d wall could be realized only for higher derivative conformal scalar. In summary, in this section for constant bulk potential, we presented the nice realization of quantum creation of 4d de Sitter or 4d hyperbolic brane Universes living in 5d AdS space. The quantum dynamical determination of dilaton value is also remarkable.

163

One can consider the case that the dilaton field <j> has a non-trivial potential:

f

-> V{<j>) = f + $() .

(84)

The surface counter terms when the dilaton field <j> has a non-trivial potential are given in (16), which we write in the following form:

S<2) = Sf + St

- jV,4)* • VmA - ^n"9 ( ^ 5 M * ( « ) } •

(85)

Following the argument in 37 , if one replaces | | in (57) and S\ in (56) with V{<j>) in (84) and Sf in (85), we obtain the gravity on the brane induced by S*. We now assume the metric in the following form 3

ds2 = f(y)dy2 +yY,

gij(xk)dxid^,

(86)

i,j=0

as in 44 and 0 depends only on y. As the singularity usually appears at y = 0, we investigate the behavior when y ~ 0. Here we only consider the case k > 0. First one assumes that there is no singularity. Then \ (constant) when y —> 0 .

(87)

It is supposed the spacetime becomes asymptotically AdS, which is presumbly the unique choice to avoid the singularity and to localize gravity on the brane 45 . The condition to get asymptotically AdS requires *'(0i) = O,

(88)

$'(0)~/3 0 ) , fa = $ - fa .

(89)

and one assumes

Then from the equation of motion, if we also assume fa behaves as fa ~ bya (a > 0) ,

(90)

164

one obtains a —1

,

(91)

a (92) Eq.(91) requires 0 < a < 1 and/or a > 1 and Eq.(92) tells that /3 cannot vanish and b should be positive, which tells, from the equation of motion that 4> increases when y ~ 0. In 49 , it was considered the following example as a toy model:

/»*(*) = - | ^ + ^ - I * » + g .

(93)

Using the numerical calculations, it was confirmed that there is no any (curvature) singularity and the gravity on the brane can be localized. Hence, we presented examples of inflationary and hyperbolic brane-worlds as analytical solutions in d5 dilatonic gravity when brane CFT quantum effects are also taken into account. 6

Discussion

In summary, we reported the results on various topics in d5 gauged supergravity with single scalar and arbitrary scalar potential in AdS/CFT set-up. In particulary, the surface counterterms, finite gravitational action and consistent stress tensor in asymptotically AdS space is found. Using this action, the regularized expressions for free energy, entropy and mass are derived for d5 dilatonic AdS black hole. From another side, finite action may be used to get the holographic conformal anomaly of boundary QFT with broken conformal invariance. Such conformal anomaly is calculated from d5 gauged SG with arbitrary dilatonic potential with the use of AdS/CFT correspondence. Due to dilaton dependence it takes extremely complicated form. Within holographic RG where identification of dilaton with some coupling constant is made, we suggested the candidate c-function for d4 boundary QFT from holographic conformal anomaly. It is shown that such proposal gives monotonic and positive c-function for few examples of dilatonic potential. We expect that our results may be very useful in explicit identification of supergravity description (special RG flow) with the particular boundary gauge theory (or its phase) which is very non-trivial task in AdS/CFT correspondence. We show that on the example of constant dilaton and special form of dilatonic potential where qualitative agreement of holographic conformal

165

anomaly and QFT conformal anomaly (with the account of radiative corrections) from QED-like theory with single coupling constant may be achieved. The role of brane quantum matter effects in the realization of de Sitter or AdS dilatonic branes living in d5 (asymptotically) AdS space is reported. (We are working again with d5 dilatonic gravity). The explicit examples of such dilatonic brane-world inflation are presented for constant bulk dilatonic potentials as well as for non-constant bulk potentials. Dilaton gives extra contributions to the effective tension of the domain wall and it may be determined dynamically from bulk/boundary equations of motion. The main part of discussion has dealing with maximally SUSY Yang-Mills theory (exact CFT) living on the brane. However, qualitatively the same results may be obtained when not exactly conformal quantum matter (like classically conformally invariant theory of dilaton coupled spinors) lives on the brane. An explicit example of toy (fine-tuned) dilatonic potential is presented for which the following results are obtained from the bulk/boundary equations of motion: 1. Non-singular asymptotically AdS space is the bulk space. 2. The brane is described by de Sitter space (inflation) induced by brane matter quantum effects. 3. The localization of gravity on the brane occurs. The price to avoid the bulk naked singularity is the fine-tuning of dilatonic potential and dynamical determination (actually, also a kind of fine-tuning) of dilaton and radius of de Sitter brane. Note also that in the same fashion as in ref.37 one can show that the brane CFT strongly suppresses the metric perturbations (especially, on small scales). One can easily generalize the results of this report in different directions. For example, following to brane-world line and taking into account that it is not easy to find new dilatonic bulk solutions like asymptotically AdS space presented in this work one can think about changes in the structure of the boundary manifold. One possibility is in the consideration of a KantowskiSachs brane Universe. Another important question is related with the study of cosmological perturbations around the founded backgrounds and of details of late-time inflation and exit from inflationary phase in brane-world cosmology (eventual decay of de Sitter brane to FRW brane). The number of other topics on relations between AdS/CFT and brane-world quantum cosmology in dilatonic gravity maybe also suggested. Acknowledgements The work of S.O. has been supported in part by Japan Society for the Promotion of Science, that of S.D.O. by CONACyT (CP, ref.990356 and grant 28454E) and by RFBR and that of O.O. by CONACyT grant 28454E.

166

Appendix A

Remarks on boundary values

Prom the leading order term in the equations of motion 0=

_ ^a*fa^,M

_ ^ (y C ^ ^ ) ,

(94)

which are given by variation of the action (95)

s

l 6 k

'^oL.S ^ { -t>" (95)

with respect to (f>a, we obtain d$((0)a

= 0.

(96)

The equation (96) gives one of the necessary conditions that the spacetime is asymptotically AdS. The equation (96) also looks like a constraint that the boundary value (o) must take a special value satisfying (96) for the general fluctuations but it is not always correct. The condition <j> = <^>(0) at the boundary is, of course, the boundary condition, which is not a part of the equations of motion. Due to the boundary condition, not all degrees of freedom of (/> are dynamical. Here the boundary value (0) is, of course, not dynamical. This tells that we should not impose the equations given only by the variation over 4>(o)- The equation (96) is, in fact, only given by the variation over (0). In order to understand the situation, we choose the metric in the following form 2

ds = G^dx^dx"

d I2 . . 2 = —p~ dpdp + y_.9ijdxidx:' ,

gij = p~1gij ,

(97)

»=i

(If g^ = rjij, the boundary of AdS lies at p = 0.) and we use the regularization for the action (95) by introducing the infrared cutoff e and replacing f dd+1x-+

jddx

J dp ,

[

ddx ( • • • ) - » f ddx (•••)! _

.

(98)

167 Then the action (95) has the following form:

(99) Then it is clear that Eq.(96) can be derived only from the variation over (0) but not other components ^ (i = 1,2,3, • • •)• Furthermore, if we add the surface counterterm S^

^1) = ~16^G^" f /

dd

*y-W0V)'---'^(°))

(10°)

to the action (95), the first (0) dependent term in (99) is cancelled and we find that Eq.(96) cannot be derived from the variational principle. The surface counterterm in (100) is a part of the surface counterterms, which are necessary to obtain the well-defined AdS/CFT correspondence. Since the volume of AdS is infinte, the action (95) contains divergences, a part of which appears in (99). Then in order that we obtain the well-defined AdS/CFT set-up, we need the surface counterterms to cancell the divergence. References 1. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B 428, 105 (1998). 2. M. Gunaydin, L.J. Romans and N.P. Warner, Phys. Lett. B 154, 268 (1985); M. Pernici, K. Pilch and P. van Nieuwenhuizen, Nucl. Phys. B 259, 460 (1985); B. de Wit and H. Nicolai, Nucl. Phys. B 259, 211 (1987). 3. N.R. Constable and R.C. Myers, hep-th/9905081. 4. J. Distler and F. Zamora, hep-th/9911040. 5. K. Behrndt and D. Lust, hep-th/9905180, JEEP 9907, 019 (1999). 6. D. Freedman, S. Gubser, K. Pilch and N.P. Warner, hep-th/9906194. 7. L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, hep-th/9909047. 8. K. Behrndt, E. Bergshoeff, R. Halbersma and J.P. Van der Scharr, hepth/9907006. 9. G. 't Hooft, gr-qc/9310026. 10. G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15, 2752 (1977). 11. J.D. Brown and J.W. York, Phys. Rev. D 47, 1407 (1993). 12. V. Balasubramanian and P. Kraus, hep-th/9902121. 13. R.C. Myers, hep-th/9903203.

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171 Q U A N T U M GROUP SUQ(2) A N D PAIRING IN NUCLEI S. S. S H A R M A A N D N. K. S H A R M A Departamento

de Fisica e Departamento Londrina, Londrina, E-mail:

de Matemdtica, 8d051-970, PR, [email protected]

Universidade Brazil

Estadual

de

A scheme for treating the pairing of nucleons in terms of generators of Quantum Group SU,(2) is presented. The possible applications to nucleon pairs in a single orbit, multishell case, pairing vibrations and superconducting nuclei are discussed. The formalism for performing BCS calculations with g-deformed nucleon pairs is constructed and the role played by deformation parameter q analyzed in the context of nucleons in a single orbit and for Sn Isotopes.

Pairing of nucleons manifests itself in the energy Gap in even-even nuclei, odd-even staggering, moment of inertia of deformed nuclei, low lying 2 + states, ground state spins and decay properties of nuclei. Quasi spin operators, the generators of group SU(2) have been an interesting artefact for studying nucleon pairing since the time that Racah and Talmi l pointed out the group symmetries of the zero range pairing interaction model. In this talk, a more general scheme for treating the nuclear pairing problem in terms of generators of quantum group SUg(2) is presented. The quantum group SU g (2), a (/-deformed version of Lie algebra SU(2), has been studied extensively 2>3'4, and a g-deformed version of quantum harmonic oscillator developed 5 ' 6 . The quantum group SU ? (2) is more general than SU(2) and contains the later as a special case. The underlying idea in using the zero coupled nucleon pairs with (/-deformations is that the commutation relations of nucleon pair creation and destruction operators are modified by the correlations as such are somewhat different in comparison with those used in deriving the usual theories. The (/-deformed theories reduce to the corresponding usual theories in the limit } - > ! • We first introduce the seniority scheme and the quasi-spin operators in section I. The q-deformed nucleon pairs are denned in section II which also contains a brief review of seniority scheme based on q-deformed nucleon pairs 7 . Section III contains the formulation of random phase approximation (RPA) with q-deformed nucleon pairs (boson approximation) for nuclei with no superconducting solution and RPA with q-deformed quasi-particle pairs (quasi-boson approximation) for superconducting nuclei 8 . In section IV, the formalism for BCS theory with q-deformed nucleon pairs is presented and it's application to the case of Sn nuclei discussed9. The Nucleon pairing in a

172

single j shell has also been treated by Bonatsos et. al 1 0 - u ' 1 2 by associating two Q-oscillators, one describing the J = 0 pairs and the other associated with J ^ 0 pairs. In their formalism, Q-oscillators involved reduce to usual harmonic oscillators as Q —> 1 and the deformation is introduced in a way different from ours. 1

Quasi-Spin operators and Seniority Scheme

Creation and destruction operators for a zero coupled nucleon pair in single particle orbit j are, Z0 = — ) = ( A > x A ' ) ° v2

T0 = -i= (B j x B j )° v2

and

where A-jm

= Qjm;

£>jm

=

a

v'-l

j,—m\

a

jmajm

+ ajmajm

=

*•• V - V

With number operator defined as n

°p

=

/_^ajmaim

'

(2)

m

and putting ft = ^ ± 1 [Z 0 ,2o] = ^

;we

n

can verify that

- 1;

[nop,Z0} = 2Z 0 ;

[nop,J^\ = -2Zo~ .

(3)

We can use quasi spin operators defined as

Sj+=

J^ {-l)]-m a\ma)_m

;

jm>0

Sj- = ^

(-lY~m

a

j-majm

,

(4)

jm>0

and for a single particle orbit j identify

s+ = Vsiz0, S- = Vnz^-

s0= {n°p2 n).

(5)

The operators S+, 5_ and So are the generators of SU(2) and satisfy the commutation relations of angular momentum operators, [5+,5_] = 25 0 ;

[S0,S±] = ±S±.

(6)

173

In seniority scheme an n nucleon state with v unpaired particles (v being the seniority) is represented by \n,v), we have ZQZQ \V,V) = 0. The states with p pairs of nucleons and n = v + 2p can be constructed as N{Z0)p\v,v).

\n,v) = Choosing a pairing Hamiltonian H = \n,v) is 13 E(n,v) 2

A

=

2

+

AZQZQ

, the pairing energy in state

(n - v) (2fl - n - v + 2).

(7)

Nucleon Pairs with q-deformation

To construct nucleon pairs with q-deformation, we next examine the generators of quantum group SUq(2). The operators S+(q) , S-(q) , and So(q) satisfy the commutation relations [S+(q),S-(q)]

= {2So(q)}q ;

[5 0 ( 9 ), S±(«)] = ±5±(g),

(8)

where

Expressing the creation and annihilation operators for 9—deformed nucleon pair as Z0(q) = ~S+(q);

ZM

= ~=S-(q),

(10)

the commutation relations for q—deformed nucleon pair creation and destruction operators are found to be , {nov — fi}„ P [Z0(q), Z0(q)] = '", n

[nop, Z0(q)} = 2Z0(q);

[nop,^>{q)] =-2T0{q).

(11)

The pairing Interaction Hamiltonian is now written as H{q) = AZo(q)Zo(q). Using q = eT, (r ^ 0) the pairing energy in seniority scheme for q-deformed pairs is P

^

/„ „x ( n , W )

"

2A sinh (pr) sinh [(fl - « - p + 1) r] (2J- + l ) s i n h 2 ( r ) •

(12)

Application to various isotopes in single particle orbits lfi, and lgs. have shown a good agreement with experimental ground state energies for small

174

values of deformation parameter 7 . The formalism for realizing a multishell calculation was also developed and applied to Calcium isotopes. It was found that in general weakly interacting heavily deformed nucleon pairs reproduced the spectra very similar to that produced by strongly interacting weakly deformed nucleon pairs. However, depending upon the distance from the closed shell, the energy spectra could shrink or expand with increase in deformation 7 . 3

R P A with q-deformed nucleon pairs and q-deformed Quasi-particle pairs

Using the q-deformed pair creation and destruction operators of Eq. (11) we derived the Random Phase Approximation equations for the pairing vibrations of nuclei. For nuclei with no superconducting solution, the boson creation operator that links the ground state of the nucleus \A, 0) to the excited eigen state v of the A+ 2 nucleon system with Jn = 0 + is denned as

*-?^(M)-?^(M)

<->

such that \A + 2,v)=R»+\A,0)

,

R"\A,0)=0.

(14)

We use the indices mn(ij) for single-particle(hole) levels and Rv = (-R+) • The equations of motion are set up for R+ using single-particle plus pairing Hamiltonian and the RPA equations for the system obtained using the commutation relations of eq. (8). The dispersion relation 1

G

_ V^ {ttn}q ^{2en-huu)

V^ {ftj}q Z-<(2tj-hwv)

y

(,t| '

along with the nomalization condition easily yields a graphical solution. A similar procedure is followed for constructing the solution for two-hole phonon states such that \A-2,»)

= R»+\A,0) ,

fi"|4,0)=0,

(16)

where

^?^(M)-?y"(M)-

(17)

The two phonon states

\A,v,ri = KK\A0)

(18)

175

are the excited 0 + states of the nucleus with excitation energy E{Q+) = hLJv + hujtl.

(19)

The q-deformed RPA when applied to study the pairing vibrational states in the nucleus 2 0 8 Pb showed that for r = 0.405 the experimental excitation energy of the double pairing vibration state and the transfer cross section for two neutron transfer are well reproduced 8 . For superconducting nuclei, one has to construct the quasi-boson creation and destruction operators from q-deformed quasi-particle pair creation and annihilation operators. The set of coupled equations

[hju - 2Em)xvm = -G^/{ft r a } 9 £ J{nJq [x; « « J + v2mv2p) p

-Y;{u2mv2p+v2mul)]

(20)

and

(tujv + 2Em)Y^ = G V / { f t m } ? £ V / ^ « [Y; KU2P + vlvl) p

2

2

2

-X;(u mv p+v mu2p)]

(21)

can be solved using standard procedure to furnish the roots E = huv . For testing the formalism, q-deformed boson and quasi boson approximation calculations for 20 nucleons in two shells were performed, and compared with exact shell model results. The deformed boson approximation results for r = i0.104 and deformed quasiboson approximation energies for r = 0.15 overlap the exact calculation results in a wide region away from the phase transition region. The deformation effectively results in including the correlations left out in normal approximate treatments. One can expect, therefore, that in a realistic calculation deformation parameter can be used as a quantitative measure of correlations left out in an approximate treatment in comparison with the exact results. 4

Gap equation in Q B C S and the Ground State Energy

Pairing effect can not be interpreted as a contribution to an average static potential (as in Hartree Fock) or contribution to average vibrating singleparticle potential (as in RPA). It is analogous to Superconductivity in metals. In 1959 Belyaev14 successfully applied to nuclei the Bardeen-Cooper-Schrieffer (BCS) theory originally formulated to explain superconductivity in metals 15 . In view of the usefulness of formulating the nucleon pairing problem in terms

176 of the generators of SU,(2), we are encouraged to formulate a q-deformed version of BCS theory or qBCS. Following the idea of building correlations in to the theory by using pair generators satisfying g-commutation relations, we next present the g-analog of BCS theory (gBCS) for nuclei. The formalism when applied to the case study of 1 1 4 _ 1 2 4 Sn nuclei elucidates the role played by g-deformation in these nuclei. For N nucleons in m single particle orbits, we consider the trial wave function ,

*=

•••**

where for the orbit j ,

\ n=0

n

\n!(%Qjl- n ) 1! .

|n) ;ttj =

2j + l

(22)

and |n> =

"{n*-"VI [w* {<WJ

i 2

(5 i + (g)) n |0) (S

is the normalized wave function for n zero coupled nucleon pairs with qdeformation occupying single particle orbit j . Using a variational approach with the single particle plus pairing Hamiltonian for g-deformed pairs given by

H = £ ernlp - G £ > + {q)Ss- (
(23)

jr,

and the gap parameter defined as, A(q)

=G(9

J^Sr+(q)

Y^GurVr{ttr}q

= we obtain the occupancies, /

*\=£Ar(«z) (24)

177

gap parameter / A(«) = ^ G { n , - } , 0 . 5 i

ti ~ *)'

^

(e;-A)

2 +

(26)

(A(,)i^y

and consequently the gap equation

y , ( £ ; - A ) J f l 5 + ( A ( 5 ) {(),•>,)

'

To include the effect of terms containing UjV? left out earlier, we now replace the chemical potential A by „2

A(«) = A +

^

" ' ( { n , } , - n,- + I ) .

(28)

The ground state BCS energy, (* \H\ * ) is m

( 2 e i fii u i "

Ebcs(q) = £

G

^ W ,

(&i}g

~ % + l))

=i

(A(g)) G

(29)

We notice that in a very natural way, the SU, (2) symmetry introduces in the interaction energy, a q dependence which is linked to the j-value of the orbit occupied by the zero coupled nucleon pairs. 4-1

Single orbit with 2£l degenerate states and Sn nuclei

For N nucleons in a single orbit with an occupancy of 20, the ground state wave function is ^ = $ j and Ebcs(q) is Ebcs{q) = ejN - G {%}, ^

( 2 {ilj}q - N + ^j

.

(30)

The exact energy of the N nucleon zero seniority state, Eexact=ejN-G'j(2nj-N

+ 2),

(31)

178

can be reproduced (Et,cs(q) = Eexact) strength G' such that

by choosing q value and the pairing

G'ilj (2»j - N + 2)

~{0,} 9 (2{n j } ? -iV+f) for the choice ej = 0 . 0 . In Ref. 9 the single orbit limit of gBCS is applied to nuclear sdg major shell with Q, — 16, and 4,10,14,20,24,30 valence nucleons occupying degenerate Ids, Ogz ,2s i, ldg ,and 0/tu orbits. The intensity of pairing strength required to reproduce Eexact decreases with increasing q and ultimately G —> 0 for all cases. It is also found that the strongly coupled zero coupled pairs of BCS theory may well be replaced by weakly coupled (/-deformed zero coupled pairs of qBCS theory. To get more clues as to whether it is possible to replace the pairing interaction by a suitable commutation relation between the pairs determined by a characteristic q value for the system at hand, real nuclei have also been examined in Ref. 9 . There has been an increased interest in the experimental and theoretical study of Sn isotopes, more so after the observation of heaviest doubly magic nucleus g^Snso in nuclear fragmentation reactions 16,17 . We examined the heavy Sn isotopes with N = 14,16,18, 20, 22, and 24 neutrons outside 5o°Sn50 core. The model space includes Ids ,0gi,2si, Ida, and Ohii, single particle orbits, with excitation energies 0.0, 0.22, 1.90, 2.20, and 2.80 MeV respectively. The pairing correlation function D = A(q)/\/G as a function of G for the cases where deformation parameter takes some typical successively increasing values varying from 1.0 to 1.7 shows some interesting features. In 5o0Sn7o, pairing correlations are found to increase as q increases while the pairing strength G is kept fixed. For q = 1.0 that is conventional BCS theory the pairing correlation vanishes for G < G c ( ~ 0.065 MeV) as expected. We find D going to zero for successively lower values of coupling strength, for example Gc ~ 0.04 MeV for q = 1.3 as the deformation q of zero coupled pairs increases. We may infer that the qBCS takes us beyond BCS theory. The sets of G,q values that reproduce the empirical A for 5o°Sn70 are next used to calculate the gap parameter A and the ground state BCS energy EN, for even isotopes 114 ~ 124 Sn and compared with the experimental values of A in Ref.9. The results of qBCS for Sn isotopes are not much different from BCS as far as the Gap parameter A is concerned. The ground state binding energies are however lowered by the deformation. The pairing correlations, measured by D = A(q)/VG, are seen to increase as q increases (for q real) while the pairing strength G is kept fixed, in Sn isotopes. It is immediately

179

seen that q parameter is a very good measure of the pairing correlations left out in the conventional BCS theory. The underlying g-deformed nucleon pairs show increasingly strong binding as the value of q is increased. It opens the possibility of obtaining the exact correlation energies by choosing appropriately the combination of G, q values. The results of our study of qBCS are consistent with our earlier conclusions 7 ' 8 that the g-deformed pairs with q > 1 (q real) are more strongly bound than the pairs with zero deformation and the binding energy increases with increase in the value of parameter q. In contrast by using complex q values one can construct zero coupled deformed pairs with lower binding energy in comparison with the no deformation zero coupled nucleon pairs 8 . In general the pairing correlations in N nucleon system, measured by D = A ( g ) / v G , increase with increasing q (for q real) and gBCS takes us beyond the BCS theory. The formalism can be tested for several other systems, for example metal grains, where cooper pairing plays an important role. 5

Acknowledgments

S. Shelly Sharma and N. K. Sharma acknowledge support from Universidade Estadual de Londrina. References 1. Racah, G. and Talmi, J. Physica 18, 1097 (1952). 2. M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11, 247 (1986). 3. Woronowicz, Publ. RIMS (Kyoto University) 23 , 117 (1987); Commun. Mat. Phys. I l l , 613 (1987). 4. Pasquier, Nucl. Phys. B295, 491 (1988); Commun. Mat. Phys. 118, 355 (1988). 5. Macfarlane A. J., J. Phys. A 22, 4581 (1989). 6. Biedenharn L. C , J. Phys. A 22, L873 (1989). 7. S. Shelly Sharma, Phys. Rev. C 46, 904 (1992). 8. S. Shelly Sharma and N. K. Sharma, Phys. Rev. C 50, 2323 (1994). 9. S. Shelly Sharma and N. K. Sharma, Phys. Rev. C 62,034314 (2000). 10. D. Bonatsos, J. Phys. A 25, L101 (1992). 11. D. Bonatsos, C. Daskaloyannis and A. Faessler, J. Phys. A 27, 1299 (1994). 12. Dennis Bonatsos and C. Daskaloyannis, Prog. Part. Nucl. Phys. 43, 537 (1999).

180

13. R. D. Lawson, Theory of the Nuclear Shell Model (Clarendon, Oxford, 1980). 14. Belyaev, S. T., L. Dan Vidensk. Selsk. Mat.-fys. Medd. No. 11, 31 (1959). 15. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 16. M. Lewitowicz et al., Phys. Lett. B332, 20 (1994). 17. R. Schneider et al, Zeit. Phys. A 348, 241 (1994).

181

SOME TOPOLOGICAL CONSIDERATIONS A B O U T D E F E C T S O N N E M A T I C LIQUID CRYSTALS M. S I M O E S A N D A. S T E U D E L Departamento de Fisica, Universitario,

Universidade Estadual 86051-970, Londrina E-mail: [email protected]

de Londrina, (PR),Brazil.

Campus

In this study, a nematic liquid crystal is used as an example to show analytically how an originally bi-dimensional system avoids a configuration with a singular pole. The system escapes into the third dimension with the simultaneous creation of a branch cut. It is presented an introductory study of some topological properties of defects in nematic liquid crystal. Special detach is given to the planar instabilities of these objects and to the corresponding properties of the escape into the third dimension of the disclinations with topological charge S = l and S = l / 2 . As a final result we show that the tri-dimensional pole with charge S = l / 2 is topological equivalent to the Mobius strip.

1

INTRODUCTION

The study of defects and patterns is usually considered as a traditional trademark of the physics of the condensed matter 1 ' 2 that has been widespread for all physics; it is impossible to talk about critical phenomena, disordered systems, cosmological phase transitions, frustrated media, convective instabilities, patterns, biological shapes, etc, without the use of the concept of symmetry breaking and their fellows, the defects. Words like dislocations, textures, grain boundaries, etc, are so common that probably no one remembers that the first observations of these kind of structures were not made in crystalline materials, but in a very special kind of liquid; the liquid crystals. The liquid crystals 3 ' 4 were discovered more than a century ago, in 1888, when Priedrich Reinitzer attributed to some organic compounds, derived from the cholesterol, the absolutely unusual property of having two melting points. Immediately it was observed that these kind of substances were rich in textures. The first systematic study of these textures appeared in 1904, when Lehmann gathered together a huge number of observations 5 . In 1922 Friedel, for the first time, observed that these textures are rich of geometry and gives the first correct description of this character of these defects6. The reason for the early observation of these structures in liquid crystals is due to the fact that they can be easily detected by a simple optical microscope, while the defects in metals, for example, only were definitively ascertained in the 50's with the use of the electron microscope.

182

The aim of this work is to use the liquid crystals to give an explicit example of how the imposition of global geometrical properties can lead to the formation of singularities. We will present the axial polar defects 7 ' 8 and we will show how, to avoid the infinite energy involved on some singularities, the system becomes unstable and relax. Furthermore, depending on the topological properties of the axial singularity, this relaxation can create a new kind of topological singularity. The next two sections are a rather overview on the issue of axial disclinations and it is devoted to non-specialists, or students, of the subject. In these sections some geometrical aspects of the axial disclinations will be exhibited, and fundamentally we will pave the way for analytic study of the out-of-plane-tilt to be made in the last section. There, the theory of the out-of-plane-tilt will be presented and it will be analytically demonstrated that the 5 = 1 / 2 pole is associated with a branch-cut with the topology of the Mobius strip. 2

AXIAL DISCLINATIONS AND T H E M I C R O S C O P I C N A T U R E O F T H E L I Q U I D CRYSTALS

The name liquid crystal arises when these kind of liquids where first observed through an x-ray apparatus. It was observed that they presented some interference patterns characteristic of crystalline materials. This happens because the orientations of their molecules are not randomly distributed as in a usual liquid, in a crystal the molecules have both positional and directional correlation and, normally, during the melting they lose both of these symmetries. A liquid crystal is a liquid in which during the crystalline to liquid phase transition only the center of mass correlation between the molecules is lost; the molecules remains directionally correlated. As a result, at each point of a sample there is a direction defined by the mean orientation of the molecules at the neighborhoods of that point. This set of directions defines a vector field in which the preferred direction, at each point, is called director. Of course, the origin of this unusual behavior is due to the peculiar form of the molecules of these materials. They are strongly asymmetric, existing two main kinds of molecular forms; materials with rod-shaped molecules, in which one molecular axis is much longer than the other two, and materials with disklike molecules, namely, one molecular axis much shorter than the other two. The origin of the orientational order becomes evident when we realize that it may exist a temperature interval in which the molecules, even without a positional correlation, may prefer to stay aligned with their neighbors than rotate freely. This kind of molecular arrangement is known as nematic order. The singularities of the vector field constructed by the average directions

183

of these molecules are known as defects. For the usual crystalline lattice the defects are known as dislocations and are related to the symmetry breaking of translations along the lattice 1 . As, in the liquid crystals, the lattice does not exist their defects are connected the symmetry of rotation, known as disclination, of the director. A presumed characteristic of this structure is that when we turn around the axial center "O" in a closed loop the director undergoes a complete rotation of 2-K. Nevertheless, this does not need happen in the general case. An important aspect of the nematic domains concerns the local rotational symmetry properties of the director. Contrarily to what happens in a usual vector field, the director field of a nematic material is not symmetric by a local rotation by an integer multiple of 2TT. It is symmetric by a rotation of an integer multiple of IT (this symmetry is know as IT symmetry). This property can be easily understood if we realize that microscopically the molecule can be taken as a hard-rod for which the long axis of symmetry does not have a preferred direction and, so, any 7r rotation leaves it invariant. The simplest singularity that can be found with this symmetry has the property that when we turn the pole, in a closed path, the director direction changes by 27r. To this pole is attributed the topological charge 5 = 1. Moreover, the simplest non-trivial pole is the one for which the director only returns for its original orientation if we follow a path the encircles the pole two times. This is the bi-dimensional axial pole with topological charge 5 = 1/2. The essential characteristic of this structure is that when we turn around the axial center " 0 " in a closed loop the director undergoes a rotation of 7r. There is an enormous number of different bi-dimensional polar axial disclinations that can be constructed using the 7r symmetry. They are fully characterized by two variables; by the topological charge 5, which corresponds to the number of times that the director rotates when the polar center is encircled by a single loop, and by the inclination,

5 = 0, ± 1 / 2 , ± 1 , ± 3 / 2 . . .

where ip is the angle of rotation of the director, 9 is the polar angle, and tp0 is the initial inclination of thew director. A full discussion of the topological properties of these defects can be found in the quoted references. We suggest4 for some beautiful illustrations of some of these defects.

184

3

THE BRANCH-CUT

Up to now we have described the axial poles as bi-dimensional structures. But, as a really bi-dimensional structure is not easy to construct and, probably, they do not exist in nature. In the three dimensional version of the 5 = 1 axial disclination the singular point becomes a singular line, along which the director direction is not defined. Using the theory of the elastic interaction between the domains of the nematic liquid crystal, to be presented ahead, it is easy to show that the energy stored in this singular line is infinite4 and, as any disturbance that should reduce this infinite energy will immediately do it, this configuration is unstable 7 ' 8 . So, it can be shown that in the neighborhoods of the axial line the director escapes into the third dimension creating a new kind of configuration, an out-of-plane-tilt, in which the singular line is avoided. In the case of the axial disclination 5 = 1/2 the situation is not so simple; in order to relax the energy stored along the singular line new kinds of singularities must necessarily appear in the sample. Meyer 8 , using a graphical (non analitical) argument, demonstrated that for the disclination with 5 = 1 / 2 the escape into the third dimension is necessarily associated with a line of branch cut. That is, in a complete turn around the line of disclination, the director can not change continuously at every points. There must be a line along which the arrow describing the up or down direction director changes abruptly. Mayer called this line by branch cut. In the next section we will study an analytical expression for this out-of-plane-tilt and shown that this name justified. 4

T H E OUT-OF-PLANE-TILT

The simplest assumption about the nature of the interaction between the molecules of a nematic liquid crystals is to assume that there is an elastic force between neighbors molecules that try align them. When the most general form for this interaction is considered, and it is assumed that it must be in accord with the 7r symmetry, it can be show that it must have the form F[nz] = ^j

T(n,dn)dV

(1)

where JF(n, dn) = Ku

(v-nj

+ K22 (n • (V x n))

+K33 (ft x (V x n))

-L(n-(Vxn)^)

,

(2)

185

where Ku, K22, K33 and L are the elastic constants related to elastic distortions of the nematic medium. By the way, let us remember that an important characteristic of these parameters is that it can be shown4 that all elastic textures of the nematic medium can always be written as a combination of these four elastics terms. Therefore, this expression takes care of all possible kind of bulk deformations in the nematic material. As our aim is to describe the axial quoted above, a proper coordinate system must be used. Due to the cylindrical form of these defects the director components will be described by nx = rcos8(x,y,z),

n2 = 1 — r2,

ny = rsin6(x,y,z),

(3)

and the spatial coordinates will be represented by x — Rcosp,

y = Rsiiap,

z = z.

(4)

As simple as it can appear the substitution of Eq. (3) and Eq. (4) in Eq. (1) leads to a so huge calculation that, up to our knowledge, no one has yet done it. In order to make it we used analytical computation and have found that the free energy can be written as

where 1 F0 = < 2 r » * >

»„o 1 dnz\ { t )

2 R 2 {

I ,,,

„,

, x _ , A „ , d6 {(^+^nz>)+2Lrdz

+ 2 r 4 i ? 2 ( g ) 2 (K22 + K;2nz>)\ +

( ^ )

i^s+Kss+K^

nZ*

+ (AT2-3 + Kn nz2) cos (2(0 -
{jp)'{(Kb+

**»*) 2

+ ((Ku - K22 nz ) - K33 r2) cos (2(6 - tp))}

+K^2 + Ki3 nz2 } cos (2(9 -
186

+r4jR2

Gl) 2 Ws + ^ W

+ ((-K11+K22nz2) +2r4R^J-

+

{Kf2^nZr2

+2nzr

+ K22nz2)

{{-Kn

| nzr2R^

+ 2 ^

+ KZ3r2)}

2

cos

(2(6-ip))

+ K33 r2} sin (2(0 -
(K+2 + K21 cos (2(6 -
" +^3"i^2) ^

}sin(2(
i i ^ { - ^ ( ^ + + ^ 1 cos2(0-^)) +

+ # r 2 J J ^ sin ( 2 ( 0 - * > ) ) } } , d# and 1

\(

T

n

„

2dO\

dnz dip

r, 2 (

r

nzRr2

n

2„

-L-2r2K?„ V dnz ( T , o d^

f

2/

„ dnz /

r

<16 \

23

„ 2 r , - d
d6

— — dz J dR „T dnz

2

r^

/ d9 \ 1

OT ^

2d6\dnz

1

where we have used Kf- = ifji + Kjj and 2
187 p -» p + 2-n , 9 -> 6 + 2irS .

So, as the phases of F0 are determined by the terms cos (2(0 — p)) and sin (2(9 — p)), in a complete turn one has 2(

2(

+ 2(5 -

1)2TT.

So, the phase of F0 will change by an integer multiple of 2ir and. Therefore Fa —> Fo.

By another side, the phases of -F\ are determined by the terms cos(0 -
1)2TT,

and, as 5 can be semi-integer, a complete turn leaves to a change in F\ of the form F l -> ±F1, where the positive number results when S is aninteger, 5 = 0, ± 1 , ± 2 , . . and the negative one results when 5 is a semi-integer, 5 = ± 1 / 2 , ± 3 / 2 , . . Therefore, while a complete turn around the axial desclination does not change the sign of Fo, the same does not happen with Fi. Nevertheless, the important aspect of the disclinations with semi-integer topological charge is that the simultaneous change (p-9)^(p-9) + (Snz —> — nz ,

1)2TT ,

make the Free-energy T invariant. So, the axial disclination with 5 = 1/2 is topological equivalent to the Mobius strip. That is, to return to the initial position with a unique turn a rotation and an inversion must be combined. Otherwise, with a double rotation the initial configuration is restored. 5

Conclusion

In this work we have used the bi-dimensional pole with charge 5 = 1/2 of a nematic liquid crystal to show analytically how the instability of this singularity creates an escape into the third dimension. It is shown that this out-of-plane-tilt creates a new pattern with the topology of the Mobius strip.

188

Acknowledgments The author M.S. thank CNPq for partial support. References 1. Ashcroft, Neil W.; Mermin, N. D.; Solid State Physics (Holt-Saunders Int. Ed. 1981). 2. C.Kittel, Introduction to Solid State Physics (John, Wiley & Sons, inc. 1976.) 3. M. Kleman. Defects in Liqid Crystals in Advances in Liquid Crystals. (Edited by Glenn H. Brown, Academic Press, New York 1975). 4. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 2nd edition, 1993.) 5. Collings, P.J. and Hird, M. Introduction to liquid Crystals (Taylor & Francis, 1997) 6. Friedel, E.C.R. Acad. Sci., Paris 180, 269 (1925). 7. P.E. Cladis, M. Kleman, J. de Physique 33, 591 (1972). 8. R.B. Meyer, Phyl. Mag. 27, 405 (1973).

189

N O N - L I N E A R REALIZATIONS A N D B O S O N I C B R A N E S P. WEST Department of Mathematics, King's College, London, UK Email: [email protected] In this very short note, following hep-th/0001216, we express the well known bosonic brane as a non-linear realisation. The reader may also consult hepth/9912226, 0001216 and 0005270 where the branes of M theory are constructed as a non-linear realisation. The automorphisms of the supersymmetry algebra play an essential role.

1

The Theory of Non-linear Realizations

We first briefly review the theory of non-linear realizations set out by Coleman, Wess and Zumino and extended by Volkov. Rather than consider a general group we restrict ourselves to the (super) group G whose generators can be divided into the two sets K_ and H_. The sets K_ and H_ are both supgroups of G and H_ is the automorphism group, of K_. The generators in each of the above two classes are further divided as K_ = {K, K'} and R = {R, R'}. The generators K and R both form subalgebras of the Lie (super) algebra G whose associated groups we denote by K and H respectively. The generators of H are an automorphisms of K. The division of the generators of G into these four classes corresponds to: K , unbroken generators associated with positions in (super) space-time, K' , spontaneously broken generators associated with positions in (super) space-time, R , unbroken automorphism generators, R' , spontaneously broken auotmorphism generators. The automorphism generators include those of the Lorentz group or its covering group the corresponding Spin group, in some cases internal group generators, but also other generators which are not usually considered. From now on we will drop the prefix (super), but it is to be understood to be present. As will be clear, the objects associated with the group G carry an underlined indices while those associated with the subgroup G carry no underline. The decomposition of the former into the latter and the remainder is achieved using unprimed and primed indices respectively. We wish to consider the coset

190

G_/H. For simplicity we will use an exponential description of group elements and we may then choose the coset representatives to be g = eXKeX>.K>e*'.R>

(1

1}

In this equation • denotes the relavent summation over the indices. Under any rigid group transformation go we find that S-> 9o9 = 9 = e*Keji'K'e*'R'h0

(1.2)

where h0 is an element of H. The Cartan forms are given by g-1dg = F-K

+ F'-K'+LJ'-R'+GJ-R.

(1.3)

Under equation (1.2) the Cartan forms transform as gdg = hoig^dg)^1

+ hodh^1 .

(1.4)

It can happen that the Cartan forms carry a reducible representation of H, in which case, certain of the forms can be set to zero. This is the so called inverse Higgs effect. It has the effect of eliminating some of the Goldstone fields in terms of the others. The action or equations of motion are to be constructed from the Cartan forms in such a way that they are invariant under the transformations of equation (1.2). The conventional interpretation of the above equations is to regard the X as the coordinates of (super) spacetime and to take the fields X' and ' to depend on them. This leads to a field theory on the coset space G/H. This approach has been almost universally adopted. However, when considering branes it is instructive to consider a more general possibility. The brane is moving through the coset space G_/H_ with tangent group H_ and sweeps out a submanifold that has the dimensions of the coset G/H and a tangent space group H. We therefore consider the representatives of the coset G_/H of equation (1.2) to depend on the variables £ which parametrises the embedded submanifold. Since the Cartan forms involve the exterior derivative d they are independent of the coordinate system used. The Cartan forms associated with K are given by F • K = d£ • F • K and similarly for the other Cartan forms. We can think of the F in this equation as the vielbein on the embedded submanifold. The covariant derivatives of the Goldstone fields associated with K' are defined by F' K' = F- AX' -K' = d£- F-1 • F' • K'.

(1.5)

The A X ' are independent of reparamenterisations in the paprameters £. A similar construction can be made for the covariant derivatives of 4>'. When identifying the fields that can be set to zero, i.e. use the inverse Higgs mechanism, we must not only maintain the G invariance of equation (1.2) and

191

also reparameterisation invariance. In effect, this means setting only those covariant derivatives of the Goldstone fields in equation (1.5) that transform in a covariant manner under /i 0 to zero. The equations of motion, or action, are to be constructed from the vielbein and the covariant derivatives of the Goldstone fields that remain. In this way one can find a formulation of brane dynamics that is reparameterisation invariant and also invariant under the rigid G_ transformations of equation (1.2). From this approach we can recover the more conventional approach by using the reparameterisation invariance to choose static gauge, i.e. X = £ for those coordinates that lie in the brane directions. 2

Bosonic Branes

In this section we will show that the dynamics of bosonic p-branes in a flat background in D dimensional space-time arises as a non-linear realization in the sense of the previous section. We take G = ISO(D — 1,1) with K_ = {Pn} and H_ = {Jnm} where n,m = 0 , 1 . . . ,D — 1 and G = ISO(p,l) with K = {Pn} and H — {Jnm,Jn'm'} where n,m = 0 , 1 , . . . , p + 1 and n',m' = p + 1,... ,D — 1. This is to be expected as the presence of the p-brane clearly breaks the background space-time group ISO(D — 1,1) to ISO(p, 1) x SO{D -p-1). The Lie algebra of ISO{D - 1,1) is given by {.Jnrni Jpq\ ~

Vnp^mq

~ VrnQ^np

' Vnq^mp

* Vmp^nq

•>

yZ.l)

We can write the coset representatives in the form g(X,<j>) = exp(XnPn

+ Xn'Pn,)exp{cj>nm'

= expiX^PjJexpW

Jnm.)

• J) .

(2.4)

We distinguish X from X' by the indices they carry, in other words the prime on the X is understood to be present and we just write Xn'. We also drop the prime on

+ Qnm'Jnm,

+ wn'm' Jn.m, + wnmJnm

(2.5)

which we may express as g"ldg = exp(-cf) • J)(dX^Pn)exp((t> • J) + exp(- • J)dexp( • J) .

(2.6)

192

A straightforward calculation shows that en = dX^^rrJ1 = -2nm'dXml + dXn + ... , / " ' = dXm^nkn'

= -2
nnm' wn'm'

=

(^n'^m1 _ ^

= d(j>nm' ,

^ ^

^

=

(0P'» d( £ p ™ _ ( n <_>. m ) ) ,

(2.7)

where S ^ i s defined by exp(-(t>-J)Pnexp((t>-J) = Qn-Pm = P„ + 2 0 I f P 2 L + . . . and + . . . means higher order terms in <^im-. Under a group transformation, gog(X, <j>) = g(X, (j))ho, the Cartan forms transform in accord with equation (1.4). The effect of taking P„ transformations in go is to simply to shift the X- while the Cartan forms are left invariant. Writing h0 = 1 + rnm Jnm, we find that en = en - 2eprpn,

fn' = / " ' , finm' = ftnm' + 2rpnQp

wnm = wnm - 2{rnpwpm

m

' - 2r pTO 'fi p " , (2.8)

- ( n o m)) + drnm ,

(2.9)

to lowest order in rnm. Similar results hold for Jn'm'- The fields e™ and fn transform as expected under the Lorentz group SO(p, 1) x SO(D — p — 1). Clearly, we can set fn = 0 and preserve SO(l,D-l) and reparameterisation symmetry. At the linearized level, examining equation (2.7) we find it implies that dXn' = 2n'mdXm or ^ m-

dXm

dp

•

V-™)

If we choose static gauge this equation becomes, / Wn

r)Xn' = Q^ •

(2-10)

While solving for / " = 0 to all orders may be complicated it is clear that its content is to solve for 4>nm in terms of dmXn . What is really of interest to us is the non-linear form of e n once we have solved this constraint/" = 0 . Examining equation (2.7) we find that the vector f— = (e n , fn ) is related by a Lorentz transformation to the vector (dXn,dXn ) = dX-. As such, epnr]nmeqm = dvX*dqX*%aBl

(2.11)

193

since / " = 0 . The above expression is invariant under go transformations. A worldvolume reparameterisation and group invariant action is therefore given by f dp£ det{e) = f d?ZsJ-dettdpX^X^nrn)

.

(2.12)

In other words, the well known generalisation of the Nambu action for the string to a p-brane follows in a straightforward consequence of taking the nonlinear realization of ISO(D — 1,1) with subgroup SO(p, 1) x SO(D — p - 1). Clearly, had we not included the Lorentz group in our coset and introduced the corresponding Goldstone bosons the veilbein on the brane would have been trivial and we would not have found the above action.

194

CALCULATION OF BOSONIC M A T T E R FIELDS O N A N iV-SPHERE F.L. WILLIAMS Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA E-mail: [email protected] We solve the spectral problem and the Klein-Gordon equation for a massive and massless field theory on an n-sphere, using a theory of hypergeometric equations developed by A. Nikiforov and V. Uvarov.

1

Introduction

In the paper, 1 the authors consider among other issues the Klein-Gordon equation on a 3-sphere equipped with the spatial section of a Robertson-Walker metric. Explicit eigenvalue computations are given, but the corresponding fields are not explicitly obtained because of difficulties involved in solving a certain radial equation. The purpose of this paper is to indicate how to solve this radial equation not only in the case of a 3-sphere, but in fact for a sphere of arbitrary dimension n > 3. Our method is to apply various changes of variables that eventually transform these radial equations to differential equations of hypergeometric type, in a generalized sense. We then apply a theory developed by Nikiforov and Uvarov2 to solve the latter equations. By this theory we obtain moreover a natural quantization condition (from a single, elegant point of view) which yields the explicit form of the eigenvalues. Thus we obtain explicit formulas for the spectrum and for the corresponding matter fields, especially in the bosonic case. Using the spectral part of these results, one can analyze the path-integral partition function, as is done in 1 for the special case n = 3, and in particular consider its behavior with respect to the spherical radius. We reserve this matter for another occasion. 2

The Laplace-Beltrami operator and the Klein-Gordon operator

To set up the Klein-Gordon equation in a concrete form we need a formula for the Laplace-Beltrami operator in suitable coordinates on an n-sphere X =

195

Sn(a) with radius a > 0, a formula which generalizes formula (2.3) in. 1 n+l +1

X^ix^ixx,...,xn+1)

2

G R" 1 HxH = J2A

=

fl2

}

C2-1)

is assigned the metric induced by the inclusion X —> K n + 1 . We take n > 3 and we choose the following set of coordinates on the domain W = (0, 2TT) X (0,7r) x • • • x (0,7r) x (0, a) where the Cartesian product of (0, IT) with itself is taken n — 2 times: xi = r sin # n _i sin 6n-2 • • • sin #2 sin #i , X2 = r sin # n _i sin 0„_2 • • • sin #2 cos 6\ , x3 = r sin #„_! sin 6n-2 • • • sin #3 cos #2 , i

(2-2)

a:„_i = r s i n 0 n _ ! cos# n _ 2 , xn = rcos0 n _x , xn+i

-

y/a2

r2.

-

We will write y = (61,62, • •• , #„_i,r) for the typical element of W. The function h : W -> X given by h(y) — x — [x\,... , xn+i) for the Xj in (2.2) is a diffeomorphism of the open set W in W1 to an open set U in X. The line element on X ds2 = ^2gijdyidyj

(2.3)

is determined by the components g^ of the metric tensor, which in terms of the coordinates (2.2) are given on U by 9jj(Hv)) 9n-i,n-i(h(y))

= r2 s m 2

0j+i ' ' ' s m 2 ^n-i

for 1 < j < n - 2,

2

= r, a2

9nn(h(y)) = -5

with ^y = 0

1 J"

=

1

1 2

fOT K =

(2-4)

~~2 '

for i ^ j .

In particular for n = 3 we write 0 = 6\, 6 — 62, (2/1,2/2,2/3) = (4>,9,r) and obtain in (2.3) ds2 = J2gjjdy2

= r2(sm9 d<j>2 + d62) + — ^ - j ,

(2.5)

196

which is the spatial section of the Robertson-Walker space-time metric, up to a scale factor a2(t) related to the "radius of the universe at time t." For
9 = \9ii\,

(2-6)

the Laplace-Beltrami operator A on the curved space X is given locally by ^/detff f-' dVj =

dVj

n i d d nrr- E a ~ "Sdet 9 9jj d^Vi~ Vdetff jr[ dVj

(27)

(since glj = 0 for j ^ i), say on the open set U = h(W). Using the formulas in (2.4) one computes the expression for A in (2.7) and obtains the following formula. T h e o r e m 2.1 For f a C°° function on the n-sphere X = Sn(a) with radius a>0,

W -» 1 where W d= (0,2n) x (0,7r) x • • • x (0,TT) X (0,a)

let f* = foh: def

as above with h(y) = x = (xi,... ,xn+\) for y = (61,62, ••• ,6n-i,r) and for the Xj given by (2.2), 1 < j < n + 1. Then ,A,x,L/^

r2\d2f*,,

(,

/(n-1)

e W

n r \ 5/* + ^(An_1/')(y)

(2.8)

where n-2

1 jP_ 2 sin fy+i • • • sin 6n-i dO2 2

(j-l)co8gj d_ 2 2 sin 6j sin 0 J + i • • • sin 0n-\ 06j

(n-2)cosgn_1 d sin^! 00„_i

82 d62n_x-

K

-y;

One can also write A„_i =

. n_2„ ^ sin" Sin"-2^!^-! n-2

+

2

(9 n _i " ^n-x ,

V E ^ - Ji_ ,1 . ^ 22 , ^

sin

0,- sin 0 j + 1

19

„2fl2 0 .W^~ *M;• •o;• sin n _i 50,-

(2 10)

-

197 A„_i is in fact the Laplace-Beltrami operator on the (n — l)-sphere Sn unit radius expressed in terms of the coordinates X\ = sin#„_i sin# n _2 • • -sin62 sin#i , X2 — sin On-i sin 0 n -2 • • • sin 62 cos #i , X3 = sin 9n-\ sin Qn-v • • • sin 63 cos 62 , .

* of

(2-H)

J„_i = sin# n _icos0 n _ 2 , xn = cos#„_i onSn~l. For a scalar field of mass m we take Ld=-A

+ m2

(2.12)

as the Klein-Gordon operator, and we consider the Klein-Gordon equation L(j) = \<j>

(2.13)

for a bosonic field ^ o n l . It is natural to propose a separation of variables (h(6u ... ,0„_i,r)) = R(r)Z(Ou... Setting 6 = (81,...

,en-X).

(2.14)

,0 n _i), y = (0,r), we obtain by Theorem 2.1

(A0(My)) = ( l - J ) R"(r)Z(6)

+ (

^

- ^

R'(r)Z(6)

+ ^ (z A n _ i Z ) ( 0 ) . (2.15) r On the other hand let Pi be a spherical harmonic of degree £; ie. Pe is the restriction to 5 n _ 1 of a harmonic, homogeneous polynomial on Rn of degree £, where I > 0 is an integer. Then for the choice of Z given by Z(6U...

,0„_i) d = P / ( i i , . . . ,xn)

(2.16)

for the i j in (2.11) we have A n _ i Z = -£{£ + n -2)Z. The sequence {-(.{£ + n - 2)}^>0 actually provides for all of the eigenvalues of A n _ i . We deduce from (2.15) that (Acf>)(h(y)) = ( l - J ) R"(r)Z(9)

+ (~^

- ^ )

R'(r)Z(9)

\^zAR{r)m.

(2,7)

198 That is, equations (2.13), (2.14) mean that the function R(r) is subject to the differential equation 1

R"(r) +

n-

1

nr R'(r) ~75 £(e + n»

2)

2

'-+m2 R(r) = -XR(r)

(2.18)

on (0, a), where t > 0 is an integer. If we define p(x) = R(xa)

on (0,1)

(2.19)

then equation (2.18) is transformed to the equation (l-x')p"{x)

+

In-

1

+ n - 2) 2 ,' p(x) —= + mV

— nx p'(x)

= -\a2p(x)

(2.20)

1

on (0,1), which for n = 3 is exactly the radial equation (2.5) of. It is clear now that the main point is to obtain solutions of equation (2.20). As indicated in the introduction this can be done by a series of changes of variables. These will ultimately transform (2.20) to a generalized type of hypergeometric equation—namely to an equation that has the form a2(x)$"{x)

+

CT(X)T(X)&(X)

+ a(x)$(x)

=0

(2.21)

where cr(x), a (a;) are polynomials in x of degree at most two and T{X) is a polynomial in x of degree at most one. A beautiful theory exists for solving such an equation. 3

Solutions of the radial equation (2.20)

Towards the transformation of equation (2.20) to an equation of the form (2.21) we write (2.20) as (1 - x')p"(x)

+

fn-1

— nx P'(x) +

^=jp(x)

(3.1)

for x £ (0,1) and for / S ^ -t[t

+

n-2),

Udef 7

^' _ ( A - m 2 ) a 2 .

(3.2)

on (0,1),

(3.3)

If we set y(x) = xp(x)

199 then (3.1) is transformed to the following equation for y(x):

(l-x2)y"(x)

+

n —3 + (2 - n)x y'(x) +

C

A

xz

y{x) = 0

(3.4)

for x £ (0,1) and for C = f / ? - n + 3,

(3.5)

A = n - 2 - 7.

Next for t £ (0,7r/2), since x = sin* € (0,1), we can set v(t) =

(3.6)

y(smt).

The equation (3.4) for y(x) is then transformed to the equation v"(t) + (n-3)

(cot t)v'(t) +

C + A v(t) = 0 sin 2 i

(3.7)

on (0,7r/2) for v{t). Finally set $(r) = u(cot _ 1 r)

(3.8)

for r > 0

-1

where we note that c o t r 6 (0,7r/2) for r > 0. Then equation (3.7) yields the following equation for <3?(r): a 2 (r)$"(r) + cr(r)T(r)*'(r) + ^ ( r ) * ^ ) = 0

(3.9)

where cr(r) = 1 + r ,

r(r) = (5 — n)r, 2

?(r) = [ C ( l + r ) + A]

(3.10)

= [(/? - n + 3)(1 + r 2 ) + n - 2 - 7]

by def. (3.2), (3.5)

= [(-£(* + n - 2) - n + 3)(1 + r 2 ) + n - 2 - 7]. Thus by (3.9), (3.10) equation (2.20) (or {32)) is indeed transformed to an equation of the form (2.21) of hypergeometric type. One is now in a nice position to apply a theory of Nikiforov and Uvarov2 which provide for solutions of (2.21) (and of (3.9) in particular) under a natural integrality condition, which we interpret as a quantization condition. Details of this theory can also be found in Chapter 4 of.3 There are actually two integrality conditions; these arise according to sign choices ± . They are namely def

-y = \P = (p + t)(p + e + def

(3.11)

n-l),

7 = \> ^ (p - £){p -e + 3-n)

+

(2-n)

(3.12)

200

where p = 0,1,2,3, — For physical reasons we focus only on the first condition (3.11). Theorem 3.1 Define *p(r)

= (1 + r 2 ) ( " + ' + n - 2 ) / 2 i ^ ( l + r 3)-«+<»-i)/a)

(3.13)

for r > 0, p = 0 , 1 , 2 , 3 , . . . . Then for - 7 = Ap in (3.11), ie. for A = (p + e)(p + £ + n- l)/a2 + m2 (by (3.2)), $ p (r) is a solution of equation (3.9). A proof of Theorem 3.1, based on the theory developed in, 2 is given in.4 Working backwards through the transformations (3.8), (3.6), and (3.3) we obtain by Theorem 3.1 the following main result. Theorem 3.2 For p = 0 , 1 , 2 , 3 , . . . define pp on (0,1) by

(

(314)

* *>^*'(^J for $ p given in (3.13). Then for - 7 = Ap in (3.11), ie. A = (P + *)(P + < + n - 1 )

+

^

^

pp(x) is a solution of equation (2.20) on (0,1). Hence by (2.19) / ^ ( r ) ^f -r * p (Va\~r2]

,

0 < r < a,

(3.16)

is a solution of equation (2.18). Using spherical harmonics Pi as in (2.16) and the solutions Rp in (3.16) we obtain desired solutions 4> = <j>pt and eigenvalues A of the Klein-Gordon equation (2.13): pt(rxi,... ,rxn,\/a? A=

-r2)

= pi{h(6i,... , 0 n _ i , r ) ) = Rp(r)Pe(xi,...

(p + e)(p + t + n-l) 5 a2

,

,xn),

2

1- m , (3.17)

as in (2.14); see (2.11). Also see Appendix for a tabulation of the functions $„ in (3.16) for 1 < p < 10. For p = 0 we have $„(r) = (1 + r 2 ) - ( f + 1 > / 2 = > Po(x) = xl =>• i? 0 (r) = (r/a)e. We conclude with remarks on the quantization condition (3.11) which is central for Theorems 3.1, 3.2. The generalized hypergeometric equation (2.21) can be reduced to a simpler form a{x)w" + T(X)W'{X)

+ nw{x) = 0

(3.18)

201

as shown in, 2 which we call a canonical form. Here r(x) is a polynomial in x of degree at most one and \i is a scalar. This can be done in such a way that solutions $ of (2.21) can be obtained from those of (3.18): $ = , like r and fi, is constructed from the data a, a, f. Since degree a < 2 and degree r < 1 \ d-£f , Xp = -pr is a scalar for p = 0,1,2,3, —

P(P ~ 1) _„ a

/„ 1Q x (3.19)

Consider the remarkable condition H = Ap.

(3.20)

Given (3.20) one can construct a polynomial solution wp of (3.18) with degree wp < p. Then by (i), $ p = <j>wp is a corresponding solution of (2.21). In the present case of equations (3.9), (3.10) (a special case of (2.21)) it is shown in4 that ti = A +

C+-^--

(n-3Y

+A

*/*

(3.21)

where A, C are given in (3.5). Then condition (3.20) is shown to reduce to condition (3.11), for certain sign choices, and one is thus lead to the solutions $ p in Theorem 3.1. For alternate choices of signs one obtains condition (3.12). The integrality condition (3.20) is not only sufficient for the existence of non-zero solutions w of (3.18) as we have noted but, conversely, it is also necessary, assuming that w is subject to an appropriate growth condition at infinity. Condition (3.20), which has lead to the form of A in (3.17), also provides for the quantization of energy in quantum mechanics—including the explicit computation of energy levels from a single, elegant point of view. 2 ' 3 Acknowledgments The author extends special thanks to Volker Ecke for his skillful preparation of the manuscript, and in particular for setting up a Mathematica Program that explicates the first ten functions $ p in the Appendix for the radial solutions RP(r). Appendix Table 1 provides a tabulation of the functions $ p in (3.16) for 1 < p < 10, as defined in equation (3.13) of Theorem 3.1.

202 Table 1. Functions $ p for 1 < p < 10.

*p(r)

- l + 2£ + n ) r ( l + r 2 ) " ~ 3 ) - l + 2£ + n ) ( l + r 2 ) ^ ( - 1 + (2£ + n) r2) - 1 + 2£ + n) (1 + 2£ + n) r (1 + r 2 ) ~ 2 ~ 5 (_3 + (2£ + n ) r 2 )) - 1 + 2£ + n) (1 + 21 + n) (1 + r 2 ) ^ ~ 3 + (2 + 2£ + n) r 2 ( - 6 + (2£ + n) r 2 )) - l + 2£ + n ) ( l + 2£ + n ) ( 3 + 2£ + n ) r ( l + r 2 ) ~ 3 ~ 5 15 + (2 + 2 £ + n) r2 (-10 + (2£ + n) r 2 ))) - 1 + 2£ + n) (1 + 2£ + n) (3 + 2£ + n) (1 + r 2

- 1 5 + (4 + 2£ + n) r (45 + (2 + 2£ + n)r

2

2

) ^

(-15 + (2£ + n) r 2 )))

- 1 + 2 £ + n) (1 + 2 £ + n) (3 + 2 £ + n) (5 + 2 £ + n) r (1 +

r2)"4"^

-105 + (4 + 2£ + n) r2 (105 + (2 + 2£ + n) r 2 (-21 + (2£ + n) r2)))) - 1 + 2£ + n) (1 + 2£ + n) (3 + 2£ + n) (5 + 2£ + n) (1 + r 2 ) " ^ 105 + (6 + 2 £ + n) r 2 (-420 + (4 + 2 £ + n) r 2 (210 + (2 + 2 £ + n) r2 - 2 8 + ( 2 £ + n)r 2 )))) - l + 2£ + n ) ( l + 2£ + n ) ( 3 + 2£ + n)(5 + 2£ + n)(7 + 2£ + n ) r 1 + ' T - 2 ) _ 5 ~ * (945 + (6 + 2£ + n) r 2 (-1260 + (4 + 2£ + n) r2

378 + (2 + 2£ + n).r 2 (-36 + (2£ + n) r2))))) _l

+ 2£

+r

2

+ n ) ( l + 2£ + n ) ( 3 + 2£ + n)(5 + 2£ + n)(7 + 2£ + n)

) ^ ^ (-945 + (8 + 2£ + n) r 2 (4725 + (6 + 2£ + n) r2 (-3150

+ (4 + 2£ + n) r 2 (630 + (2 + 2£ + n) r2 (-45 + (2£ + n) r 2 )))))

203

References 1. V.B. Bezerra, E.R. Bezerra De Mello, and A.C.V.V. De Siqueira, Bosonic and Fermionic Partition Functions on S3, Modern Physics Letters A, Vol. 13, No. 11, 899-909 (1998). 2. A. Nikiforov and V. Uvarov, Special Functions of Mathematical Physics, A Unified Introduction with Applications, Translated from the Russian by Ralph Boas, Birkhauser, Basel, Boston, 1988. 3. F. Williams, Notes on Quantum Mechanics: An Introduction for Mathematicians, Manuscript submitted for publication to the Am. Math. Society. 4. F. Williams, Remarks on the Klein-Gordon Equation in a RobertsonWalker Universe, Manuscript to be submitted for publication to the International J. of Differential Equations and Applications.

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