FROM FIELD THEORY TO
QUANTUM GROUPS Birthday volume dedicated to
Jerzy Lukierski
Editors
B Jancewicz
J Sobczyk Institute ofTheoretical Physics University of Wroclaw, Poland
,IIit World Scientific
Singapore • New Jersey· London • Hong Kong
JERZY LUKIERSKI
Contents
.Jerzy Lukierski - A Scientist and Teacher
xi
List of Publications by Jerzy Lukierski
xv
Chapter I. Quantum Groups: General Formalism
Contractions, Hopf Algebra Extensions and Covariant Differential Calculus ..... ....... .... ............ .......... ........... ... ... J. A. de Azcarraga and J. C. Perez Bueno
3
The Linear Difference Derivatives and Some Q-Special Functions M. Klimek
29
The K-Weyl Group and Its Algebra P. Kosinski and P. Maslanka
41
Rational Solutions of Yang-Baxter Equation and Deformation of Yangians . . .... . . .. . . . . ... ... . . . . . . . .. .. . . . . .... . .. . . . . .. .. S. M. Khoroshkin, A. A. Stolin and V. N. Tolstoy
53
Chapter II. Quantum Groups: Applications
Large N Matrix Models and q-Deformed Quantum Field Theories .. ......... ........... ......... ........... ........... ....... I. Ya. Aref'eva
79
99
Quantum Group Covariant Systems M. Chaichian and P. P. Kulish The Renormalization Group Method and Quantum Groups M. A. Martin-Delgado and G. Sierra
113
Remarks on the Quasi-Bialgebra Structure in Quantum Mechanics A. Nowicki
141
Chapter III. Supersymmetry
Lagrangian Models of Particles with Spin: The First Seventy Years A. Frydryszak ix
151
x
Contents
D = 1 Supergravity and Spinning Particles J. W. van Holten
Superalgebra Structures on 50(3) Tensor Operator Sets P. Minnaert and M. Mozrzymas The Supersymmetric SL(2,C) Kac-Moody Algebra and the Supersymmetric Korteweg-De Vries Equation Z. Popowicz
.
Chapter IV. Miscellanea The Group of Diffeomorphisms and Its Cnitary Reali7:ation in QFT Z. Haba
.
Chiral Systems on Group Manifolds Z. Ilasiewicz Quantum Potential and Quantum Gravity J. Kowalski-Glikman The General Form of the Lagrange Function for Classical Two Particle Equations of Motion Covariant under Galilei Transformations J. Lopuszanski and P. C. Stichel Greensite-Halpern Stabilization of A k Singularities in the N ---> 00 Limit J. Maeder- and W. Riihl Some Aspects of Soliton Unwindings B. Piette and W. J. Zakrzewski BV-Algebras of W-Strings K. Pilch
.
.
Xl
Jerzy Lukierski
a scientist and a teacher
At May 21, 1996 Jerzy Lukierski, ordinary professor at the Institute of Theoretical Physics of the University of Wroclaw, will be sixty years old. !lis friends and colleagues dedicate this volume as their homage to him. As far as I am concerned I know Jerzy Lukierski since approximately 40 years; the first time I met him when he was still a student of the University. He impressed me in the course of years when our friendship grew by his nonstandard and straightforward behaviour and warm personality. Jerzy Lukierski got his Ph.D. at our University in May 1961 for his dissertation Theory of spin ~ fields in the isotopic Minkowski space accomplished under the guidance of Professor Jan Rzewuski. He got his habilitation degree in 1967 also at University of Wroclaw for his work Relativistic multiple poles in quantum field theory. He became extraordinary professor at our University in 1974 and ordinary professor in 1983. Professor Lukierski is an outstanding scientist, well known specialist in the theory of fields as well as the group-theoretical and geometrical foundations of the theory of fundamental interactions. His work is characterized by original, brilliant ideas, close links with the up-to-date scientific developments and the importance of considered problems. The most significant achievement in his scientific career is - in my opinion the first in scientific literature model of quantum deformations of the Poincare algebra, invented by him and his collaborators in 1991-92. For this achievement he got the award of the Polish Minister of National Education in 19!H and the Maria Sklodowska-Curie award in 1995. At present Lukierski is in the first line of scientists working in these topics and has been leading a large international group of researchers. This group has been exploring the consequences of the theory of quantum groups in describing the four-dimensional symmetries of space-time as well as in dynamical physical theories, i.e. classical mechanics, quantum mechanics and field theory. Hut also his earlier results carry a considerable scientific weight. Let me mention some of them, the most important ones. I. In his work in 1967-77 Lukierski investigated the description of the four-dimensional Green functions in the quantum field theory [50-54, 56] and the formulation of the renormalization group in terms of quantized field operators [76, 81 83]. He considered the distributional structure of Green functions in the equal time limit and explored the behaviour of the commutator of the renormalized fields for small intervals of time. These results were exploited in the definition of the renormalization constants as singularities of the Grecn functions. Also the transformations of renormalization group described by Callan-Symanzik equation were given by him in terms of the
xii renormalized field operators. 2. In the same period of time, 1967 -78, Lukierski proposed the general theory [48, 49, 60, 79] of non-stable particles and resonances in quantum field theory and explored their models [62, 63, 70, 79, 90]. He gave also the generalized formalism of the scattering matrix for resonances and interacting multi particle states [73]. 3. In his later work, Lukierski's interests are centered upon problems of supersymmetric field theory, namely: i) In 1978 (in collaboration with V. Rittenberg) he formulated the so called coloured supersymmetry [89] which allows the description of the internal symmetry being the product of flavour and colour symmetries. ii) In 1979-83 he presented new field theory models describing composite supersymmetric gauge fields [97, 98, 101], composite gravity as well as supergravity [123, 126, 138]. In the papers [97, 101], the models of composite gluons and quarks in supersymmetric chromodynamics were given. iii) Several papers in 1981- 88 were concerned with models of supersymmetric particles and contained the prescription of the first and second quantization of these models. In particular, Lukierski, together with J .A. de Azcarraga, presented in 1981 for the first time the covariant model of a superparticle with nonzero mass (118] and in 1988 the first model of a particle with double supersymmetry (the spinning superparticle) [155, 157, 163, 169, 170]. Both these ideas are quoted and pursued in the world literature up to present time. 4. J. Lukierski with A. Nowicki generalized the supersymmetric formalism of Penrose's twistors to the case of N-extended supersymmetry [149, 156, 178] and gave the full classification of the supersymmetric extensions of the Korteweg-de Vries equations [154] (for N = 1,2,3,4; for N = 2 inspection of new terms, for N = 3,4 derivation of new equations). 5. The consecutive work of Lukierski develops a new method of quantization of cohomological complex constraints [157, 162, 175]. It is a variant of BRST (Becchi, Rouet, Stora and Tyutin) formalism describing the generalized quantization of Gupta and Hleuler with hoJolllorphic constraints. Recently (1990-95) the interest of Lukierski is focused upon the so-called quantum groups. Here arc the most important - in my opinion results of Jerzy Lukierski. i) There was given a system of quantum-deformed (q-defonned) boson as well as fermion creation and annihilation operators covariant under the quantum supergroup. The description of quantum deformations of the supersymmetric oscillator was obtained. This work has been written in collaboration with P. Kulish and M. Chaichian [166, 173]. ii) The first model of quantum four-dimensional Poincare algebra with the structure of the noncocommutative Hopf algebra was given in 1991. The representation theory of this model was investigated and the first attempts to provide applications
XIII
were presented (modification of the relativistic kinematics, deformed Klein-Gordon and Dirac equations, corrections to the Lamb shift as well as to the anomalous magnetic moment g-2, etc.). These results were obtained in collaboration with H. Ruegg, A. :'-lowicki, W. Riihl and V. Tolstoy [176, 179, 183, 195, 202J. iii) Two types of quantum deformations of the four-dimensional conformal algebra were proposed: first one in collaboration with A. Nowicki obtained in 1992 [185J and second very recently in collaboration with P. Minnaert and M. Mozrzymas [223J. iv) There was obtained first in the literature quantum deformation of the four-dimcnsional Poincare supcralgebra (collaboration with A. Nowicki and J. Sobczyk [196]) as well as four-dimensional Poincare supergroup (collaboration with P. Kosillski, P. MaSlanka and J. Sobczyk [200, 209, 211]). When looking at this list of recent achievements the results quoted in ii) seem to me to be of particular importance as far as the possible physical implications are concerned. If one replaces the standard classical relativistic symmetries by the corresponding quantum ones (described by the so called quantum Lie algebras and quantum groups) one gets quite new fundamental approach to the problem of spacetime structure at small distances. It appears that in this new approach a new fundamental constant with dimension of mass (or length) is added to the well known two constants: h (Planck constant) and c (the light velocity). This constant should control the physical events at small distances where the modification caused by quantnm geometry becomes essential. In the theory of deformed relativistic symmetries proposed by Lukierski and his collaborators such a constant appears naturally as a fundamental mass parameter 11:. It should be stressed that from physical point of view the appearance of this parameter II: distinguishes favourably the above mentioned theory from other approaches to the quantum deformations of the space-time symmetries. Besides, only for this type of deformation with the fundamental parameter 11:, it is possible to introduce such a deformation of the relativistic symmetry which leaves the three-dimcnsional nonrelativistic symmetries classical (undeformed). It should be cmphasized that at present some comparisons with the experimental data were performcd and the most rcstrictive estimates imply that the values II: > 10 12 GeV are admitted. This estimate means that the K-modification of the relativistic symmetries is permitted at the distances r < 10- 26 cm (a natural assumption seems to be that II: ::::: Planck mass::::: 10 19 GeV which corresponds to r ::::: 10- 33 cm). The papers of Lukierski, especially the last ones, caused considerable interest among specialists in the whole world. This was testified by the growing number of quotations in the leading intcrnational scientific journals as well as by many invitations to significant conferences dcvoted to the topics pursued by Lukierski. To give an example, Lukierski already in 1978 was invited to the Rochester conference in Tokyo as a chairman of the session named "New Developments". lie was also invited to many plenary talks at the conferences - mostly on applications of group-theoretic and differential-geometric methods in physics. The importance of the scientific results
XIV
of Lukierski is also affirmed by the fact that the quantum deformations of relativistic symmetries with a fundamental mass parameter are at present the subject of investigation in many other research centers in Poland (L6di, Warsaw, Cracow) as well as abroad. In the last 15 years Lukierski led a vast collaboration with well-known scientists abroad. His collaborators were e.g. .l.A. de Azcarraga, S. Aoyama, M. Chaichian, W. Heidenreich, A. Isaev, P. Kulish, P. Minnaert, P. Presnajder, V. Rittenberg, H. Ruegg, W. Ruhl, V. Tolstoy, P. Vindel and J. van Holten. Nowadays Lukierski plays a leading conceptual role in a considerably big international team working on quantum group problems. He is author or co-author of 223 original contributions published mostly in outstanding scientific journals (Physical Review D, Physics Letters 13, Nuclear Physics 13, Journal of Mathematical Physics, A nnals of Physics, Letters in Mathematical Physics, Nuovo Cimento, Journal of Physics A etc.) as well as in the proceedings of international conferences, schools and workshops (see the attached list of publications) . .lerzy Lukierski was a considerable man as far as the teaching process is concerned. Under his guidance 14 young scientists got their Ph.D. His chair of Elementary Particles and High Energy Physics, comprising mostly his former or actual co-workers and pupils, is the most active scientific group in the Institute of Theoretical Physics of our University. Lukierski was director of three International Winter Schools of Theoretical Physics in Karpacz. The last one, in 1994, was named "Quantum Groups: Formalism and Applications" and was definitely a scientific success. In 1977-90 Lukierski was scientific deputy director of the Institute of Theoretical Physics of our University. He supervised also for many years the postgraduate studies at our Institute. Since September 1990 he is a director of the Institute. Approaching sixty, he is a scientist of plenty of energy and new ideas. His moral standards are high. In private life, Jerzy is happily married with Elibieta, talented artist (26 years younger than him), and a father of two their children: Robert and Ursula. He is always friendly, frank and open towards other people, in particular towards younger ones who need his advice. I wish him a happy birthday. Jan LopuszaTiski Institute of Theoretical Physics University of Wroclaw
LIST OF PUBLICATIONS by JERZY LUKIERSKI
1. J. Lukierski, On a Full Gcomctrization of Conservation Laws in Gurseys Formalism, Nuovo Cim. 13,410 414 (1959)
2. J. Lukierski, On the Interaction of Isovector Components,
Bull. Acad. Polan. Sci. 7,577--581 (1959)
3. J. Lukierski, Two Quantum Interaction in Relativistic Helium Atom, Zcsz. Nauk. Univ. Seria Mat. Fiz. Chern., 155-178 (1959) 4. J. Lukierski,
The Generalization of Dirac Equation I, Acta Phys. PoloIl. 19,499-513 (1960)
5. J. Lukierski, The Global Symmetry in Weak Interaction, Bull. Acad. Polan. Sci. 8, 803 806 (1960)
6. J. Lukierski, The Isorepresentation of LeptolJs, Bull. Acad. PaloIl. Sci.IlI, 8, 553 557 (1960)
7. J. Lukierski, On the AntiIinearizatiolJ of Klein-Gordon Equation for 4-Spinors,
Bull. Acad. Polan. Sci. 9,317-321 (1961)
List of publications
XVI
8. J. Lukierski, The Generalization of Dirac Equation II,
Acta Phys. Polan. Sci. 20,517-535 (1961)
9. J. Lukierski, The Spinor Space as an Euclidean Complex Space,
Acta Phys. Polon. Sci. 20, 945-967 (1961) 10. J.Lukierski, The Electric Charge as lsovector, Acta Univ. Wratisl. 12,87-91 (1962)
II. J. Lukierski, The Mass as lsovector,
Acta Univ. Wratisl. 12,77-86 (1962) 12. J. Lukierski, The Natural Coordinate System in Minkowski's Isospace for 1/2 Spin Particle,
Progr. of Theor. Phys. 27, 262-272 (1962) 13. J. Lukierski, Gauge Transformation in Quantum Field Theory,
Nuovo Cim. 29,564-568 (1963) 14. J. Lukierski,
On the Interaction of Lepton Fields with Vector Bosons, Acta Univ. Waritsl. 17, 89100 (1963) 15. J. Lukierski, The lnvariance of Free Baryon Lagrangeans with Respect to the Minkowski [sogroup, Acta Univ. Wratisl. 17,101-109 (1963)
XVII
16. J. Lukierski, Gauge Properties of Propagators in the Quantum Theory of Non-Abelian Vector Gauge Fields, Matscience Institute, Madras 1964
17. J. Lukierski, The Quantum Variables and the Proper Choice of the Subsidiary Condition for Non-Abelian Gauge Fields, Matscience Institute Madras 1964
18. J. Lukierski, Renormalizability of Higher Spin Theorics" Nuovo Cim. 38,1407-1411 (1965)
19. J.Lukierski, Covariant Generalization of Planc Solutions of Itterated Wave Equation, Bull. Acad. Polan. Sci. 14,459-462 (1966)
20. J. Lukierski, Degencracy of Physical Vacuum in Vector Meson Thcories with Conservcd Current, Nuovo Cim. 45,93-107 (1966)
21. J. Lukierski, Lagrangcan Formalism for 2J+I-Component Higher Spin Theory, Bull. Acad. Polon. Sci. 14, 697 -700 (1966)
22. J. Lukierski, The Canonical Formalism for Double Pole, Bull. Acad. Polon. Sci. 14,337-343 (1966)
23. J. Lukierski, The Commutation Relations for Doublc Pole Field, Bull. Acad. Polan. Sci. 14, 275-278 (1966)
List of publications
XVlll
24. J. Lukierski, The Electromagnetic Potentials in Landau Gauge, Bull. Acad. Polon. Sci. 14, 569-573 (1966)
25. J. Lukierski, The Generalization of LSZ Asymptotic Condition for Double Pole, Bull. Acad. Polan. Sci. 14,273 (1966)
26. J. Lukierski, The Generalization of Plane Waves for Iterated
J(lein~Gordon
Equation,
Bull. Acad. Polon. Sci. 14, 453-458 (1966)
27. J. Lukierski, The Space of States for Relativistic Double Pole Field, Acta Phys. Polon. 30, 457 (1966)
28. J. Lukierski,
Z3
=
0 and Equal Commutation Relations,
Nuovo Cim. 49,312 315 (1967)
29. J. Lukierski, Complex Double Pole and Decay Law, Bull. Acad. Polan. Sci. 15, 217-222 (1967)
30. J. Lukierski, Complex Double Pole in the Propagator and Spectral Condition, Bull. Acad. Polon. Sci. 15,211-216 (1967)
31. J. Lukierski, The Formulation of QED with Strong Lorentz Condition I Heisenberg Picture, Acta Phys. 1'010n. Sci. 31,63-94 (1967)
32. J. Lukierski, The Operator Formulation of QED in Landau Gauge, Suppl. Nuovo Cim. 5,739 748 (1967)
XIX
33. J. Lukierski, Theory of Free Relativistic Multipole Field 1. Classical Part, Acta Phys. Polan. 32,551 577 (1967)
34. J. Lukierski, Theory of Free Relativistic Multipole Field IJ. Quantum Part, Acta Phys. Polon. 32,771-799 (1967)
35. J. Lukierski, Translational Invariance and Energy-Momentum Conservation Law for Unstable Particles, Nuovo Cim. 47,926- 929 (1967)
36. J. Lukierski, Complex Higher Order Poles and Generalized Unstable Particles, Acta Phys. Polan. Sci. 14,247-255 (1968)
37. J. Lukierski, T. Cukierda, Covariant Theories of Antisymmetric Massive Tensor of Rank Two" Bul\. Acad. Polan. Sci. 16,657-662 (1968)
38. J. Lukierski, T. Cukierda, Equivalence of Two Formulations of Massive Spin One Theory, Nuc\. Phys. B5, 508-514 (1968)
39. J. Lukierski, Logarithmically Divergent Wave Renormalization Constant and ET Canonical Limit, Bul\. Acad. Polon. Sci. 15, 817822 (1968)
40. J. Lukierski, Modification of Solutions in the Froissart Model and MaBsles Second Order Pole, Acta Phys. Polon. 33,969-980 (1968)
xx
List of publications
41. J. Lukierski, Nonuniqueness of the Canonical Formalism in the Presence of Indefinite Metric, Nuovo Cim. 54A, 583-587 (l968)
42. J. Lukierski, On the Gencralization of Haag-Ruelle Theory of Asymptotic Fields, Nuovo Cim. 54A, 500-503 (1968)
43. J. Lukierski, Rcnormalization Fourdimensional Dcrivative Coupling Model, Bull. Acad. Polon. Sci. 16,219-226 (1968)
44. J. Lukierski, The Definition of Creation and Annihilation Operators of Covariant Unstablc Particle, Hull. Acad. Polon. Sci. 16, 343-348 (1968)
45. J. Lukierski, The Gcneralized Scattering Amplitudes for Covariant and Local Unstable Particles, Bull. Acad. Polon. Sci. 16,431-436 (1968)
46. J. Lukierski, T. Cukierda, Thc Quantized Cauchy Data and Particle States for Theories with Higher Order Derivates, Acta Phys. Polon. 34,913-928 (1968)
47. W. Karwowski, J. Lukierski, N. Sznajder, Field Theory Describing Interacting Scattering, Nuovo Cim. SeT. X63, 509-528 (1969)
48. J. Lukierski, W. Karwowski, N. Sznajder, A Field Theory Describing Interacting Two-Particle Subsystems, II. Dcscription of ElaBtic Scattering, Nuovo Cim. 60A, 509 523 (1969)
XXI
49. J. Lukierski, A Field Theory Describing Interacting Two-Particle Subsystems General Formalism, Nllovo Cirn. 60A, 353-374 (1969) 50. J. Lukierski,
Analytic Representations of Two-Point Functions with Non-Canonical Light Cone Singularities I,
J. Math. Phys. 10,918-924 (1969) 51. J. Lukierski,
Causal Propagator as Boundary Value of the Analytic Function in Coordinate Space, Nucl. Phys. BU, 151-159 (1969) 52. J. Lukierski,
Equal Time Commutators for the Renormalized Current in Quantum Electrodynamics, Nuovo Cim. Lett. 1, 259-262 (1969) 53. J. Lukierski,
On the Definition of Equal Time Commutator by Means of Time Ordered Functions, Bull. Acad. Polon. Sci. 17,81-86 (1969) 54. J. Lukierski,
Second Order Polarization and the Analytic Regularization Space, Bull. Acad. Polon. Sci. 17,37-44 (1969) 55. J. Lukierski,
Complex Lee- Wick Poles as Chata, Nuovo Cim. 66, 807 (1970) 56. J. Lukierski,
Equal Time Singularities in Renormalized QFT, Nllovo Cim. 68, 667 (1970)
JI1
Coordinate
List of publications
XXII
,57. J. Lukierski, T. Cukierda, Field--Theoretic Description of Masslcs Particlcs with Highcr Spin and Dcfinite Parity I. Integer Spin,
J. Math. Phys. 11,46 (1970) 58. J. Lukierski, L. Turko, Lagrangian Formulation of Zachariassen Model with a CDD Pole, Nuovo Cim. 66,402 (1970) 59. J. Lukierski, M.Oziewicz, Thc Field Theorctic Description of N-O Sector in Lec Model Using the Field Opcrator with Continuous Spcctrum of Asymptotic Energics, Bull. Acad. Polon. Sci. 18,695 (1970) 60. J. Lukierski, Hilocal Field Operator Describing Arbitrary Two-Particle State by Mcans of One-Particle States,
Nuovo Cim. Lett. 2,784-788 (1971) 61. J. Lukierski, Gcncralization of the Bjorkcn Asymptotic Expansion for Non-intcgcr Asymptotic Powers, Rep. Math. Phys. 2,181-198 (1971) 62. J. Lukierski, L. Turko, Lee Model with Additional Fermi Interaction and thc Composite Nature of V- Particle, Acta Phys. Polon. B2,741 752 (1971)
63. J. Lukierski, M. Oziewicz, Lce Model with V-Particle Having Continuous Mass Spectrum and EnergyConserving V-NO Vertex, Uull. Acad. Polon. Sci. 19,245-248 (1971)
XXIll
64. J. Lukierski, J. Lopuszariski, The Definition of Interacting and Asymptoting Currents in Quantum Field Theory of Currents, Rep. Math. Phys. 1, 265 284 (1971) 65. J. Lukierski, M. Oziewicz, Lee Model with V-Particle Having Mass Spectrum of Asymptotic Masses, Acta Phys. Polan. Sci. B3, 231-246 (1972) 66. J. Lukierski, Field Theory Describing Arbitrary Two-Body Scattering Amplitute as a Horn Term, Fortschr. cler Phys. 21,85112 (1973) 67. J. Lukierski, Sz. Sznajder, The Definition of the Product and its Applications, Rep. Math. Phys. 4, 65-75 (1973) 68. J. Lukierski, Asymptotic Logarythmic Behaviour and the Complex Dimensionality Parameter, Nuovo Cim. 20A, 667 669 (1974) 69. J. Lukierski, Covariant MeIIin Transform and the Group Quantization, Phys. Lett. 53B, 89-92 (1974) 70. J. Lukierski, A. Brzeski, Covariant Wavc Equations for Unstable Particle, Nuovo Cim. Lctt. 9,205-209 (1974)
71. J. Lukierski, W. Sienkiewicz-J!:drzejewska, Gencralizcd Frcc Field and the Representations of Weyl Group, Journ. Math. Phys. 15,344-349 (1974)
xxiv
List of publications
72. J. Lukierski, M. Oziewicz, Lagrangean Formulation of the Relativistic Dynamics Describing an Interacting Two-Particle System,
Bull. Acad. Polan. Sci. B22, 711-718 (1974) 73. J. Lukierski,
On the Scattering Formalism for Interacting Multiparticle System, Nuovo Cim. 23A, 716-735 (1974)
74. J. Lukierski, L. Turko, The Solution of Van Hove Model Describing Nuclear Diffraction for Pions,
Bull. Acad. Polon. Sci. B22, 611-616 (1974) 75. J. Lukierski, W. Sienkiewicz-J ~drzejewska, Generalized Free Fields and the Representations of Weyl of Group 11. Reducible Representations,
Journ. Math. Phys. 16,901-905 (1975) 76. J. Lukierski, A. Ogielski, Global Scale Transformations for Renormalized Field Operators,
Phys. Lett. 58B, 57--61 (1975) 77. J. Lukierski, L. Rytel,
On the Algebraic Consistency Conditions in Canonical Theory, Nuc!. Phys. B85, 311 316 (1975) 78. J. Lukierski, L. Turko, On the Field-Theoretic Formulation of Photon-Hadron Interactions,
Bul!. Acad. Polon. Sci. 23, 623-627 (1975) 79. J. Lukierski, A. Brzeski, Wave For Unstable Particle and Resonances: General Considerations and Soluble Models,
Acta Phys. Polan. Sci. B6, 577-598 (1975)
xxv
80. J. Lukierski, Lagrangean Model of Conformal-Invariant Interacting QFT,
Nuovo Cim. Lett. 16,312-316 (1976)
81. J. Lukierski, A.Ogielski, Henormalization Group and Scale Invariance in Terms of Asymptotic Fields,
Phys. Lett. 64B, 336-340 (1976) 82. J. Lukierski, Renormalization Group and Scale Transformations for Renormalized Field Operators,
Phys. Rev. D14, 3412-3429 (1976) 83. J. Lukierski, A.Ogielski, Scale Transformations of Renormalized Field Operator Discussion of a Soluble Model, Ann. Phys. 100, 196-226 (1976)
84. J. Lukierski, Z. Haba, Stochastic Method in the String Model,
Memorial Volume to 60-th Birthday of Prof. J. Rzewuski, Wroc1aw, 37-51 (1976) 85. J. Lukierski, M.Oziewicz, Relative Time Dependence as Gauge Freedom and Bilocal Models of Hadrons,
Phys. Lett. 69B, 339 (1977) 86. J. Lukierski, Renormalization Group Transformations as Symmetry Mappings,
Fortschr. Phys. 25, 765-788 (1977) 87. J. Lukierski, Z. Haba, Stochastic Description of Extended Hadrons,
Nuovo Cim. 41A, 470--486 (1977)
List of publications
XXVI
88. J. Lukierski,
Superconformal Anomalies, Phys. Lett. 70B, 183-186 (1977) 89. J. Lukierski, V. Rittenberg,
Color-de Sitter and Color-Conformal Superalgebras, Phys. Rev. l8D, 385-:J89 (1978) 90. J. Lukierski,
Complex Mass and Field Operator for Unstable Particles, "Group Theoretical Methods in Physics", Proc. VII Int. Col!., Austin 1978, Lecture Notes in Physics bf 91,256-58 (1979) Springer Verlag
91. J. Lukierski, Complex and Quaternionic Supersymmetry, "Supergravity", Proc. of Stony Brook Workshop, 1979, Eds. Freedman et al., 301 309 (1979) North Holland Compo 92. J. Lukierski,
Fundamental Fermionic Coordinates and Quark Variables Model, Czech. J. Phys. B29, 11-59 (1979) 93. J. Lukierski, W. Sienkiewicz-JI;drzejewska,
Generalized Free Fields Conformal Group and Five-Dimensional QFT, Bul!. Acad. Polon. Sci. 27, 33-10 (1979) 94. J. Lukierski,
Graded Orthosymplectic Geometry and OSp(4:1) Invariant Fermionic sigmamodels, Lett. Math. Phys. 3, 135 (1979) 95. J. Lukierski,
Graded Orthosymplectic Group OSp(8: 1) and Fundamental Fermionic Twistor Variables, Nuovo Cim. Lett. 24,309-315 (1979)
XXVII
96. J. Lukierski, Quaternionic Superspace and the Supersymmetry Extension of Sp(n) Group,
Bull. Acad. Polon. Sci. 27, 24:l 248 (1979) 97. J. Lukierski, B. Milewski, Supersymmetric Dynamics on Pro-QCD Level with Elementary Quarks and Composite Gluon,
Phys. Lett. 93B, 91-94 (1979) 98. J. Lukierski, Composite Gauge Fields and Riemannian Geometry,
"Group Theoretical Methods in Physics", Proc. IX Int. Coil. Cocoyoc, Mexico 1980, Ed. K.B. Wolf, Lecture Notes in Physics, 135, 584-93 (1980) Springer Verlag 99. J. Lukierski, L. RyteI, Extcnded Supcrsymmetry in Five Dimcnsions and its Super Poincare Limit,
Bull. Acad. Polon. Sci. 28,57-61 (1980) 100. J. Lukierski, Fields Operator for Unstable Particles and Complex Mass Description in Local QFT,
Fortschr. der Phys. 28,259-271 (1980) 101. J. Lukierski, Four Dimensional Quaternionic o--models, "Field-Theoretic Methods in Particle Physics", Proc. of Kaiserslautern Summer School of Physics, Ed. W. Ruhl, 361--371 (1980) Springer Verlag 102. J. Lukierski, Quarks and Fermionic Geometry in Hadronic Matter at Extreme Encrgy DCIlsity,
"Hadronic Matter at Extreme Energy Density", Proc. of Workshop Erice, Italy 1978, Ed. 1.\'. Cabibbo and L. Sertorio, 18, 7-199 (1980) Plenum Press
xxviii
List of publications
103. J. Lukierski, Quaternionic Superspaces and Supersymmetric Extensions of Quaternionic Groups,
"Teordiko-gruppovyc metody v fizike", Proc. Int. Sem. Zvenigorod 1979, I, 76-82 (1980) Nauka, Moskva
104. J. Lukierski, Quaternionic and Supersymmetric (J"-models in Differential Geometric Methods in Mathematical Physics,
"Differential Geometrical Mcthods in Mathematical Physics", Proc. of the ConL at Aix--cn--Provcllce and Salamanca 1979, Ed. P.L. Garcia et al., Lecture :'-Iotes in Mathematics ,836,221-245 (1980) Springer Verlag
105. J. Lukierski, L. Rytel, Renormalized Operator Form of Quantum Action Principle, Ann. of Phys. 124,282-400 (1980)
106. J. Lukierski, Superco/J[orl11al Group and Curved Fermionic Twistor Space,
J. Math. Phys. 21,561-67 (1980)
107. J. Lukierski, Supersymmctric Generalization o[ Riemannian Symmetric Pairs, "Group Theoretical Methods in Physics", Proc. IX Int. CoIl. Cocoyoc, Mexico 1980, Ed. K.R. Wolf, Lecture Notes in Physics, 135, 580-583 (1980) Springer Verlag
108. J. Lukierski, Supersymmetric Pre-QCD Dynamics, "Unification of Fundamental Paticle Interactions", Ettore Majorana Int. Scicnce Serics, VII, Eds. J. Ellis et al., 701-710 (1980) Plenum Press
109. J. Lukierski, B. Milewski, Dynamical Gauge Fidds in Four Dimensions [rom sigma-models, Phys. Lett. 100B, 321 -326 (1981)
XXIX
110. J. Lukierski, L. Rytel, Extcnded Supcrsymmetry in Five Dimensions and its Supcr- Poincare Limit,
Bull. Acad. Polon. Sci. Phys. Astr. 28,57-61 (1981) Ill. J. Lukierski, Supersymmetric IT-models and Composite Yang Mills Theory,
"Developments in the Theory of Fundamental Interactions", XVII Karpacz Winter School on Theoretical Physics, 1980 , Ed. L. Turko,3, 189-212 (1981) Harwood Acad. Pub\. 112. A. Frydryszak, J. Lukierski, N=2 Massive Matter Multiplet from Quantization of Extended Classical Mechanics,
Phys. Lett. 117B, 51-56 (1982) 113. A.K. Kwasniewski, J. Lukierski, Functional lntegration around Onc-instanton Solutions llP(1) sigma-model,
III
Four Dimensional
Nuovo Cim. 70a, 371-378 (1982) 114. J. Lukierski, A. Din, W. Zakrzewski, Classical Solutions of Two-dimensional Model with Interacting Bosons and Fermions,
Nllcl. Phys. B194, 157-171 (1982) 115. J. Lukierski, L. Rytel, Geometric Origin of Central Chargcs,
Journ. of Phys. A15, L215-L220 (1982) 116. J. Lukierski, Nonlinear Realizations of Extended Supersymmetrics with Central Charges,
Czech. Journ. of Phys. B32, 504--520 (1982) 117. J.Lukierski, A.Nowicki, Sllperspillors and Graded Lorcntz Groups in Three, Four and Five Dimensions,
Fortschr. der Phys. 30,75-98 (1982)
List of publications
xxx
118. J. Lukierski, J .A. de Azcarraga, S'upersymmetric Particles with Internal Symmetries and Central Charges, Phys. Lett. 113B, 170174 (1982)
119. Z. Hasiewicz, J. Lukierski, P. Morawiec, Seven-Dimcnsional De Sitter and Six-Dimcnsional Conformal S'upcrsymmctries., Phys. Lett. 130B, .1S--60 (1983)
120. B. Jancewicz, J. Lukierski, Editors of "Qauntum Theory of Particles and Fields, Birthday Volume Dedicated to Jan Lopuszanski, pp. 262 (1983) World Scientific Pub!., Singapore
121. J. Lukierski, M. Nowotnik, Analytic Reprcsentations of Global and Local Supersymmetry, Phys. Lett. 125B,152 456 (1983)
122. J. Lukierski, Classical Mechanics in Supcrspace and its Quantization, "Supersymmetryand Supergravity", XIX Karpacz Winter School on Theoretical Physics, 1983 ,Ed. B. Milewski, 4.16-465 (1983) World Scientific Pub!., Singapore
123. J. Lukierski, Composite Gravity and Composite Supergravity from Nonlincar Realizations of Supersymmetry, Phys. Lett. 121B, 13.1-140 (1983)
121. J. Lukierski, A. Nowicki, Euclidean Superconformal Symmetry and its Relat,ion with Minkowski Supersymmctrics, Phys. Lett. 127B,10 1.1 (1983)
125. J. Lukierski, L. RyteI, Extended Supcrsymmetry in Fivc Dimensions with Central Charges, Phys. Rev. 27D, 2351-2357 (1983)
XXXI
126. J. Lukierski, From Composite Yang-Mills Theory to Composite Supergravity, "Quantum Theory of Particles and Fields", Birthday Vol. Dedic. to J. Lopuszariski, Eds. H. Jancewicz et aI., 71-77 (1983) World Scientific Publ., Singapore 127. J. Lukierski, A. Nowicki, Quaternionic Supergroups and D=4 Euclidean Supersymmetries, "Supersymmetry and Supergravity", XIX Karpacz Winter School on Theoretical Physics, 1983 , Eds. B. Milewski, 541-559 (1983) World Scientific Publ., Singapore 128. J. Lukierski, P. Minnaert, Seven-Spheres from Octonions and Geometric Torsion, Phys. Lett. 129B, :l922-396 (1983) 129. J. Lukierski, J .A. de Azcarraga, Supersymmetric Particles in N=2 Superspace: Hamiltonian Dynamics,
Phase-space Variables and
Phys. Rev. D28, 13371345 (1983) 130. Z. Hasiewicz, J. Lukierski, Supersymmetrized D=ll De-Sitter-Like Algebra from Octonionic., Phys. Lett. 145B, 65-69 (1984) 131. Ling LieChau Wang, J. Lukierski, Z. Popowicz, Supersymmetric Algebra of Non-local Charges in Graded Chiral Models, Lett. Math. Phys. 8,81 (1984) 132. J. Lukierski, P. Minnaert, Seven Spheres from Octonions, "Group Theoretical Methods in Physics", Proc. XIII Int. ColI. Trieste 1983, Eds. G. Denardo et aI., Lecture Notes in Physics 201, 287-294 (1984) Springer Verlag
List of publications
XXXII
133. Z. Hasiewicz, J. Lukierski, Fourfold Gradings on Superalgebras and Structure Equations Describing Extended Superspaces.,
Phys. Lett. 155B, 347-351 (1985)
134. J. Lukierski, A. Nowicki, AlI Possible Dc-Sitter Superalgebras and the Presence of Ghosts,
Phys. Lett. 151B, 382-386 (1985)
135. J. Lukierski, J.A. de Azcarraga, Nonlinear SuperficJd Realization of Extended Supersymmetries,
Class. QuantuIlJ Gravity 2, 683-691 (1985)
136. L. Lukierski, J.A. de Azcarraga, Preon Dynamics and the Supersymmetric Extension of Local Current Algebra, Phys. Rev. D31, 912--916 (1985)
137. J. Lukierski, Two Supergeometries from Fourfold Gradings of Superalgebras,
Supp!. Rend. C. Mat. di Palermo, Seric II-no 9, 11-19 (1985)
138. J. Lukierski, Composite Models from Gravity and Supergravity with Hidden Supersymmetries, "Field and Geometry", XXII Karpacz Winter School on Theoretical Physics, 1986 , Ed. A. Jadczyk, 658-672 (1986) World Scientific Pub!., Singapore
139. J. Lukierski, J .A. de Azcarraga, P. Vindel, Covariant Quantization of the D==l N==2 Supersymmetric OscilIator,
"Fields and Geometry", XXII Karpacz Winter School on Theoretical Physics, 1986 , Ed. A. .Jadczyk, 646-657 (1986) World Scientific Pub!., Singapore
140. J. Lukierski, Euclidean Superalgebras for 3
:s D :s 10,
"Supersymmetry and its Applications: Superstring, Anomalies and Supergravity", Proc. of the Workshop, Cambridge 1985, Eds. G.W. Gibbons et al., 463--481 (1986) Cambridge University Press
XXXlll
141. J. Lukierski, A. Nowicki, Quaternionic Supergroup and D==4 Euclidean Extended Supersymmetries, Annals of Phys. 166, 164188 (1986)
142. J. Lukierski, J .A. de Azcarraga, Superfields and Canonical Methods in Superspace, Modern Phys. Lett. AI, 293-302 (1986)
143. J. Lukierski, J.A. de Azcarraga, Supersymmetric Particle Model with Additional Bosonic Coordinates, Z. Phys. C: Particle and Fields 30, 221-227 (1986)
144. J.A. de Azcarraga, J. Lukierski, P. Vindel, Superfield Commutators for D==4 Chiral Multiplets and their Applications, Czech. Journ. Phys. B37, 401-411 (1987)
145. J.A. de Azcarraga, J. Lukierski, P. Vindel, SupcrsymIllctric Extension of thc D==4 Bilocal Current Algcbra, Phys. Lett. B195, 195-201 (1987)
146. J. Lukierski, W.J. Zakrzewski, Euclidean Supersymmetrization of lnstantons and Self-dual Monopoles, Phys. Lett. 189B, 99 104 (1987)
147. J. Lukierski, M. Mozrzymas, L. Rytel, Geometrization of Planck Length in Composite Gravity and Supcrgravity, Modern Phys. Lett. A2,
261~267
(1987)
148. J. Lukierski, Holomorphic and Real Euclidean Supersymmetries in Three and Four Dimensions, Czech. Journ. Phys. B37, 359-372 (1987)
XXXIV
List of publications
149. J. Lukierski, Supersymmetric Extension of Twistor Formalism,
"Field Theory, Quantum Gravity and Strings II", Proc. Meudon and Paris VI, Eds. H.J. de Vega et al., Lecture Notes in Physics 280, 137-156 (1987) Springer, Heidelberg 150. J.A. de Azcarraga, J. Lukierski, J. Niederle, Contractions Yielding New Supersymmetric Extensions of the Poincare Algebra,
Preprint SISSA (Scuola Internazionale Superiore di Studi 1\ vanzati, Trieste, Italy 131 EP (October 1988) 151. J .A. de Azcarraga, J. Lukierski, P. Vindel, Covariant Operator Formalism for Quantized Superfields,
Fortschr. der Phys. 36,453 - 478 (1988) 152. J.A. de Azcarraga, J. Lukierski, Gupta-Bleuler Quantization of Massive Superparticle Models in 6, 8 and 10 Dimcnsions,
Phys. Rcv. D38, 509 - 513 (1988) 153. J .A. de Azcarraga, J. Lukierski, Superalgebras with Grassman-algebra-valued Structure Constants from Superfields,
J. Math. Phys. 29,797 - 801 (1988) 154. M. Chaichian, J. Lukierski, N=l Super WZW and N=1,2,3,4 Super-DdV Models as D=2 Current Superfield Theories,
Phys. Lett. B212, 451 - 466 (1988) 155. J.Kowalski-Glikman, J.W.van Holten, S.Aoyama, J.Lukierski, The Spinning Superparticle,
Phys. Lett. B201, 487 - 491 (1988)
xxxv
156. J. Lukierski, A. Nowicki, General Superspaccs from Supcrtwistors, Phys. Lett. B211, 276 - 280 (1988) 157. S. Aoyama, J. Kowalski-Glikman, J. Lukierski., J.W. van Holten, Covariant BRST Quantization of the Four-Dimensional Supcrparticle, Phys. Lett. B216, 133 136 (1989) 158. M. Chaichian, D. Leites, J. Lukierski, N
= 6 From
Central Extension of Doubly lnfinite Superalgebras,
"Functional Integration, Geometry and Strings", XXV Karpacz Winter School on Theoretical Physics, 1989 , Eds. Z. Haba, Progress in Phys. 13, 421-432 (1989) Hirkhauscr 159. M. Chaichian, J.A. De Azcarraga, J. Lukierski, Euclidean Supersymmetry with Different Self-Dual and Anti-Dual Sectors,
Phys. Lett. B222, 72-78 (1989) 160. M. Chaichian, D.A. Leites, J. Lukierski, General D Algebras,
=I
Local Supcrcoordinate Transformations and their Supcrcurrent
Preprint Universite dc Geneve, Dept. de Physique Theorique, Gcneve, UGVA·· DPT/ 1989/06-617 (1989) 161. M. Chaichian, D. Leites, J. Lukierski, New N=6 lnfinite·Dimensional Superalgebl'a with Ccntral Extension, Phys. Lett. B225, :347 351 (1989) 162. Z. Hasiewicz, J. Kowalski-Glikman, J. Lukierski, J.W. van Holten, sl( III : R)-CollOIIlologies in BRST-Gupta-Bleuler Quantization, Phys. Lett. 217B, 95-97 (1989) 163. J. Kowalski-Glikman, J. Lukierski, Massive Spinning SuperparticIc, Modcm Phys. Lett. A4, 24372445 (1989)
XXXVI
List of publications
164. J.A. de Azcarraga, A. Frydryszak, J. Lukierski, Supersymmetry Currents and WZ - Like Terms in (Supersymmetry
F Models,
Phys. Lett. B247, 289-294 (1990)
165. M. Chaichian, D.A. Leites, J. Lukierski, General D=l Local Supercoordinate Transformations and their Supercurrcnt Algebras,
Phys. Lett. B236, No 1,27-32 (1990)
166. M. Chaichian, P. Kulish, J. Lukierski, q-Deformed Jacobi Identity, q-Oscillators and q-Deformed Infinite-Dimensional Algebras.,
Phys. Lett. B237, 401-406 (1990)
167. A. Frydryszak, J. Lukierski, Spinning Superparticle Models - Recent Developments,
Preprint Wroc!aw University ITP UWr 752/90 (1990)
168. W. Heidenreich, J. Lukierski, Quantized Supert wistors, Higher Spin Superalgebras and Superspingletons., Mod. Phys. Lett. A5. 439-451 (1990)
169. J. Lukierski, Spinning Superparticle Model,
"Selected Topics in QFT and Math. Physics", 1'roc. Int. ConL, Liblice (CSRR), June 1989, Eds. J. Niederle et a!., 295-308 (1990) World Scientific Pub!., Singapore
170. J. Lukierski, Spinning Superparticles and Infinite Towers of Ilelicities,
"High Energy Physics and Cosmology", 1'roc. of Summer Workshop, July 1989, Eds. J.e. Pati et a!., 395 402 (1990) World Scientific Pub!., Singapore
171. J.A.de Azcarraga, J.Lukierski, J.Niederle, Contractions Yielding New Supersymmetric Extensions of the Poincare Algebra,
Reports on Math. Phys. 30,33-40 (1991)
XXXVII
172. M. Chaichian, P. Kulish, J. Lukierski, Supercovariant q-oscillators,
"Nonlinear Fields: Classical Random Semiclassical", XXVII Karpacz Winter School on Theoretical Physics, 1991 ,Eds. P. Garbaczewski et al., 336-345 (1991) World Scientific Publ., Singapore 17:1. M. Chaichian, P. Kulish, J. Lukierski, Supercovariant Systems of q-Oscillators and q-Supersymmetric Hamiltonians,
Phys. Lett. B262,43-48(1991) 174.
M.Chaichian, P. Presnajder,
A.P.Isaev,
J.Lukierski,
Z.Popowicz,
q-Deformations of Virasoro Algebra and Conformal Dimensional,
Phys. Lett. B262, 32-38 (1991) 175. Z. Hasiewicz, .J. Kowalski-Glikman, J. Lukierski, J. W. van Holten, BRST Formulation of the Gupta-Bleuler Quantization Method,
.Journ. Math. Phys. 32,2358 - 2364 (1991) 176. J. Lukierski, A. Nowicki, H. Ruegg, V.N. Tolstoy, Q-Deformation of Poincare Algebra, Phys. Lett. B264, 331(1991) 177. J. Lukierski, A. Nowicki, Quantum Deformations of Poincare Algebra, Proc. of the XXV Int. Symp. Ahrenshoop, Berlin, September 1991, ed. H.H. Kaiser, DESY, 10-18 (1992) 178. J. Lukierski, A. Nowicki, Quaternionic Six Dimensional (Super) Twistor Formalism and Composite (Super) Spaces, Modern Phys. Lett. A6, 189-197 (1991) 179. J. Lukierski, A. Nowicki, H. Ruegg, Real Forms of Complex Quantum Anti-de-Sitter Algebra Uq(Sp(4:C)) and their Contraction Schemes,
Phys. Lett. B271, 321-328 (1991)
xxxviii
List of publications
180. V.K. Dobrev, J. Lukierski, J. Sobczyk, V.N, Tolstoy, q-Deformed Conformal Superalgebra and its Hopf Subalgebras, Preprint ICTP (Inter. Centre for Theor. Phys. Trieste), /92/188, pp. 19 (1992)
181. R. Gielerak, J. Lukierski, Z. Popowicz, Editors of "Quantum Groups and Related Topics", Proceedings of the First Max Born Symposium, 27-29 September 1991, Wojnowice, Math. Phys. Studies, Vo!. 13, pp. 274 (1992) Kluwer Acad. Pub!.
182. J. Lukierski, K-
Deformation of (Super) Poincare Algebra,
"Mathematical Physics X", Proc. of lAMP Conf. Leipzig, 30.07-9.08 (1991), cd. K. Schmudgen, 274-280 (1992) Springer Verlag
183. J. Lukierski, A. Nowicki, H. Ruegg, New Quantum Poincare Algebra and ",-Deformed Field Theory, Phys. Lett. B293, :344- 352 (1992)
184. J. Lukierski, A. Nowicki, Quantum Deformations of D = 4 Poincare Algebra, "Quantum Groups and Related Topics", Proceedings I Max Born Symposium 1991, Eds. R. Gielerak et aI., 13, 3344 (1992) Kluwer Acad. Pub!.
185. J. Lukierski, A. Nowicki, Quantum Deformations of D = 4 Poincare Deformed D = 4 Conformal Algebra,
and Weyl Algebra from Q-
Phys. Lett. B279, 299--307 (1992)
186. J. Lukierski, A. Nowicki, H. Ruegg, Quantum Deformations of Poincare Algebra and the Supersymmetric ExtenSIOns,
"Topological and Geometrical Methods in Field Theory", Proc. of Int. Conf. in Turku 26.05-1.06 1991, Eds. J. Mickelsson ct a!', 202-226 (1992) World Scientific Pub!., Singapore
XXXIX
187. J. Lukierski, A. Nowicki, Real Forms of Uq (OSp(112)) and Quantum D = 2 Supersymmetry Algebra,
J. Phys. A; Math. Gen. 25, L161-L169 (1992) 188. Y. Frishman, J. Lukierski, W.J. Zakrzewski, Quantum Group a Models - qN
= 1 Case,
"Differential Geometric Methods in Theoretical Physics", Proc. XXI Int. ConL Tianjin-China 1992, Eds. C.N. Yang et al., Int. J. Mod. Phys. (Proc. Suppl.), 3A, 354-358( 1993) World Scientific Publ., Singapore 189. Y. Frishman, J. Lukierski, W.J. Zakrzewski, Quantum Group a Models,
J. Phys. A: Math. Gen. 26,301312 (1993) 190. J. Gawrylczyk, J. Lukierski, Quantum KP Equation from Central Extension of W
(X)
Algebra,
Preprint Wroclaw University ITP UWr 822/93, pp. 12 (1993) 191. J. Lukierski, P. Minnaert, A. Nowicki,
D
= 4 Quantum
Poincare-Heisenberg Algebra,
"Symmetries ill Science VI", Proc. of Symp. August 1992, Bregenz (Austria), Ed. B. Gruber, 469476 (1993) Plenum Press 192. J. Lukierski, A. Nowicki, H. Ruegg, D
=4
Quantum Poincare Algebras and Finite Difference Time Derivatives,
"Spinors, Twistors, Clifford Algebras and Quantum Deformations", Proceedings II Max Born Symposium 1992, Eds. Z. Oziewiczet.al., Fundamental Thear. of Phys. 52, 257-266 (1993) Kluwer Acad. Publ. 193. J. Lukierski, A. Nowicki, H. Ruegg, K
deformed Poincare Algebra and Some Physical Consequences,
"Symmetry VI", Proc. of Symp. August 1992, Bregenz (Austria), Ed. B. Gruber, 477-488 (1993) Plenum Press 194. J. Lukierski, A. Nowicki, J. Sobczyk, All Real Forms of Uq (sl(4; C) and D = 4 Conformal Quantum Algebras, J. Phys. A, Math. Gen. 26,4047-4058 (1993)
xl
List of publications
195. J. Lukierski, H. Ruegg, W. Riihl, From K-Poincare to K-Lorentz Pseudogroup: a Deformation of Relativistic Symmetry,
Phys. Lett. B313, :357··366 (1993) 196. J.Lukierski, J.Sobczyk, A.Nowicki, Quantum D=4 Poincare Superalgebra,
J. Phys. A: Math. Gen. A26, L1099-Lll09 (1993) 197. J. Lukierski, A. Nowicki, R. Ruegg, Quantum Deformations of D = 4 Poincare and Conformal Algebras,
"Quantum Symmetries", Proc. Int. Workshop on Math. Phys., Clausthal, July 1991; Eds. H.-D. Doebner et.al., 304-326 (1993) World Scientific Publ., Singapore 198. J. Lukierski, A. Nowicki, H. Ruegg, Quantum Poincare Algebra,
"Group Theoretical Methods in Theor. Phys." ,Proc. XIX Int. ColI. Eds. M.A. del Olmo ct aI., Ciemct. Annales de Fisica, 139-144 (1993) I
199. J. Lukierski, A. Nowicki, H. Ruegg, Quantum Poincare Algebra with Standard Real Structure,
"Infinite Dimensional Geometry in Physics", Proc. of XXVIII Karpacz Winter School on Theoretical Physics, 1992 , Eds. R. Gielerak et al., Journ. of Geom. and Phys., 11, 425·436 (1993) 200. P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, Quantum Deformation of the Poincare Supergroup and K-dcformed Super-
space, J. Phys. A. Math. Gen. 27,6827-6837 (1994) 201. J. Lukierski, A. Nowicki, H. Ruegg, D = 4 Quantum Poincare Algebras,
"Particles and Fields", Proceedings of the First German· Polish Symposium, Rydzyna, 28 April - 1 May, 1992 , Eds. H.D. Doebner et al., 229-236 (1994) World Scientific Pub!., Singapore
xli 202. J. Lukierski, H. Ruegg, Quantum ,,-Poincare in Any Dimension,
Phys. Lett. B329, 189-194 (1994)
203. J. Lukierski, H. Ruegg, Quantum Deformation of Inhomogeneous Rotation Algebras,
"Quantum Groups", Proc. Symp., Clausthal, July 1993, Ed. H.-D. Doebncr, Springer Verlag (1994)
201. J. Lukierski, H. Ruegg, A. Nowicki, Quantum Deformations of Nonsemisimple Algebras: The Example of D=4 Inhomogeneous Rotations,
J. Math. Phys. 35,2607-2616 (1994)
205. J. Lukierski, H. Ruegg, V.N. Tolstoy, A. Nowicki, Twisted Classical Poincare Algebras,
J. Phys. A: Math. Gen. 27,2389-2399 (1994) 206. J. Lukierski, A. Nowicki, H. Ruegg, V.N. Tolstoy, Twisting Poincare Algebras,
"Quantization and Infinite-Dimensional Systems", Proc. of the Twelfth Workshop on Geometric Methods in Physics, July 1-7,1993, Bialowieza, Poland Eds. J.P. Antoine et aL, 265 272 (1994) Plenum Press, New York and London
207. J. Gawrylczyk, J. Lukierski, Symplectic Structures from Central Extension of Woo Algebra and KP Equation,
Modern Phys. Lett. AlO, 273 278 (1995)
208. M. Klimek, J. Lukierski, ,,-Deformed Healization of D = 4 Conformal Algebra,
Acta Phys. Polonica, B26, 1209-1216 (1995)
209. P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, K,-Deformation of Poincare Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure,
J. Phys. A. Math. Gen. 28,2255-2264 (1995)
List of publications
xlii
210. P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, Quantum Deformation of the Poincare Supergroup, "Quantum Groups. Formalism and Applications", XXX Karpacz Winter School on Theoretical Physics, 1994 , Eds. J. Lukierski et aI., 353 358 (1995) Polish Scientific Publishers PWN, Warszawa
211. P.Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, The Classical Basis for ,,-deformed Poincare Algebra and Superalgebra, Mod. Phys. Lett., AIO, 2599-2606 (1995)
212. J. Lopuszanski, J. Lukierski, Jan Rzewuski - In Memoriam, Acta Physica Polonica, B26, 1195 1200 (1995)
213. J. Lukierski, H. Ruegg, W.J. Zakrzewski, Classical and Quantum Mechanics of Free ,,-Relativistic Systems, Annals of Phys. 243,90-116 (1995)
214. J. Lukierski, H. Ruegg, W.J. Zakrzewski, Classical and Quantum Mechanics of Free ,,--Relativistic Systems, "Quantum Groups. Formalism and Applications", XXX Karpacz Winter School on Theoretical Physics, 1994 , Eds. J. Lukierski et aI., 539-554 (1995) Polish Scientific Publishers PWN, Warszawa
215. J. Lukierski, Differcnt Bases for" -Deformed Poincare Algebra, "Symmetries in Scicnce VIII", Proc. ConL August 1994 Bregenz (Austria), Ed. B. Gruber, Plenum Press (1995) ; in press
216. J. Lukierski, H. Ruegg, V.N. Tolstoy, Quantum"
Poincare 1994,
"Quantum Groups. Formalism and Applications", XXX Karpacz Winter School on Theoretical Physics, 1994 , February 1994, Eds. J. Lukierski et aI., 359 -378 (1995) Polish Scientific Publishers PWN, Warszawa
xliii 217. J. Lukierski, Quantum Deformations of D = 4 Supersymmetries,
"Quantum Groups and their Applications in Physics", Proc. of Varenna School, Varenna July 1994, Eds. L. Castellani, J. Wess, Springer Verlag (1995) ; in press
218. J. Lukierski, J. Sobczyk, A. Nowicki, H. Ruegg, V.N. Tolstoy, Quantum Deformations of Space-Time (super)Symmetries with Fundamental lvlass Parameter,
"Group- Theoretic Methods in Physics", Proc. of Yamada Conf., Toyonaka (Japan), July 1994, Ed. A. Arimaet.a!., 307··313 (1995)World Scientific Pub!., Singapore 219. J. Lukierski, A. Nowicki, V.N. Tolstoy, Twisted Poincare Algebras,
"Quantum Groups. Formalism and Applications", XXX Karpacz Winter School OIl Theoretical Physics, 1994 , Eds..J. Lukierski et a!., 379-388 (1995) Polish Scientific Publishers PWN, Warszawa 220. J. Lukierski, Z. Popowicz, J. Sobczyk, Editors of: "Quantum Groups, Formalism and Applications",
Proceedings of the XXX Karpacz Winter School of Theoretical Physics, Polish Scicntific Publishers PWN, Warszawa 1995 221. J. Lukierski, A. Nowicki, J. Sobczyk, ,,-deformation of D = 4 Poincare Superalgebra,
Proc. of III-th Wigner Symposium, Oxford, September 1993, Eds. L. Royle, A. Solomon, (1994) World Scientific Pub!., Singapore 222. J. Lukierski, P. Minnaert, M. Mozrzymas, New Quantum Deformations of 1) = 4 Conformal Algebra,
Czcch.
JOUrIl.
Phys., (1995) ; in press
22:3. J. Lukierski, P. Minnaert, M. Mozrzymas, Quantum Deformations of Conformal Algebras Introducing Fundametal Mass Parameters,
Preprint Bordeaux University, LPTB 956, June 1995; Preprint hep-qalg 9507005 , Phys. Lett. B (1996), ; in press
xliv
List of publications
224. P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk,
Bicrossproduct lIopf Superalgebras and D = 4 K--deformed Poincare Supergroup,
.J. Math. Phys. (1996) ; in press
Chapter 1. QUANTUM GROUPS: GENERAL FORMALISM
From Field Theory to Quantum Groups
CONTRACTIONS, HOPF ALGEBRA EXTENSIONS AND COVARIANT DIFFERENTIAL CALCULUS J .A.
DE AZCARRAGA AND
J .C.
PEREZ BUENO
Departamento de Ffsica Te6rica and IFIC, Centro Mixto Univ. de Valencia-CSIC 46100-Burjassot (Valencia), Spain e-mail: azcarragatevalvx.ific.uv.es.pbuenoatlie.ific.uv.es
Abstract: We re-examine all the contractions related with the Uq (su(2)) deformed algebra and study the consequences that the contraction process has for their structure. We also show using Uq (su(2))xU(u(I)) as an example that, as in the undeformed case, the contraction may generate llopf algebra cohomology. We shall show that most of the different llopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual 'group' versions. The bicovariant differential calculi on the deformed spaces associated with the contracted algebras and the requirements for their existence are examined as well.
1.
Introduction
As is well known, the standard Wigner-inonii contraction P] of simple Lie algebras with respect to a subalgebra leads to algebras which are the semidirect product of the preserved suhalgebra and the resulting Abelian complement. Other types of contractions involving powers of the contraction parameter, first discussed in [2], may lead to a central extension structure. Due to the singular nature of the contraction process, (non-simple) groups/algebras which are the direct product/sum of two groups/algebras may not retain this direct product structure after the contraction limit if the contraction affects suitably the central trivial extension; one may refer to these groups as being pseudoextended [3,4] when the extension is trivial but behaves non-trivially under the contraction. A well known example is the direct product P x U( I), P being the Poincare group, for which a suitable limit leads to the centrally extended Galilci group [1,3-5].
4
J.A. de Azcarraga and J.e. Perez Bueno
One of the interests of non-commutative geometry is to provide a rationale for possible deformations of the spacetime manifold, which becomes a non-commutative algebra. By extending standard Lie group arguments about quotient spaces, it is natural to associate these spacetime deformations with the deformation of inhomogeneous groups, which are non-simple. Since the standard deformation procedure [6-8] applies to the simple algebra/group case, the contraction of deformed simple algebras suggests itself as a possible way of obtaining deformed inhomogeneous algebras. This process usually requires involving the deformation parameter q into the contraction [9,10], and is rather complicated; in fact, the contraction of deformed algebras/groups is besieged by the appearance of divergences (the contraction is not always possible or the R-matrix diverges), and a complete theory is still lacking. Clearly, the difficulty lies in having a well defined contraction process in both the algebra and coalgebra sectors. Contraction is not, however, the only way of finding deformations of inhomogeneous groups. Much in the same way we may construct Lie groups out of two by solving the corresponding group extension problem (which always has a solution for Abelian kernel, precisely the semidirect extension, see e.g. [4]), we may look for a similar direct construction for Hopf algebras without thinking of obtaining them by contraction. Such a construction already exists for certain cases, and leads to the bicrossproduct and cocycle bicrossproduct structure of Hopf algebras of Majid [11,12] (see also [13-15]; a summary of Majid's theory is given in Appendix B). For instance, the K-Poincare algebra of Lukierski et al. [16], which is obtained from Uq (so(2, 3)) by a contraction involving the deformation parameter q written as q = exp 1/ KR, where R is the (de Sitter radius) contraction parameter so that [K] = L -1, has been shown [17] to possess such a bicrossproduct structure. In this paper we intend to re-examine in this new light the simplest contraction examples, including the earliest ones [9,10] (several of them discussed from various points of view in [18-26]). We shall also look at the notion of central extension pseudocohomology for Hopf algebras, and find that the contraction process generates Hopf algebra extension cohomology as it docs for its undeformed Lie counterpart. We shall discuss both the 'algebra' and 'group' aspects of the deformed Hopf algebras, and study whether they lead to a bicrossproduct or cocycle bicrossproduct structure. In contrast, the problems associated with the contraction and the it-matrix behaviour will not be discussed here. In fact, the constructions presented in Sec. 5 may be considered as a way of avoiding the search for an R-matrix. It would be interesting to perform a more general analysis of the consequences of the contraction process for the structure of the resulting deformed Hopf algebras. We hope to report on this elsewhere [27]. The analysis of the differential calculus on the 'spaces' associated with the inhomogeneous deformed groups is also of importance; this has been recently made for K-spacetime algebras in [28,29]. It was shown there that the demand of covariance for the differential calculus required to enlarge the spacetime algebra by an element
Contractions, Bopf Algebra Extensions
5
related to a central extension of the \lopf algebra; this phenomenon will also appear here for certain cases (Sec. 6).
2.
Contractions of Uq (su(2)) The well known Uq (su(2)) deformed Bopf algebra is defined by (q = e
Z )
(2.1 ) Let us consider the different contractions of Uq (su(2)).
(1) Uq (£(2)). The standard contraction procedure with respect to the I10pf subalgebra generated by J3 , implying the redefinitions J I = c 1 PI , J2 = c 1 P2 , J3 = J, leads ([26]; see also [19]) to
111\2
= exp( -zJ) ® P I,2 + P1,2 ® exp(zJ)
S(PI ,2)
= -exp(zJ)P1,2exp(-zJ)
,
S(J)
,
11J
= -J
= J 181 1 + 1 181 J ;
f(J,Pd
(2.2)
=0
This deformation of the Euclidean algebra is a Hopf algebra where the deformation only appears at the coalgebra level, and will be denoted Uq ( £(2)).
(2) Uw (£(2)). A second contraction, leading to another deformation of the Euclidean algebra. may bc pcrformed. This contraction [9,10] requires writing previously q = exp(fwj2) since it is not performed with respect to a Bopf subalgebral : it is performcd with respect to J 2 , which is a Bopf subalgebra only for q = 1. The rcdefinitions J I = C I ]>2 • J 2 = J , h = f - I PI , Z = fwj2 in (2.1) lead to [9,10] the 1 It may be worth mentioning that contracting with respect to a Hopf subalgebra (as in the case (1) above) is not a sufficient condition to define a contraction without involving the deformation parameter in it. The above is, in fact, a rather exceptional case.
6
J.A. de Azcarraga and J.e. Perez Bueno
Uw ( £(2)) Euclidean lIopf algebra 2
t1J=exp(-wPJ/2)Q0J+JQ0exp(wPJ/2)
;
S'(PI •2 )=-P1,2
w S(J) = -exp(wPI /2)Jexp(-wPJ/2) = -J + 2P2
;
,
t(J,PI,2) = 0
(2.3) Besides the above, we may consider two 'non-standard' contractions (i.e. involving higher powers of the contraction parameter t). They are obtained by extending to the deformed case the generalized contraction in [2]. (3) Uw(9( 1+ 1)). A third contraction leads to a deformation of the Galilei algebra (the (1 + 1) version of the (1 + 3) deformed Galilei algebra in [30)). We make the redefinitions 3 z = ~, J I = a-- I c l V, J 2 = _c 2 X, J 3 = CIa- Xt. By taking the limit t - t 0, we get
[X,V]
=0
(2.4)
S(X)
=
-X , S(V)
((Xt,X, V)
=
-exp(wXt)Vexp{-wXtl = -V +wX
=0
We will denote this deformed Galilei algebra by Uw(9(1 + 1». In the w- t 0 limit, eg. (2.4) gives the lIopf structure of the cnveloping algebra Uw(9(1 + 1» of thc Galilei Lie algcbra. (4) Uw(HW). Finally, there is another contraction of Uq (su(2». It is obtained by making in (2.1) the redefinitions J I = c l Xq , J 2 = c l Xp , J 3 = t- 2 := and 2lf we want to look at PI, P2 as deformed translation generators. [P;] = L -1, it is sufficient to take [{] = L- I , [w] = L. 3The parameter u , [u] = T L -1/2, is introduced to give standard dimensions to the generators of the Galilei algebra ([{] L -1/2, lW] T), but it disappears after the contraction.
=
=
7
Contractions, Hopf Algebra Extensions
z
= w(2/2. 4
,1Xq,p
=
The result is the w-deformed Heisenberg-Weyl Uw(HW) Hopf algebra
exp( -w:= /2)
(9
Xq,p
+ Xq,p 12) exp(w:= /2)
"'( -) = -.=-
...J"::
;
(2.5) (denoted Heisenberg quantum group H (I)q in [9,10]). By making the change of basis Xq,p = exp( -w:=/2)Xq,p the Uw(lIW) algebra takes the form - X ] [..::, p,q = 0
,1Xq,p = Xq,p
@
1 + exp( -w:=)
in the undeformed limit U(llW) are recovered. 3.
1[V _'\q, XJ_ P -
w -+
@
Xq,p
exp (-2w:=)
2w
,1:= = 1 @ := + := I8i 1
(2.6)
0, the standard expressions for the Hopf strueture of
Structure of the Uq (su(2)) contractions
As mentioned, the bicrossproduct [11, 12] of Hopf algebras (see Appendix B) may be used as an alternative construction of deformed Hopf algebras when the undeformed ones are not simple. Non-simple algebras may arise from contraction, a process which for ordinary Lie algebras leads to a semidirect product algebra. Thus, it is worth exploring whether the above deformed Hopf algebras are the (right-left) bicrossproduct H~ A of two Hopf algebras H and A or have a cocycle bicrossproduct structure. The notation H~ A, for instance, indicates that A is a right H-module algebra for the right action a : A @ H -+ A , 0'( a, h) == a <J h , and that H is a left A-comodule coalgebra for the left coaction fJ : H -+ A I8i H (H is a left quantum space); a and /1 must also satisfy certain compatibility conditions [11, 12J. (1) Let us first consider Uq (E(2)), eqs. (2.2). At the algebra level it has a semidireet structure. However, if we take A as the undcformed Hopf algebra generated by PI , P2 and H as that generated by J we see that with independence of fJ, we cannot reproduce ,1(PI ,2) in Uq (E(2)); in fact, PI, P2 in (2.2) do not generate a Hopf subalgebra of Uq (E(2)). Thus, Uq (E(2)) has not a bicrossproduct structure. 41f one wishes to have q and p with dimensions of length and momentum (and [X q ] = L -1, [Xp ] = (rnol1lenlum)-I) it is sufficient to modify the redefinitions to read J I = (-I X q , h (-IAX p , h = (-2A':: , z = w(2/2A, with [e] = L- 1 , [A] = [rnol1lenlul1I]L- 1 , [w] = action, ['::] = action-I; A disappears in the final expressions (2.5).
=
J.A. de Azca.rraga and J.C. Perez Bueno
S
(2) Let us now look at Uw (£(2)), eqs. (2.3). The redefinitions
Px=F'1
,
F'y=exp(-wPJ/2)P2
,
,J'=exp(-wPJ/2)J,
(3.1)
allow us to write Uw ( £(2)) in terms of (Px , Py , J ') in the form
LlJ'=J'Q0I+exp(-wPx )0J' .':i(Px ) = -Px
,
S(Py) =
;
(J,Px,y)=O
;
- exp(wPx)Py , S(f) = - exp(wPx)J'.
(3.2) If we now take for A the commutative non-cocommutative Hopf translation subalgebra Uw (Tr(2)) of (Px , Py) contained in (3.2) and 11. is the commutative and cocommutative algebra generated by J', the bicrossproduct structure 11.f>4 A of (3.2) is exhibited if
since it may be seen that the compatibility axioms 5 (B.10), (B.11), (B.12), (B.13) and (13.11) are satisfied and that (B.16), (R17), (R1S), define the coproducts, antipodes and counits in (3.2). This shows that Uw (£(2))= U(u(l))f>4 Uw (Tr(2)). (3) Consider now the deformed (1 + 1) Galilei 1I0pf algebra u,;,(9 (1 + 1)) of (2.1). It was found in [29] (for the (1 +3) case) that it is also endowed with a bicrossproduct structure. To show this, we make the redefinitions
With them, the Uw(Q(l + 1)) Hopf algebra takes the form
[XL, Vj LlXt Ll V
= -x ,
[X, V]
= X t iI9 1 + 1 lSi X t
= wX 2 ,
LlX
= V iI9 1 + exp( -2wXt ) iI9 V
S(X t ) = -Xl
,
S(X)
,
=X ;
[X, Xtl
=0 ;
iI9 1 + exp( -2wXtl iI9 X
(V, X, Xt)
= -exp(2wXtlX ,
=0
Sty)
,
;
= -exp(2wXtlV
,
(3.5) SThe formulae (B.-) refer to the corresponding (B'.-) ones given in Appendix B; they may be found in the original papers [11] or in the Appendix of [29] (there with the same numbering).
Contractions, Hopf Algebra Extensions
9
(which is eg. (6.1) in [29] for C;;" with 1/2"- = w). The bicrossproduct structure is summarized in the definitions of the action a and the coaction j3 (A is the Abelian, non-cocommutative Bopf subalgebra UdT r(2)) generated by X and XI, and 1t is given by the commutative and cocommutative Hopf algebra generated by V) a(X, V) == X
<J
V := [X, V] = wX 2
j3(V) := exp( -2wXtl ® V
.
(3.6) It may be shown that the bicrossproduct conditions arc verified and hence that
Uw(9(1 + I)) =U(u(I))I>
(3.7) This induccs the appropriate coproduct for X p (idcntified as X p 0 I in 1t 0 A) and antipode (eg. (2.6» from (B.16) and (B. IS), respectively; the commutators in (2.6) follow from (B.15). 4b) The cocycle bicrossproduct structure is constructed from the undeformed Hopf algebras A = U(:=) (generated by:=) and 1t = (Tr(2)) (id. Xq,X p). Since we wish to obtain a deformation of a ccntral cxtension algebra the action a must be trivial. We take
and
~
: 1t 01t
--+
A (antisymmetric cocycle), lj; : 1t
~(Xq,Xp) = -~(Xp,Xq) =
--+
I • _ sinh(w:=) -exp(-w.::) • 2 w
A IX: A given by (3.9)
(i.e. lj; trivial, lj;(h) = 1®lt(h)) plus (B.19) and (B.2S). Thus, the deformed charactcr of the resulting algcbra (and hence w) enters in this case through j3 and ~ only. Sincc 6Note that the expressions (2.4), (3.5) and (3.6) may be obtained from standard contraction of their analogous ones in the Euclidean case (2.3), (3.2) and (3.3).
10
.I.A. de Azc
a and 1/J are trivial and A and H have the cocommutative Hopf algebra structure
associated with the Abelian enveloping algebras U('=:), U(Tr(2)), it is not difficult to check that the compatibility conditions (8.14), (B.25), (8.26) and (8.27) are fulfilled. Moreover, (B.29) and (8.30) reduce using (8.19) to (hQ9a)(g@b)=hg0ab+10~(h,g)ab
L1(h@I)=h010101+ll/9h(I)0h(2):8I1
,h,gEH ,
h,gEH.
(3.10) (3.11)
Denoting the clements (h 1/9 I) and (g 0 I) in H 0 A by hand g, eq. (3.10) leads to [h,g] = 2~(h,g) ([h,g] = 0 in H) so that the commutators in (2.6) are recovered for ~ given by (3.9). Similarly,
L1(Xq,p) = Xq,p Q9 I (3: I 0 I + 1 @exp(-w'=:) 0 Xq,p 01 == Xq,p 0 I + exp( -w'=:) 0 Xq,p (3.12) plus ,1('=:) = '=::811 + 1 (3:'=:. With ((Xq,p) = 0 = ((.=:), the Hopf algebra structure of Uw(HW) is obtained by adding the antipode as defined by (2.6).
4.
The co cycle extended Euclidean Hopf algebras Up (E(2)), Up(E(2))
Consider Uw ( £(2)) xU( u( I)). This Hopf algebra has a trivial central factor and, as such, it might have been obtained from Uq (su(2))xU(u(I)) by contraction, since the redefinitions given in Sec. 2 (2) (and (3.1)) do not affect the U(u(l)) part ('=:). However a generalization of the pseudocohomology mechanism [3,4] mentioned in the introduction may also be used here to obtain non-trivial extensions of Hopf algebras by contracting trivial products (see [29] for the case of the deformed extended (1+3) Galilei Hopf algebra). We now find two deformations of the centrally extended Euclidean algebra using this procedure. (a) Up ( E(2)). Consider the Uq (su(2)) xU( u( I)) Hopf algebra generated by (JI, J 2 , J 3 , '=:') given by eqs. (2.1) plus the U( u( I)) relations L1'=:'='=:'01+1(3:'=:' The redefinition J 3 = J [
1,
,
5('=:')=-'=:'
J;; +.=:'
,
(('=:')=0
;
[,=:',all]=O
. (4.1)
[9] leaves (2.1) and (4.1) unchanged but for
J 1 = sinh(2z(J; + .=:')) 2 2sinh(z)
(4.2)
Contractions, Hopf Algebra Extensions
II
Because (Jh.h] involves :=', we refer to Uq (su(2))xU(u(1)) in the form (4.2) as a pseudoextension (the trivial direct product structure is disguised beneath the election of the generators). To obtain a non-trivial Hopf algebra extension from it, we now make a rescaling involving :=',
(4.3) redefine z as z = by
p(2
[N,Xd = X 2 .1Xi
and take the limit (
---->
O. The resulting Hopf algebra is given
2p:= [X I, X]2 = sinh2 p'
,
= exp(-p:=)@Xi+Xi®exp(p:=) (i = 1,2)
S(X 1,2)
= -Xl,2
,
S(N)
= -N
,
5(:=)
, .1N
= -:=
;
[:=, al~
=0
;
= N ® I + I @N t(Xl,2,N,:=)
;
= O. (4.4)
This Hopf algebra will be denoted by Up (£(2)). It is convenient to make in (4.4) the change Y; = exp( -p:=)Xi
(4.5)
This modifies only
.1Yi = exp( -2p:=)
Q9
Y;
+ Y; ® I
(4.6)
S(Y;) = - exp(2p:=)Y;
which reproduces the Heisenberg-Weyl Uw(llW) Hopf algebra of (2.6) with 2p = W. al) Up ( £(2)) has the bicrossproduct structure Hl>tl A, in which A is the deformed Heisenberg- Weyl Up ( HW) Hopf subalgebra in Up ( £(2)) generated by (Y1 , Y2 ,:=) with primitive coproduct for := and .1(Y;) given in (4.6), and H is the commutative and cocommutative algebra generated by N. The right action ~ of N on A is then designed to reproduce the commutators in Up ( £(2)) :=~N=O
,
(4.7)
and the coaction (J is taken to be trivial, (J( N) = I ® N, since the coproducts in both
H and A are already those in Up (£(2)). a2) The cocycle extension structure of UP(£(2)) is achieved by taking A generated by := [ego (4.1)] and H as the undcformed Euclidean algebra U( [(2)). The action a
12
J.A. de Azcarraga and J.C. Perez Bueno
of'H on A is trivial (we want := to be central), and so is the map 1/J ((8.22), (B.23)); the antisymmetric cocycle ~ and coaction fJ are given by (d. (4.6))
~(})Y2) = ~(1 8p
- exp(-4p:=)), 11(Y;)
= exp(-2p:=) 0 Y;,
I1(N)
=
10 N
(4.8)
(the coaction on N is trivial). We may check that all relations (B.19)-(B.26), (8.27)7, (8.28) are fulfilled and that (B.29)-(B.30) then reproduce (4.6); thus, U p (£(2)) has a cocycle bicrossproduct structure. (b) U p(£(2)). Consider again the algebra Uq (su(2)) x U(u(I)) given by eqs. (2.1) plus the relations (4.1) for the central u(l) generator, now denoted ~'. The redefinition J 1 = J{ +~, leaves (4.1) and (2.1) unchanged but for
(4.9)
If we \lOW make the rescaling (4.10) and set z =
{3
p,
in the limit
[J',Px] = Py ,
t -+
0 we obtain the Hopf algebra Up( £(2)) given by
[J',Py] = -Px ,
[px,Py] = ~
dJ' = .I' &J 1 + 1 (;1 .1' + p(~ 0 Px - Px 0
,
[~,alfj
=0
;
t) , (4.11 )
This algebra has a cocycle exte\lsion structure. To show this, we make the nonlinear change (4.12) This modifies only
d.l
= J®l+10J+2pt0Px
,
S(J)
= -J+2p~Px (4.1:1)
Contractions, Hopf Algebra Extensions
13
If A is taken as the Hopf subalgebra generated by ~ and 1-l is the undeformed Euclidean Hopf algebra U(E(2)), the algebra (4.11), (4.13) is obtained as the rightleft cocycle bicrossproduct with 0: and 1/1 trivial and f3 and ~ defined by
(4.14) (a') Let us go back to the case (a) above. If we make the redefinitions IV = = i(X'fiX2 ), the Up (£(2)) algebra in the basis (4.4) takes the form [10,24]
iN, A = i(X\Jt 2), A+
[A, A +]
= -i sinh 2p2'
[2', all]
2p
!1N = IV (91
+I
®
tv ,
!1A = exp( -p2') (9 A + A ® exp(p2')
=0
,
(4.15)
(b') Similarly, the redefinitions j = iJ, A = i Px:;[, , A+ =;= i Px -:;:, take the algebraUp (£(2)) in the basis (4.13) to the form
[J,A]=-A
,
[.J,A+l=A+
,
[A,A+l=-i~
!1J = J (91 + 1 @ j + v'2P~ ® (A + A+) !1A = 1 ® A + A ® 1 ,
!1A+ = 1 ® A+
[.=, all]
= 0
,
+ A+ ® 1
5(A) = -A
,
(4.16)
5(A+)
= -A+
The algebras (4.15) [9,10] and (4.16) are a deformation of the four-generator oscillator algebra which is recovered in the limits p -> 0 [(4.15)], P -> 0 [(4.16)]. Eqs. (4.15) or (4.16) do not, however [10], define the algebra of the q-oscillator [31--33]. The oscillator algebra may be obtained by contraction using the finitedimensional representations of su q (2) [34]. To derive it directly, without resorting to the su q (2) representations, consider the four generators algebra Uq (su(2)) xU(u(1)) with [J,h] = ±.h , [J+,1-] = [2J]q == sinh2zJ/sinhz , [2',alfj = O. Now, we
14
J.A. de Azcarraga and J.C. Perez Bueno
perform the redefinitions 8 J+ = [2/f]~/2ii+ , J_ = [2/(g12 a , J = N - ~/f; this means that a, ii+ and N are independent generators. Assuming q real, z > 0, the contraction leads to [N, a] = -a , [N, a+] = a+ , [a, a+] = q-2N; the familiar q-commutator relations [N, a] = -a, [N, a+] = a+ , [a, a+]q = q-N follow for a = qN/2a , a+ = a+qN/2.9 The coproduct in Uq(su(2))xU(u(l)), however, does not have a limit and this explains why the Hop! structure for the q-oscillator (as defined by these relations) is lost (for recent references on this point, see [35,36]).
5.
The dual case: structure of the deformed Hopf group algebras
The previous deformed algebras may be dualized making use of the bicrossproduct construction. The dual of a bicrossproduct Hopf algebra is also a bicrossproduct Hopf algebra; thus, if 1{ and A are Hopf algebras from which the bicrossproduct 1{~ A is constructed, then their duals H and A lead to the dual bicrossproduct II ",
= x 0 1+ 1® x
8(x,y) = (-x,-y)
L1y=y01+10y f(X,y)=O
x,y E A
[X,y] = -wy
(5.1 )
and H is generated by
+ 1 ®
(5.2)
8Notice that, wcrc it not by the q-bracket [x]q = (qX - q-X)!(q - q-l), these redefinitions would be equivalent to those in (4.3); this exhibits once more the non-commutative nature of many contraction! deformation diagrams. 9The above oscillator algebra, where N is treated as an independent gcnerator, has a non-trivial central element, z q-N+l([N]q - a+a) and many irreducible representations (for 0 < q < 1)[34] unequivalent to the Fock space ones with vacuum state and number operator N, [N]q = a+a, for which z = o.
=
Contractions, Bopf Algebra Extensions The duals
fJ
15
and ex of ex and !~ are found to be
~(x) =
x ()\: cos 'P
+ y C9 sin
(3(y) = -x
ri(x C9
Q)
sin
(9
cos
(5.3)
n(y C9
,
The compatibility conditions (8' .IO)-(B' .14) are satisfied, and (B' .15), (B' .16), (8' .17) and (B' .18) determine the Bopf structure of Funw( £(2)), [x,y]=-wy,
[x,
Ll
[y,
Llx = 1 Q) x + x ® cos
,
Lly=lQ9y-x!X:sin
, ,
(5.1)
t(
S(y) = - sin
,
(3) A discussion of the Galilei case will be presented elsewhere. (4) Consider now the case of the deformed 1/eisenberg- Weyl 'group' Funp(HW), (see [39]) dual of the algebra U p ( 1/W) as given in (4.6) (i.e., (2.6) for w= 2p). It was shown in Sec. 3 (4b) that Up(HW) could be obtained as the cocycle bicrossproduct [11,12] (Appendix B) Up(HW) = H~ A of the undeforrned algebras H = U(T1'(2)) and A = U(u(I)) by using the non-trivial ~ and ~ given (3.8), (3.9). Thus, the deformed Ileisenberg- Weyl group algebra Funp(IlW) may be found as the cocycle bicrossproduct of 1/ = 1'1'(2) and A = U( 1) using the duals n : A C9 H ---> Hand '0 : A ---> H (9 1/ of ~ and ~ respectively. Using (Yl' Y2; X) for the parameters of 1'1'(2) and U(I), < Y;, Yj >= Dij , < X >= 1 the dualization of ~ immediately leads to
-=,
n(X,Yi) == Xf>Yi
= -2PYi
or
l-ex J;4 P
[X,YiJ
= -2pYi
,
(5.5)
= 1,2. Let us now dualize ~ = '=). What was really needed in (3.9) to compute [Y1 , Y2] was the difference ~(Yl, Y2) - ~(Y2, Y1 ); the ambiguity in ~(YI' Y2) is related to the cohoundary ambiguity. A suitable election produces p
i
-
1
V'(X) = 2"(Yl C9 Y2 - Y2 is!
ytl ,
from which Ll(X) is easily found using (8'.34) since Funp(HW) is determined by fYi, Yj] = 0 LlYi
,
[X, Yi] = -2pYi
= Yi ~ 1 + 1 C·) Yi
S(Yi) = -Yi
,
{3 is trivial (0: is trivial). In all,
,
,
S(X) = -X
(5.6)
(5.7) ,
E(Yi, X) = 0
.
16
J.A. de Azcarraga and J.C. Perez Bueno
The coproduct mimics tlw familiar H W group law, and the non-commutativity is just reflected in the non-zero [X, Yi] commutator. (a) Extended Euclidean group Fun p (E(2)). The dual Funp(E(2)) of the algebra (a) given byegs. (4.6) (and (4.4)) is generated by the clements (YhY2,X,'P) « li,Yj >= 5ij , < N,'P >= 1, < :=,X >= I) for which
Ll'P = 'P (8 1 + I Q<; 'P LlY2 = I 19 Y2
,
LlYI == I ® YI
+ Y2 19 cOS'P -
YI ® sin 'P
+ YI
® cos'P
+ Y2 19 sin 'P
,
+ YI
;
,
LlX=I®X+X®1 I + 2"[YI 19 cos 'PY2
+ Y2 ::>9 sin 'PY2 -
Y2 19 cos 'PYI
Q9 sin 'Pyd
(5.8) It is not difficult to check directly that Fun p (E(2)) [(5.8)] is a Hopf algebra; we shall now obtain (5.8) by dualization in two different ways. For the dual in the basis (4.15), see
[24].
al) Fun p( ];(2)) is the bicrossproduct FunU(1) .,:lFunp(HW), where U( I) is generated by 'P and Funp(HW) is given in (5.7). To see this, it is sufficient to dualize the right action
(j(yt) = YI ::>9 cos 'P {3(X) = 1 Q9 X
+ Y2 ® sin 'P
,
fJ(Y2) = -YI Q9 sin 'P
+ Y2 19 cos 'P
,
,
(5.9) for which the coproducts and antipodes in (5.8) are obtained from (8'.16) and (8'.18). Clearly [Yi,'P] = 0 = [X,'P] since ii (~) is dual to {3, which is trivial. a2) lo'ullAE(2)) has also a cocycle bicrossproduct structure. To see this, we take A as the Hopf algebra generated by X with primitive coproduct and H as the (un deformed) Euclidean group Hopf algebra of generators (Yll Y2, 'P) with LlYi ,Ll'P , S(Yi, 'P) and t(Yi, 'P) as in (5.8). Then, since 0' and 1/J were trivial
Contractions, Hopf Algebra Extensions
17
in Sec. 4 a2), fJ and 1, are trivial (fJ(a) = a @ Ill, (a,b) = f(a)f(b)IH) and - t ff @ H , (i : A @ /I - t /I may be found from (4.8) to be
;j; : A
1;;(x) =
~[Yl iZ: cos r.pY2 + Y2 @ sin r.pY2 -
Y2
@
cos r.pYI + Yl @ sin r.pYd
,
(5.10) Thc relations (fl' .19)-(B' .28) arc fulfilled 10 and the cocycle bicrossproduct structurc of Fun p (E(2)) follows from (B'.29) (which for 1, trivial and a with primitivc coproduct leads to [a, h] = af;h) and (B'.34).
(b) Extended Euclidean group Fun p(E(2)). This is the dual Fun p(E(2)) of the lIopf algebra Up(E(2)) (scc cqs. (1.11) and (4.13)). It is gcncrated by the elcmcnts (x,y,r.p,X) « Px,x >=< Py,Y >=< J,r.p >=< ~,X >= 1) with relations [x,r.p] = [y,r.p] = [X,r.p] = [x,y]=O LX,x]=-2psinr.p
,
Llr.p = r.p 0) 1 + 1 Q9 r.p
,
LX,y]=2p(l-cosr.p) ,
Lly = 1 Q9 y + y 09 cos r.p - x @ sin r.p
S'(x)==--x
,
y Q9 sin r.py - y 09 cos r.px + x 0 sin r.px] ,
S(x)==--cosr.px+sinr.py
f(r.p,X,y,X)==-O
,
,
~[x (j) cos r.py +
LlX = 1 :i9 X + X @ 1 + S(r.p)==--r.p
+ x Q9 cos r.p + Y :i9 sin r.p
Llx = 1 Q9 x
,
S(y)=-cosr.py-sinr.px
,
,
(5.11) which define a Bopf algebra as it may be checked. Now, we take A as the Hopf group algebra gcnerated by Xand 1I as the dual un deformed Euclidcan group lIopf algcbra Fun( E(2)) (eqs. (5.4) for w ==- 0) of generators (x, y, r.p). If we now define fJ, 1, to be trivial plus
J}(X) =
~[x Q:; cosr.py +
y!Xl sinr.py -
[email protected]+
[email protected]]
, (5.12)
Xe-x=-2psill:p
,
Xf;y=2p(l-cosr.p)
,
X"r.p=O
,
lOThe only non-trivial properties are (8'.23) and (B'.25). The first one is the dual cocycle condition, verified because the dual cocycle ,p is the undeformed one, and the second one is due to the compatibility between the coproduct and the commutators.
18
.LA. de Azca.rraga and J.C. Perez Bueno
from ~ and f3 in eq. (4.14), the Hopf algebra (5.11) is recovered using (8'.29) and (13'.34), which exhibits the cocycle bicrossproduct structure of (5.11). Due to the commutators [X, xl, [X, y], there is no Hopf Funp(HW) subalgebra here and no bicrossproduct structure in contrast with the previous al) case.
6.
Differential calculus on the Euclidean and Galilean planes
We shall now introduce a covariant differential calculus [40] (sec Appendix A) on the different homogeneous spaces which can be constructed. Clearly, to have a proper action on the 'homogeneous' part, a bicrossproduct structure is needed. Let us consider now a few different cases. (1) Due to the lack of a bicrossproduct structure, the inhomogeneous part of the Uq ( [(2)) algebra (PI, P2 ) docs not constitute a Hopf subalgebra, and the construction of the space algebra as the dual of (PI, P2) cannot be performed. (2) The Euclidean plane E~ is introduced as the dual « Pi, Xj >= bij) of the translation Hopf subalgebra Uw(T r(2)) of Uw([(2)) generated by Pi (eq. (3.2)). Since Uw(Tr(2)) is commutative but not cocommutative, we obtain (eqs. (5.1))
dx = x 1291 + 1 QSix
dy=yQSiI+IQSiy
[x,y] = -wy
(6.1 )
for the E~-plane algebra associated with Uw ([(2)). Let us construct a bicovariant differential calculus on E~ which is consistent (i.e. covariant) under the action of J. The (left) action of J on E~ is defined by duality, < Px <1 J, x >=< Px, J i> X > etc., from which it follows that
Ji>X = Y
J
i>
(6.2)
Y =-x
To define a first order (J - )covariant differential calculus we have to determine all commutators [Xi, dXj] in a way which is consistent with the action (6.2) (which for instance, implies Ji>xdy = (J(1) i>x)d(J(2) i>Y)) and with the Jacobi identity. Although it is not difficult to check that the set of covariance equations (like J i> (XidxJ) - J i> (dxjx;) = J i> [Xi, dXj]) has a unique solution given by
[x, y] = -wy
[:c, dx] = 0 = [x, dy]
[y, dx] = wdy
[y, dy] = -wdx
(6.3) the above COI1lmutators do not satisfy the Jacobi identity and thus fail to provide a consistent differential calculus. This situation is not new, and has already appeared for the differential calculus on other spacetime algebras [28,29]. We now show that the solution proposed there, and which involves an enlargement of the algebra which has been found to be associated with a Hopf algebra cocycle extension [29], also applies here. We stress that this problem is associated to the deformed character of (3.2) as expressed by w, being of course absent for the undeformed Euclidean Hopf algebra U([2).
Contractions, Hopf Algebra Extensions
19
Consider the trivial extension Uw (t:(2» xU(u(1» mentioned in Sec. 4, obtained by adding the primitive Hopf algebra generated by := to Uw ( [(2)). The previous procedure applied to (Px , Py ,:=) leads now to an enlarged Euclidean algebra Ew generated by (x,y,X) « :=,X >= I) and to the additional relations
[x,x]=O=[x,y]
,
J
LlX=XI8l1+1I81X
t>
X= 0
(6.4)
.
Proceeding as before, we find that there is a unique solution for the rotation covariant differential calculus on the above enlarged Euclidean 'space' specified by (d. (6.3»
[x, y]
= -wy
[y,X]
=0
, [x, X] [y,dx]
,
=0 ,
= wdy
[x, dy]
[X,dx] = wdx
[x, dx]
= wdX
, [x, dy]
=0 ,
[x, dX]
[y,dy] = -w(dx - dX)
= wdy
[X,dX]
= wdx
[y,dX]
,
= wdy
= wdX
(6.5) and satisfying Jacobi identity. (3) We define the two-dimensional Galilean plane Gi as the dual « X, x >= 1=< X"t >, < X,t >= 0 =< X"x » ofUw(Tr(2». The commutativity (noncocommutativity) of Uw(Tr(2» implies the relations
[x, t]
= 2wx
;
Llx
= x 181 1 + 1 181 x
Llt
= t 181 1 + 1 181 t
(6.6)
for the Gi algebra. Following the same pattern of case (2) we construct a bicovariant differential calculus (covariant under the action of the 'boost' V) that satisfies Leibniz's rule and Jacobi identity. The (left) action of V on Gi is given by
V
t> X
= -t ,
V
t>
t=0 .
(6.7)
Using (6.7), we find that the covariance requirement implies the system of equations
V
t>
[x, dx] = -[t, dx]- [x, dt]
V
t>
[t, dx] = -[t, dt]
V
t>
[x,dt] = -[t,dt]
V
t>
[t, dt] = 0
+ 2wdt (6.8)
.
The unique solution linear in dx, dt that satisfies (6.8), Leibniz's rule and Jacobi identity is l l
[x, dx]
=0
,
[x, dt]
= wdx
,
[t, dx]
= -wdx
[t, dt]
= wdt
(6.9)
11 Even if there is no deformation (w = 0) there exists a non-trivial solution (see [29]) given by [x, dx] = pdt and all other commutators equal to zero.
20
J.A. de Azcarraga and J.e. Perez Bueno
Thus, this case is different from the Euclidean case E~. On Gi there is a covariant differential calculus without any additional one-form. l2 (a) It was seen (eq. (6.5)) that to define a }-covariant differential calculus on E~ it was necessary to enlarge it to Ew . Let us now show that two N -covariant calculi may be similarly constructed on Funp(HW) (eqs. (5.7)) as the dual of the Up(HW) subalgebra of Up ( £(2)) (Sec. 4( a)). The left action t> of N on (Yb Y2, X) is obtained from (4.7) and given by N
t>
Yl
= Y2
N
,
t>
Y2
= -YI
(6.10)
Proceeding as before, we find the commutators [Vi, Yj] = 0
fYi,
,
[x, dy;] = )"dy,
xl =
2PYi
[Vi, dxl = ()..
+ 2p)dYi
[X, dX] = W1X
(6.11 )
The Jacobi identity requires)., = -2p or )., - fL = -2p. The bicovariance requirement now determines two bicovariant differential calculi over Funp(HW) (on the E~ plane the coproduct of the generators was primitive, hence the differentials are bi-invariant by (A.4) and the bicovariance is trivial). We first find, using (5.7) and (A.4),
+! i1 R dX = dX ® 1 +!
i1L dX = 1 ® dX
(YI ® dY2 - Y2 ® dyIl
(6.12)
(dYI ® Y2 - dY2 ® YI)
it is easy to show that the coact ions (6.12) satisfy (A.2). If we use now (A.I) to calculate i1dx, dxl = fLi1LdX we find fL = 2).,; the same condition is obtained Ilsing i1 R • Then, (6.11) leads to ()., = -2p, fL = -4p) [Vi, Yjl
=0
,
[Vi,
[x, dy;] = - 2pdYi
and ()..
xl = 2pYi [Yi,dXl = 0
,
[X,dX] = -4p dX
(6.13)
= 2p,fL = 4p) [Vi, Yj] = 0
,
[X, dy;] = 2pdYi
[Vi, xl = 2pYi [Vi, dxl = 4pdYi
[X, dxl
= 4pdX
(6.14)
12For the differential calculus on the deformed Newtonian spacetime associated with the (I version of the deformed Galilci algebra 9. see [29].
+ 3)
Contractions, Hopf Algebra Extensions
21
Since (A.3) is satisfied, eqs. (6.13), (6.14) determine two first order N-covariant differential calculi over Ep • Acknowledgments This paper has been partially supported by the CICYT grant AEN93-187. One of us (JCPB) wishes to acknowledge a FPI grant from the Spanish Ministry of Education and Science and the CSIC. Both authors wish to thank M. del Olmo for very helpful discussions.
Appendix A:
Bicovariant differential calculus
Let A be a Hopf algebra and let Ll and E be its coproduct order bicovariaul differential calculus over A is defined [40] by d : A -> r is a linear mapping satisfying Leibniz's rule and r bimodule (r, LlL, Ll R ) i.e., the linear mappings LlL : r -> A 0 r and the exterior derivative d satisfy
and counit. A first a pair (r, d) where is a bicovariant A, Ll R : r -> r 0 A
LlR(aw) = Ll(a)LlR(w) Lldwa)
= Lldw)Ll(a)
,
LlR(wa)
= LlR(w)Ll(a)
(A.I)
(A.2)
(f 0 id)LlL = id
(A.3)
(A A)
r
where the left (right) equations in (A.2) express that is a left (right) A-comodule, (A.3) is the result of bicovariance (commutation of the left and right coaetions), and (AA) expresses the compatibility of the exterior derivative d with Ll and LlL,R. Eqs. (A.I), (A.2) and (A.3) characterize (r,LlL,Ll R) as a hicovariant bimodule over A; the addition of (AA) determines a first order hicovariant differential calculus (r, d). An element w E r is called left (right) invariant if Lldw) = I 0 w (Ll R ( w) = w Q:) I). As in the undeformed (Lie) case, the basis elements of the vector space Iinv C r of the left-invariant elements generate as a left free module.
r
Appendix B:
Bicrossproduct of Hopf algebras and co cycles
We list here for conveuience the basic formulae of Majid's bicrossproduct and cocycle bicrossproduet constructions and refer to [i 1,12] (see also [14]) for details.
22
.l.A. de Azcarraga and J.e. Perez Bueno
TIlE' expressions which characterize 'H~ A (used in Secs. 3,1) involve the mappings a: A®'H -> A (right 'H-module action), (3: 'H -> A®'H (left A-comodule coaction), 'H Q9 'H -> A (two-cocycle) and 1/J : 'H ---+ A Q9 A (hence the more detailed notation 'H{3''''~Q,~ A, see [11]). Those of the dual case ( 1I",( ~t3,,z; A when all ingredients are indicated) involve the respective dual operations; they were used in Sec. ,J. We may think of 'Hr>~ A as emphasizing the 'algebra-like' aspects and of IJ .
<4 A may be found in the original papers [11,12] (or in the Appendix of [29] with the same numbers they are referred to in the main text, also corresponding to the dual formulae for IJ ~A below). Let H and A be Hopf algebras and let
e:
e,
a) H be a left A-module algebra (H >:JA) b) A be a right H-comodule coalgebra (H lOC A) i.e., there exist linear mappings
a:AC5iH---+H
® h)
o:(a j3(a)
== at>h
(WI)
a E A, hE H
= a(1) 0 a(2)
A,
a(1) E
a(2) E
IJ
(B'.2)
such that the properties of al) 0: being a left A-module action i>:
(B',3) a't>(a~h) = a'ai>h
(B'A)
a2) H being a left A-module algebra:
(B'.5) bl)
13
being a right IJ -comodule coaction: a(1)
a(l)(l)
\1 fll(a(2)) = a ® IH
== a [(id 0 f)
\1 a(I)(2) (1 a(2) = a(l) \1 a(U) 0 a%
[((3 \1 id)
013
013=
=
id]
(id ® L1)
(B'.6) 0
13]
(B'.7)
13We use the bicrassproduct notation ~ or ~ rather tban the (right, left) crassproduct (e><:, >:I) or the (left, right) cross coproduct ()oIl,~) even if the caactians /3, /3 or the actions n, ii are trivial, and omit explicit reference to them (or to {, '" etc.)
Contractions, Hopf Algebra Extensions
23
b2) A being a right II-comodule coalgebra:
(B'.8) a [(,1 09
id)
0
rj
(I) (I)
.
(I)" (2)::>9
0) a
= (id ® id (2)
(2)
a
(I)
(1)
(2)
(2)
= a(1) (9 a(2) ® a(l) a(2)
mJl) 0
(id
® id)
({3 ® (3)
,1
== (l3c29{3)
,1] ,
(9
T
T
is the twist mapping, are fulfilled.
0
0
0
(8'.9) where l/IH is the lllultiplication in II and Then, if the compatibility conditions
(WID) ,1(ai>h) == (ai>h)(I) <X: (a[>h)(2) = (a(l;li>h(1))
is)
a(W(a(2)i>h(2»)
ij(lA) == I~) is: I~) = I A
is)
III
(J(ab) == (ab)(1)
(9
(ab)(2) = adl)b(1)
is)
,
(8'.11 ) (8'.12)
,
a(g)(a(2)i>b(2))
a(g) 0 (a(1)i>h)a(g) = a(g) 0 a(g)(a(2)t>h)
,
,
(8'.13) (13'.14)
r
are satisfied l 4, there is a Hopf algebra structure on [11 f{ == H Q9 A called the (left-right) bicrossproduct III'> ~IJ A (II ~A for short) defined by
(h®a)(g(\\:b)=h(a(1)[>g)®a(2)b, ,1/.,:(h Q9 a) qt·
h,gEII;a,bEA
= h(l) Q9 a(g) ® h(2)a(g) is) a(2)
= (Jl is) fA
,
1/\
= If{ Q9 I A
S(h 0 a) = (If{ @SA(a(1)))(Sll(ha(2))
IA)
(13'.15) (B'.16)
,
(8'.17)
,
(9
,
.
(13'.18)
In f( = II ® A, h == h IX: I A and a == If{ Q9 a; thus, ah = a(l)i>h @ a(2). There are two cases of special interest [15] (see also [II]). When i3 = I Q9 If{ i.e. (J(a) = a Q9 I Jl (trivial coaction) and A is cocommutative, J( is the scmidirect product of Hopf algcbras since then ,1n(h 0 a) = (a(1) 0 h(1)) Q9 (b(2) 0 g(2»)' When 0' is trivial, 0: = (A @ IJl (a[>h = hfA(a)) and H is commutative, J( is the semidircct coproduct of lIopf algebras since (h ® a)(g Q9 b) = hg 0 abo When 0: is trivial, (3(ab) = i3(a)(J(b) (algebra homomorphism). As for Hr>.ttl A, the above construction may be extcnded to accommodatc cocycles [11,12]. Let A and H two lIopf algebras and 0- and (J as in (B'.I), (B'.2). Then H is a left A-module cocycle algebra if (B'.3), (B'.5) are fulfilled and thcre is a linear (two-cocycle) map ~ : A ® A -+ f{ such that
(13'.19) 141f A is cocommutative and II commutative, condition (B'.14) is automatically satisfied.
J.A. de Azca.rraga and J.e. Percz Bueno
24
a(1)t>'(b(1) Q0 c(1))'(a(2) 181 b(2)c(2)) = ~(a(1) 181 b(1))'(a(2)b(2) 181 c) , Va, b, c E A, (B'.20) (cocycle condition) and (B'.4) is replaccd by
which for' trivial rcproduces (B'.4). Similarly, A is a right lJ-comodule coalgebra cocyclc if (B' .6), (B' .8), (B' .9) are fulfilled, and there is a linear map 1/J : A -+ H 181 H, 7/J(a) = ~(a)(1) 0 7jJ(a)(2), such that
t(1/J(a)(1))~(a)(2) = 1t(a)
= ~(a)(I)t(1/J(a)(2)),
.::1~(a(I))(1){>(a(W) Q9 ~(a(1))(2)a(g)
[(( 181 id)
0
~
= (id 181 () o~] ,
= ~(a(1))(1) Q9 .::11/J(a(1))(2)~(a(2))'
(B'.22) Va E A , (B'.23)
(dual cocycle condition) and (B'.7) is replaced by (1
Q9
{>( a(I)))( (,8 Q0 id)
0
;1( a(2»))
= a(g) Q9 .::1a(g)~( a(2))
(B'.24) = ((id 181 .::1),8(a(I)))(l Q9l}(a(2)))
Then, if the compatibility conditions (B'.10), (B'.12), (B'.14) and .::1(a(1)c>h){>(a(2)) = ~(a(1))[a(g)i>h(1) Q9 a(~~)(a(3)i>h(2))1 (1181 ~(a(1) 09 b(1))),8(a(2)b(2))
(B'.25)
,
= a(i;lb(g) 181 a(g\a(2)i>b(g)),(a(3) Q9 b(2))
,
(B'.26)
(which replace (B'.11)15 (H'.13)), together with .::1'(a(1) ® b(I))lNa(2)b(2)) = 1/J(a(l)) [(a(~;)i>1/J(b(1))(1))~(a(~;) 181 b(~;))181 a(W( a(3)i>1/J( b(1))(2))a(~~\ a(5)i>b(W)'( a(6) Q9 b(3»)] , (B'.27) tWa 1& b)) = ((a)t(b) , ~(1A) = III 1&111 (B'.28) hold, (A, H, ix,,8, " ~) determine a cocycle left-right bicrossproduct bialgebra Hi. ~<J,J; A. In it, the C0l11lit and unit are defincd by (B'.17) and the product and coproduct (B'.15), (H'.16) arc replaced by (h 00 a)(g ® b) = h(a(1)t>g)'(a(2) ®
b(I)) Q9
u(3)b(2)
.::1(h ® a) = h(l)1/J(a(l))(l) 181 a(g) Q9 h(2)~(a(1))(2)a(g) 181 a(3) 15With ~(u(l)
@
b(l))C 1 (U(2)
(8:
(B'.29)
, .
b(2)) = {(a){(b) (convolution invertible [11]), eq.
(B'.30) (B.26) gives
l3(ab) = a(Wb(W ® C 1 (a(l) (8: b(l»)a(g)(a(J)t>b(~~)){(a(4) @ b(J»). If A is Abelian, as is always the case in the cocycle bicrossproduct structures in the main text, this formula reduces to (B'.13).
Contractions, Hopf Algebra Extensions
For
25
It is convenient to have the explicit expression of (8'.27) in the more simple cases. 1f; trivial it reads
"b) d~ ( a ®
:=
(2») -( ~t( a p(1) ) Q9 b(I)(I») Q9 ap(2)( ) a(2)l>b(l) ~ a(3) Q9 b) (2)
(B'.31)
For t> trivial, it gives
.I.( )[.I.(b )(l)i( (I) b (I») .I'(b )(2) a(2) (2)b (2) (2)(( b) 'f/ ap) 'f/ (I) ~ a(2) ® (2) Q9 'f/ (1) ~ a(3) 181 (3)
(B'.32) For
1f;
and t> trivial, it reduces to -(
""(
(I)
"
d~ a 181 b) = ~ a(l) Q9
b(I)(I») ® a (2)) b(l)(2)"~ ( a(2) iI9 b(2) ) p
(B'.:J3)
For (trivial [((a ® b) = c(a)t(b)IHl (B'.21) reduces to (B'.4), (B'.26) to (B'.13) and (13'.29) to (B'.15). For 1f; trivial [1f;(a) = IH il91l/c(a)], (B'.24) reduces to (B'.7), (B'.25) to (B'.II) and (B'.30) to (B'.16). For (3(a):= a 181 I trivial, (B'.30) gives for the elements of A with original primitive coproduct the cocycle extension expression
d(l Q9 a) = I ® a ® I iI9 I + I 181 11811181 a + ~(a)(I) Q911811f;(a)(2) 181 1
(B'.34)
which in f{ simply reads d(a) := 1181 a + a 181 1 + 1f;(a). This was used for (5.7) [(5.6), (5.8) [(5.10)] and (5.11) [(5.12)].
REFERENCES [1] E. iniinii and E.P. Wigner, Proc. Nat. Acad. Sci. 39 (1953) 510. [2] E. iniinii, "Contractions of Lie groups and their representations" in Group theor. concepts in elem. part. physics, F. Giirsey cd., Gordon and Breach 1964, p. 391. [:1] V. Aldaya and .LA. de Azcarraga, Int. J. of Theor. Phys. 24 (1985) 141.
[4] J. A. de Azcarraga and J. M. Izquierdo, Lie algebras, Lie groups cohomology and some applications in physics, Camb. Univ. Press 1995. [5] E. J. Saletan J. Math. Phys. 2 (1961) 1. [6J V. G. Drinfel'd. in Pmc. of the 1986 Int. Congr. of Math., MSRI Berkeley, vol I, 798 (1987) (A. Gleason, ed.). [7] M. .limbo Lett. Math. Phys. 10 (1985) 63; ibid 11 (1986) 247. [8] 1"D. Faddeev, N. Yu. Reshetikhin and 1. A. Takhtajan, Alg. i Anal. 1 (1989) 178 (Leningrad Math. J. 1 (1990) 193). [9J E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini J. Math. Phys. 31 (1990) 2548; ibid 32 (1991) 1155, 1159.
26
.LA. de Azcarraga and J.e. Perez Bueno
[10J E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini, "Contractions of quantmn groups", in Lee. Notes Math. 1510, (1992) p. 221. [IIJ S. Majid J. Algebm130 (1990) 17; Israel J. Math. 72 (1990) 133. [12J S. Majid and Va. S. Soibelman J. Algebra 163 (1994) 68. [13J W. Singer J. Algebra 21 (1972) I. [14] R. J. Blattner, M. Cohen and S. Montgomery, Trans. Am. Math. Society 298 (1986) 671; R. J. Blattner and S. Montgomery, Pac. J. Math. 137 (1989):n. [15] R. Molnar J. Algebra 47 (1977) 29. [16] J. Lukierski, A. Nowicki, H. Ruegg and V.~. Tolstoy Phys. Lett. B264 (1991) 331; J. Lukierski, H. Ruegg, and V.N. Tolstoy, "K-quantum Poincare 1994", in Quantum groups: formalism and applications, J. Lukierski, Z. Popowicz and J. Sobczyk eds, PWN, Warsaw 1994, p. 3.19. [17] S. Majid and 11. Ruegg Phys. Lett. B334 (1994) 348. [18] S. L. Woronowicz ComrmlT!. Math. Phys. 149 (1992) 637. [19] P. Schupp, P. Watts and B. Zumino Lett. Math. Phys. 24 (1992) 141. [20J A. Ballesteros, E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini J. Phys. A26 (199:3) 7495. [21] A. Ballesteros, E. Celeghini, F. .I. Herranz, M. A. del Olmo and M. Santander J. Phys. A27 (1994) L369. [22] N. A. Gromov and V. I. Man'ko J. Math. Phys. 33 (1992) 1374. [23] D. Ellinas and J. Sobczyk J. Math. Phys. 36 (1995) 1404. [24J W. K. Baskerville and S. Majid J. Math. Phys. 34 (1993) 3588. [25J F. Bonechi, R. Giachetti, E. Sorace and M. Tarlini Commun. Math. Phys. 169 (199.1) 627. [26] L. L. Vaksman and 1,. I. Korogodskii, Sov. Math. Dokl. 39 (l9R9) 173. [27J J. A. de Azcarraga, M. del Olmo, J. C. Perez Bueno, in preparation. [2SJ A. Sitarz Phys. Lett. B349 (1995) 42. [29] J. A. de Azcarraga and J. C. Perez Bueno J. Math. Phys. 36 (1995) 6879. [30J S. Giller, P. Kosinski, M. Majewski, P. Maslanka and J. Kunz Phys. Lett. B286 (1992) 57. [31] A. J. Macfarlane J. Phy". A22 (1989) 4581.
Contractions, Hopf Algebra Extensions [:J2] L. C. Biedenharn J. Phys. A22 (1989) L873. [3:J] M. Arik and D. D. Coon J. Math. Phys. 17 (1976) 524. [:J4] P. P. Kulish, Theor. Math. Phys. 86 (1991) 108. [35] C. Il. Oh and K. Singh J. Phys. A27 (1994) 5907. [36J C. Quesne and]\,. Vansteenkiste, q-algj9510001. [:J7] P. Maslanka J. Math. Phys. 35 (1994) 76. [38] J. Sobczyk, Czech. J. Phys. 46 (1996) 265. [:J9] V. Hussin, A. Lauzon and G. Rideau Lett. Math. Phys. 31 (1994) 159. [40] S. L. Woronowicz Commun. Math. Phys. 122 (1989) 125.
27
From Fidd TllCory to Quantum Groups
THE LINEAR DIFFERENCE DERIVATIVES AND SOME Q-SPECIAL FUNCTIONS MALGORZATA KLIMEK
Institute of Mathematics and Computer Science, Technical University of CZljstochowa ul. Di}browskiego 73, 42-200 CZljstochowa, Poland E-mail; klimek at matinf.pcz.czest.pl
Abstract: We introduce the simplest generalization of difference and q-derivatives the linear difference derivatives and investigate their properties. They are used in explicit construction of Q-deformed independent oscillators. The applications to deformation of quantum oscillator, Legendre and Hermite equations are studied and the eigenproblem for discrete part of their spectra is solved.
1.
Introduction
The q-special functions and their application have been widely studied in the literature (see for example [1,2]). In the paper we propose the generalization of the q-derivatives and difference derivatives, used before in deformation of classical problems. We shall show that Q-commutation relation can be realized by means of the operator which we call the linear difference derivative. The creation and annihilation operators, constructed using this derivative, will then be applied to construction of the set of independent Q-oscillators. In the section 3 we consider also some classical equations deformed using the linear difference derivative. The preliminary results include the eigenstates of deformed quantum oscillator and polynomial solutions of the Legendre and Hermite equations, connected with the discrete part of their spectrum. The interesting feature of this deformation of quantum oscillator is the finitness of dicrete spectrum (33,40). The number of possible eigenvalues depends on parameters, describing the initial linear difference derivative used in the deformation.
30
2.
Malgorzata Klimek
Generalized Difference Derivatives and Their Properties
Let us remind the non-symmetric generalized difference derivative acting as follows on the real functions of real variable [3,4]:
(I) where 1; denotes the transformation of the set of real numbers 1; : JR -> JR. In the paper, we shall also consider the symmetric case given by the following formula
In the above definitions, we have used the transformation operator ( which acts on functions in the following way:
C f(x)
(+ f(x) = f(¢(x)),
= f(1;-I(X)) ,
Acting with the (-operator on the product of functions we obtain:
([J. g(x)] = U(x). (g(x). Finally, the important feature of the calculus of new derivatives is the fact that they do not commute with the transformation operator
(a~f(x) = cP2~~~)_~~x)a~u(x),
(3)
([)~J(x)
(4 )
=
cP(X~(-t-I(X)[)~U(X). 1;
X
-
X
Let us notice that when we choose ¢ as the translation (J(x) = J(x the known difference derivatives: I
fhJ(x) = Tz[J(x
+ h) -
J(x)],
I
DhJ(x) = 2h [J(x
+ Il)
+ h)
we obtain
- f(x - h)]
(5)
and far the transformation of dilatation U(x) = f(qx) we end up with q-derivatives
a J(x) = q
J(qx) - J(x) , (q - I)x
[) J(x) = J(qx) - J(q-I X) . q (q_q-l)X
(6)
Both types of derivatives are known from the theory of iterative functional equations and are also used in realizations of the quantum algebras generators [5-8]. We propose the generalization of the operators given by (5,6) (first introduced [3,4]) and we would like to present in the paper its application to deformation of
The Linear Difference Derivatives and Some Q-Special Functions
31
quantum oscillator and to construction of certain Q-deformed special functions. First we derive the realization of the Q-dcformed commutation relations. It is clear that not all transformations ¢ yield the algebraic relations of this type. We will show, however, that we need not restrict the calculus to the known difference and q-derivatives. We start from the Leibniz rule for genemlized difference derivatives: (7)
Comparing with the Leibniz rule for differential derivative, we notice the modification that the transformation operators appear on the right-hand side of both formulae. This is transferred onto the commutation rule for the derivatives and coordinate variables which we obtain from the formulae (7) taking f( x) = x and 9 an arbitrary function (8) The commutation relations can be reformulated using the properties of the transformation operators in order to compare them with the standard formula for the coordinate- and momentum operator from quantum mechanic:
(9) As we see on the left-hand side, we have now the isotopic commutation relations [9] for the redefined coordinate operator (+x, but on the right-hand side the isotopic identity (in this case the (+ -operator) does not appear. Thus the relations need to be transformed in order to obtain closed Q-isotopic commutators. This procedure can be realiwd only when the transformation operator and the generalized difference derivatives Q-commute (3,1). This condition implies the following iterative functional equations for the transformation ¢:
a",¢(x)
= const = Q
fJ",¢J = const = Q
(10)
It is easy to check that apart from dilatation and translation transformations also the linear transformation fulfills the above equation. We shall investigate from this point the properties and applications of this simplest generalization of the operators used in the literature. We call them lincur difference derivatives. They commute with the linear transformation operator in the following way:
where the ( operators describe now the linear transformation
U(x) = f(Qx
+ h)
Malgorzata Klimek
32
Taking this into account we obtain after redefinition of the momentum and coordinate operators :
(i) the isotopic Q-commutation formula for the non-symmetric case [10]: PTX - QXTP =
r- 1
(11)
with the momentum, coordinate and isotopic operators looking as follows:
T = (-8-t where we used the non-symmetric linear difference derivative which acts on functions according to general formula (I):
fjJ( )=J(Qx+h)-J(x). q, x (Q-I)x+h' (ii) the isotopic Q2-commutation relation for the symmetric case: (12)
with the following realization of the momentum, coordinate and isotopic operators
t
= (-8-t+1
where the symmetric linear difference derivative acts following the formula (2):
fJ J( ) = J(Qx + h) - J(Q-I X 1>
x
- Q-Ih) (Q_Q-I)X+(I_Q-I)h
We see that in both cases we deal with a two-parametric realization depending on the integer numbers.'> and t.
In order to restrict these algebras to the standard (that means non-isotopic) relations we choose correspondingly: for the non-symmetric case for the symmetric case
s+ t
= 0 => T = 1
s + t - 1 = 0 => T
=1
(13)
(14)
When these conditions are fulfilled we obtain the Q- and Q2-dcformed commutation relations: (15) PX-QXP=I,
Px - Q2 XP = 1
( 16)
The Linear Difference Derivatives and Some Q-Special Functions
33
with the redefined momenta and coordinate-operators looking as follows: X
= (s+JxC s =
X=
("+I X
(Qs+l x +[s+I]Qh)(+,
c s+1 =
(QS+1 X
+ [s + I]Qh) (2
(17) (18)
where
In both cases the momenta are represented by the linear difference derivatives multiplied with transformation operator. The representation of the coordinate operator depends strongly on the choice of the free integer parameter s. We obtain the simplest formulae for s = 0:
3.
Applications
3.1.
Linear Difference Derivative and Independent Deformed Oscillators
The proposed realization of the deformed commutation relations can be easily extended to the multidimensional case, where the momenta are realized using the partial linear difference derivatives and partial transformation operators acting on functions of many variables:
(19)
od(x) := [((tx;) - xit 1[(t - IlJ(x) ,
(20)
(jd(x) := [((tx;) - (C X i)tl[(t - C]f(x).
(21 )
When we take in all these partial transformation operators the same parameters
QJ = Q2 = ... = Q" we obtain two sets of commutation relations (22) (23) which are known from the model of independent deformed oscillators [5,6, II] used in the realizations of generators of quantum algebras as functions of oscillators.
3.2.
The Deformation of Quantum Oscillator
Let us consider the following iterative functional equation (24)
34
Malgorz ata Klimek
with the momen tum and coordin ate operato rs realized accordin g to formula (17). To calculat e the spectru m En we apply the techniq ue known in the theory of differential equatio ns as models with shape-in variant potentia ls [12]. It was generali zed to Hamilto nians which allow decomp osition into creation and annihila tion operato rs dependi ng on parame ters [3,13-15]. In our case it is enough to use one-par ameter decomp osition which looks as follows:
(2.5) with the explicit formulae for the creation and annihila tion operato rs
(26) It is easy to check that the following commu tation relation is fulfilled (27) where the function F depends on the transfor med parame ter A 2
A'
F(A' ) = ~+ N Q'
A'
= f(A) = QA.
It was proven in the literatu re that when the formula (27) is valid the spectru m of the Hamilto nian At(A)A(A) can be constru cted using iteration s of transfor mation of parame ter A At(A)A( A)P n = unP n n
Un
=
L: F[f(A)] = 1=]
w
(28)
1
± m( -[n]l + Q[n]Q) v"'t Q Q
where the initial value of A = ±Qf w is determi ned by conditio n (25). The eigensta tes P n are constru cted using the creation operato rs At
(29) provide d that the n-th vaccum
cPn fulfilling the equatio n of first order
(30) docs exist. To obtain this function we use the Q-comm utation relation for monom ials:
Applyin g the above relation , we can constru ct the n-th vaccum as the Q-expo nential function :
The Linear Difference Derivatives and Some Q-Special Functions
35
where I 0 > is arbitrary constant function and [2k]Q!! := [2]Q[4]Q ... [2k]Q. This construction is correct only when the series representing the function is convergent. Using the d'Alembert criterion we conclude that this is the case for
Q< I
(33)
Let us notice that the number of eigentates depends strongly on the value of characteristic parameters defining the creation and annihilation operators (17,26). For h =J 0 we obtain finite number of possible vaccums what results in finite number of eigenstates. Putting h = 0 we deal with standard Q-dcformation of quantum oscillator with infinite number of eigenstates. When both the above conditions (3:3) are fulfilled the function given by (32) is defined properly for arbitrary x E R. Like in the classical model we choose the branch of solution with the" -" sign (corresponding to the initial value of A = -w,fiJ) to obtain the function with proper asymptotic behaviour. The result of our construction is the following spectrum and eigenstates of deformed quantum oscillator constructed using the linear difference derivative: 2 2 W 1[ 1 [ ]Q (P 2 -w,l' )<1>n=- ,fiJ((jn&+Qn
<1>11 =
-
Q )<1>n,
At(A)At(f(A)) ... At(fn-l(A))cPn.
In this case the deformation consists of the modification of spectrum as well as of the entirely different construction of eigenstates. The transition from one-vaccum to many-vacua eigenstates is characteristic feature of Hamiltonians with splitting (25) and commutation relation (27) depending on transforming parameters. All the presented formulae are valid also for deformation via symmetric linear difference derivative. Let us report briefly the results. As before, the Hamiltonian is splitted into product of annihilation and creation operators depending on parameter:
p2 _ W
At (A)
2
X 2 = At(A)A(A)
= P + AX ,
+ const, w2
..
A(A)
=P
-
(35) _
TX
.
The commutation relation for these new operators is similar to the previous one (27) with changed transformation of parameter A
(36) where
w2
F(A')
,\'
= l' + Q2 '
36
Malgorzata Klimek
It implies the following spectrum and eigenstates of Hamiltonian (:JS)
(37)
The n-th vaccum,
Applying the d'Alembert criterion to these function we conclude that it is correctly defined for x E R when the corresponding conditions are fulfilled:
Q < 1,
(40)
As before, these conditions yield the finite number of eigenstates for h for h = O.
3.3.
#- 0 and infinite
The Deformed Legendre and Hermite Equations
We shall study two examples of deformation of second-order differential equations with polynomial solutions [16]. We replace the differential derivative in the operator of equation with the operator P as well as the standard variable with X given by (17). We start with the Legendre equation:
(41 ) The problem of finding the spectrum Un can be solved by the technique applied in the previous section. In this case, however, we would deal with many-parameter decomposition (2.5). The simpler way is to follow the analogy with the case of differential equation and assume that the solution is the polynomial of n-th order: n
L" =
L
ak,yk
I0 >
(42)
k=O
where the monomials X k I 0
X k I 0 >= (Qs+1 X
> look as follows
+ [s + l]h)(Qs+2 x + [5 + 2]h) ... (Qs+k x + [s + k]h) 10>
(43)
The Linear Difference Derivatives and Some Q-Special Functions
37
Throughout this section we shall denote [n]Q = [n]. In the calculation, we use the commutation relation between the operator of Legendre equation and the polynomial:
2 [P(l - X )P,
n
n-2
n
k~O
k=O
k=O
'E akXkj = 'E ak+2[k + 2][k + l]X k - 'E ak[k][k + I]X k .
(44)
It yields the following conditions for coefficients of polynomial L n :
[k][k + I] - (1n ak+2 = ak [k + 2][k + I]
0:S;k:S;n-2,
(45)
an-l ([n][n - I] - (1n) = 0, a,,([n][n
+ I]
- (1n) = O.
Solving this set of equations we obtain the explicit expression for eigenvalues
(1n = [n][n
+ I]
(16)
which are connected with the polynomials determined by coefficients given by the formulae: for n even:
ak+2
= ao
([k
+ I][k]- (1n)([k -
l][k - 2]- (1n)"'([O] - (1n) [k + 2]!
(47)
where k is an even integer number fulfilling 0 :S; k :S; n - 2; -- for n odd
ak+2
= al
([k
+ l][k] -
(1n)([k - l][k - 2]- (1n) ...([2] - (1n) [k + 2]!
(48)
with k an odd integer number fulfilling 0 :S; k :S; n - 2. Thus the deformation of Legendre equation constructed using linear difference derivative looks as follows:
(49) with the following explicit form of lowest order polynomial solutions:
Lo = ao,
L 2 = ao(1 - [3]X 2 ) I 0 > , L. = a (1 3
1
+ [2] -
[3][1]X 3 ) 1 0 > [2][3]
The next example we study is the Hermite equation written down using the linear difference derivative P and new variable X given by (17 )
(50)
Malgorzata Klimek
38
The assumption that the solution is the polynomial of n-th order yields now also the equations for its coefficients
ak+2 = ak[k
2[k]- 17n + 2][k + 1]'
an _l(2[n - 1]-
17 n )
a n (2[n] -
0~k~n-2,
(51 )
= 0,
17n )
= 0
which imply the explicit formula for eigenvalues 17 n
(52)
= 2[n]
as well as expressions for coefficients of polynomials H n for n even (2[k]- 17 n )(2[k - 2] - 17n ) ... (2[0]ak+2 = ao [k + 2]! where k is an even integer number fulfilling 0 ~ k ~ n - 2; - for n odd (2[k] - 17n )(2[k - 2]- 17n ) ... (2[1]ak+2 = aj [k + 2]!
17n )
17n )
(53)
(54)
with k an odd integer number fulfilling 0 ~ k ~ n - 2 . The polynomials determined by the above equalities solve the eigenproblem for the deformed Hermite equation: (P 2
_
2XP
+ 2[n])I1n
~ 0
(55)
with the lowest order solutions of the form:
Ho = ao,
Let us notice that the calculations for the deformation with symmetric derivative are very similar, so we have omitted them. is the orthogonality of solutions is the open problem. We should construct the scalar product and check the properties of solutions. As the definite integral for generalized difference is known [3,4], we expect to be able to use it in the construction for redefined variables X. 4.
Final remarks
We have derived in the paper the deformed commutation relations for linear difference derivatives and redefined variables using the properties of generalized difference
The Linear Difference Derivatives and Some Q-Special Functions
39
deri vat ives. It appears that the linear difference derivative is the simplest generalization of the difference allCl q-derivatives which allows us to realize the set of deformed independent oscillators. In this context it would be interesting to deform field-theoretic equations using linear difference derivative and construct the Hopf algebras of their symmetry operators in the way it was done for q-derivatives [17,18] and for difference derivatives [7,8,19] . We applied the linear difference derivatives to deformations of quantum oscillator as well as to the Legendre and Hermite equations. Using different techniques we have constructed the discrete part of their spectra and eigenfunctions. It seems that the introduced derivatives would be also useful in deformed realization of the general hypergeometric equation. This problem as well as the question of orthogonality of solutions needs further investigations.
REFERENCES [I] H. Exton, q-J/ypergeometric Functions and Applications Ellis Horwood, Chichester, 19H:l. [2J R. Askey, S.K. Suslov, Lett. Math. Phys. 29 (199:l) 12:l. and references therein. [:lJ M. Klimek, Noncanonical Extensions of Dynamical Systems Ph.D. Thesis, University of Wrodaw, 1992 (in Polish). [4] M. Klimek, J. Phys. A: Math. Gen. 26 (199:l) 955. [5] A. MacFarlane, J. Phys. A: Math. Gen. 22 (1989) 4581. [6] L.C. Biedenharn, J. Phys. A: Math. Gen. 22 (1989) L87:l. [7J .J. Lukierski, A.
~owicki,
H. Ruegg, Phys. Lett. B293 (1992) :l44.
[8J J. Lukierski, II. Ruegg, W. Riihl, Phys. Lett. B313 (1993) 357. [9J R.M. Santilli, Hadr. J. 1 (1978) 223. [10] M. Klimek, in Proceedings of the XXX Winter School of Physics: Quantum Groups Formalism and Applications, ed. J. Lukierski, J. Sobczyk, Z. Popowicz, Polish Scientific Publishers PWN, Warszawa, 1995. [IIJ T. Hayashi, Commun. Math. Phys. 127 (1990) 129. [12J D.L. de Lange, A. Welter, J. Phys. A: Math. Gen. 25 (1992) 5753 and references therein. [13] L.E. Gendenshtein, JETP Lett. 38 (1983),299
40
Malgorzata Klimek
[14] A. Arai, "Exatly Solvable Supersymmetric Quantum Mechanics", Hokkaido University Preprint in Mathematics 62/1989 [15] M.Klimek, J. Phys. A: Math. Gen. 25 (1992) Lll. [16] G.M. Murphy, Ordinary Differential Equations and Their Solutions, D. Van Nostrand Company, New York 1960. [17] R. Floreanini, L. Vinet, Quantum Symmetries of q-DifJerence Equations, Preprint Universite de Montreal, CRM-2167/1994. [18] R. Floreanini, L. Vinet, Lett. Mat. Phys. 32 (1994) 37. [19] A. Nowicki, E. Sarace, M. Tarlini, Phys. Lett. B302 (1993) 419.
From Field Theory to Quantum Groups
THE K-WEYL GROUP AND ITS ALGEBRA
Department of Theoretical Physics University of Ladt ul. Pomorska 149/159, 90-236 Mdt, Poland
Department of Functional Analysis University of Ladt ul. Ranacha 22, 90-238 Ladt, Poland
Abstract: The K-Poincare group and its algebra in an arbitrary basis are constructed. The K-deformation of the Weyl group and its algebra in any dimensions and in the reference frame in which 900 = 0 are discussed.
1.
Introduction
It is our great pleasure to contribute to this volume. In last years we had an opportunity to collaborate with Prof. Jerzy Lukierski. Our common topic has been the deformed symmetries of space-time, mainly the so called K-Poincare algebra invented by Lukierski, Nowicki and Ruegg [1]. Apart from investigating the formal properties of K-Poincare algebra and looking for its possible physical applications one of the main ideas of Prof. Lukierski is to extend the notion of K-deformation to larger groups of space-time symmetries. This idea resulted in series of papers [2-6] devoted to the K-deformation of SUSY extensions of the Poincare symmetry. The next step to be done is to look for K-deformed conformal group / algebra. This problem has not been fully solved yet but some preliminary steps were already undertaken [7,8]. Inspired by these papers and numerous discussions with Prof. Lukierski we attempt here to make it small step toward the solution of this problem. lSupported by KBN grant 2P30221706p02
42
P. Kosinski and P. MaSlanka
Classically, the conformal group in four dimensions is nothing but SO( 4,2). However, the standard (matrix) parametrization of S'O( 4,2) is not used, when SO( 4,2) is viewed as conformal group. On the contrary, the conformal group is obtained from the action of SO( 4,2) on light cone in sixdimensional space-time. But the light-cone coordinates are related in rather complicated way to Minkowski coordinates in four dimensions. Consequently, the standard parametrization of SO(4,2) is related to the "conformal" one by a complicated (even somewhere singular) change of group parameters. This poses no problem on the "classical" level. However, if we are passing to the "quantum" (i.e. deformed) case we are faced with typical ordering problems of quantization procedure. This gives some flavor of difficulties one meets trying to deform the conformal group. In the recent paper [8], Lukierski, Minnaert and Mozrzymas considered a new class of classical r-matrices on conformal algebras in three and four dimensions, which obey the classical Yang-Baxter equation and depend on dimensional parameter. An important observation concerning the d = 4 case was that the classical r-matrices obtained by them depend only on generators belonging to Poincare subalgebra of conformal algebra. Due 1.0 the fact that they obey the classical Yang-Baxter equation (and not modified one) they provide r-matrices for any algebra, containing Poincare algebra as subalgebra. One of the r-matrices considered in [8] leads to the so called [lIlll-plane deformation of Poincare algebra found, by different methods, in Ref. [9]. This deformation is similar to the standard K-deformation. The only difference is in the choice of undeformed subalgebra which is the stability algebra of light-like fourvector instead of time-like one. However, this difference is significant: in the standard case the Schouten bracket is ad-invariant but does not vanish. Therefore, the relevant r'-matrix does not provide automatically the r-matrix for any extension of Poincare algebra. Actually, the invariance is broken already after adjoining the dilatation generator D. Our aim here is to put the results of Ref. [8J in more general setting. In Sec. 2. we review the properties of Poincare group for arbitrary chosen metric and discuss the Poisson structure on it. In Sec. 3. the quantization of this classical structure is performed. The bicrossproduct form of resulting quantum group allows us to find, by duality, the relevant algebra. The Weyl group and its algebra are constructed in Sec. 4.. Finally, Sec. 5. is devoted to some conclusions.
2.
Relativity theory in an arbitrary coordinate system
Let us consider the n-dimensional linear metric space M with metric tensor 9 v (1"v=O,I, ... ,n-I) given by an arbitrary non degenerate symmetric n x n matrix (not " necessary diagonal). Poincare group P is the group of inhomogeneous transformations of the space M:
The
43
Weyl group and its algebra
K-
where the matrices A"v (Lorentz group) satisfy the condition:
It is easy to see that the Poincare algebra
P reads:
0,
[P",Pv] [M"v,P,,] [M"v,M",,]
i(9v"P" - 9,,"Pv) , i(9""M v" - 9v"M""
+ 9v"M"" -
9""M v,,)
where
(M",{3)"v = i(li"",9v{3 -li"{39v",). Now consider r E
1\ 2 P given as follows, [12]: . _
l
l--Mov AP
II _
-r
K
1J.1I,O
M"v
A
P",
(2.1)
whcre
r"v,,,, = ..!-(li"o9V '" -li vo9"C»
2K
and
K is a real deformation parameter. A calculation of Schoutcn bracket of I' with itself yields
[ r,
_
r] -
1900 - 2 M",{3
'"
{3
A PAP ,
(2.2)
K
It is not difficult to see that [I', r] is invariant, hence r is defines a structure of a Poisson Lie group on P, by the formula:
(2.3) where Xl;, XI~ arc the right- and left-invariant vector fields. It is easy to find thc following exprcssions for the invariant vcctor fields:
XC> L
X'" R
P. Kosinski and P. MaSlanka
44
In the Lie algebra basis corresponding to the above vector fields we have the following relation between the generators of the Poincare algebra and the invariant vector fields:
This enables us to calculate the Poisson brackets of the coordinate functions P:
{aU, a"}
-~((A"o K
- 8"o)A U,IJ
~(8Uoa" -
8"0 aU)'
K
{A",IJ' 3.
J1i'J
+ (Ao,IJ -
gO,IJ)g"U) ,
O.
(2.4)
The K-Poincare group and K-Poincare algebra in an arbitrary basis
If we perform the standard quantizations of the Poisson brackets of the coordinate functions on P by replacing { , } ---+ 1obtain the following set of commutation relations:
H'
-~((A"o - 8"o)A U,IJ + (Ao,IJ - gO,IJ)g"U) , K
[aU, a"]
~(8Uoa" - 8"0 aU)' K
[A"{J,Al'vJ
O.
(3.1)
This standard quantization procedure is unambiguous: there is no ordering ambiguity when quantizing the right-hand side of Eg.(2.4) due to the commutativity of A's. Since the composition law is compatible with Poisson brackets, the above commutation rules are compatible with the following coproduct:
AI'" 0A"v' Al'v (9 aV + al'
(9
I.
(3.2)
The antipode and the cOlltlit are given by:
8(Al'vl S(al')
c:(Al'vl c:(al')
o.
(3.3)
If we define the *-operation in such a way that Al'v and al' are selfadjoint clements, we conclude that the relations Eq. (3.1), (3.2), (3.3) define a Hopf *-algebra - the
The
K-
Weyl group and its algebra
45
K-Poincare group Pt<. It follows frolIl the Eg. (3.1), (3.2), (3.3) that the form of the K-Poincare group does not depend on the choice of the metric tensor g"v. The differences between the various K-Poincare groups are related to the fact that AOfi appearing in the first commutation relation of Eg. (3.1) are not the independent variables but are the linear combinations of the independent ones: AO/l = 90"A"/I' It should be stressed that the K-Poincare group can be defined as a right-left bicrossproduct [13,14]: P" = T*t><4C(L). To see this it is sufficient to define the structure maps:
A"v
(X) XV ,
-':((A"o - 8"0)Ae v + (A ov - gov)g"e). K ~oreover, while C( L) is the standard algebra of function defined over Lorentz group, T* is dcfined by the following relations:
[a",a V ] .:1 (a")
S(a")
_alJ. ,
c:(a")
o.
The bicrossproduct structure of the K-Poincare group allows us to define the dual object, the K-Poincart~ algebra P" as a left-right bicrossproduct:
where T is dual to T* and U(L) is the universal enveloping algebra of the Lorentz algebra. The duality 'j'* {=} T is defined by:
The duality betwccn thc Lorentz group and algebra is defined in the standard way:
< A"v' M°{J >= i(gO"8 fiv - gfi"8"'vl· The structure maps are defined by the following duality relations:
< t, M ofi < A l> t, M°{J >
< fJ(t), M",{J (>9 P.., >, < A (>9 t, 8(M"'/I) >,
here t is arbitrary product of a's while A is an arbitrary product of A's. Finally, using the method described in [1.1] we arrive at the following explicit form of the K-Poincare algebra:
P. Kosinski and P. MaSlanka
46
a) the commutation rules:
[Mij, Pol
0,
[M'J,Pk ] [Mia, Po]
iK(O\gOi - O\gOj)(1 - e-lf) + iWkg i,' - o\gjS)P., iKgi0(l - e-~) + igik pk, . K 00 ri (1 - 2.!:2. ) lv . ri AS l' - .!:2. -z2 kg se +, 9 vk - C
[MiO,pk]
K
-
K
. Oi i i rs p p.--9 i is p.p , +zg Pk(C- ~ -I)+-okg k K
T
2K
K
0,
[1'", Pv ]
i(g"a M V'\ _ gva M"'\
[M"v,M'\a]
+ 9 v,\ M"a
_ g"'\ M va ) ;
b) the coproducts:
+ Po I8i I ,
L1Po
I
L1Pk
Pk I8i eMij I8i I
L1M
ij
L1M iO
Q9
Po
.!:2. K
+ I I8i pk ,
+ 118i Mij , I ® Mia + Mia ® c-lf
_ .!.Mij ® p K
J
where i,j,k = 1,2,:l, ... ,n - I. Let us note that the K-Poincare algebra and group in any dimensions and for the diagonal metric tensor were obtained in 14 and 15. However the duality between the K- Poincare algebra and group was not discussed in these papers. Let us remark in the end of this section that the classical r-matrix Eq. (2.1), r = M Oi 11 pi, does not modify the coproducts for the generators Mij, forming undeformed Lie subalgebra as well as the component Po of the fourmomentum. The algebra with the generators Mij, Po describes the classical subalgebra of our K-Poincare algebra.
4.
The
K- Weyl
group and algebra
The classical Weyl group W consists of the triples (a, A, eb ), where a is an-vector, A is the matrix of the Lorentz group in n-dirnensions and b E R, with the composition law: (a", A"v' cb) * (a 'V , A'~, eb') = (A"veba'v + a!', A"vA'~, ebc b'). Its Lie algebra, the Weyl algebra
[p",Pv ] [M"v,l',\] [M"v,M,\a] [M"v,D] [p",Dj
W, reads:
0,
i(gv,\p" - 9,,'\pv ), i(g"lTMv'\ - gVlTMI'>' 0,
-iP" ;
+ gv,\M"a
- g,,>.Mvu ),
The
K-
Weyl group and its algebra
47
here M,w, 1'JI. are the generators of the Poincare algebra and D is the dilatation generator. We would like to obtain the K-deformation of the Weyl group and its Lie algebra. It is clear that the classical r-matrix for the Poincare algebra satisfying the classical Yang-Baxter equation (CYSE) is also a classical r-matrix for the Weyl algebra. It follows from the Eq. (2.2) that our classical r-matrix Eq. (2.1) satisfies the CYBE iff goo = O. This means that our r-matrix defines a structure of a Poisson Lie group on the Weyl group only in the basis in which the metric tensor takes such a from that goo = O. We shall consider only these types of matrices. In order to obtain the K-deformation of the Weyl group, we firstly find the invariant fields:
A'''' D AJI.{i 8 8AJI./3 8AJI." '
XL"
c
bAJI." a 8aJl.'
8
b
c aeb '
X (i/3 H
A{i 8 A" 8 /3 a ,,8 /l v -8A -il v8A +a--;----) - a - a ' il"v /1/3v ( a" a/3
a
xrt
aa" '
XH
a 8aJl.
JI.
a
+e
b
a
aeb'
Then, using the Eq. (2.3) we calculate the Poisson brackets of the coordinate functions on Wand perform the standard quantizations of the Poisson brackets, by replacing { , } ---+ Finally, we obtain the following set of commutation relations:
H' ].
_~((Cb A"o - 8"0)A e{i + (A o/3 - ebgo(J)g"e) , K
[a Q , a"]
~(8Qoa" - 8"oa e ) , K
0,
[A"/3' AJl.vl [AJl.v' b]
0,
[a'" b)
o.
This standard quantization procedure is unambiguous: there is no ordering ambiguity. Since the composition law is compatible with the Poisson brackets, the above commutation rules arc compatible with the following coproduct:
L1A"v L1aJl.
AJI." (gi A"v , eb A"v is: a + aJl. is: I, V
b® I
+ I ® b.
(4.1 )
48
P. KosiJlski and P. MaSlanka
The antipode and the cOllnit are given by:
S(AI'") S(al')
=:
S(b) e(AI'J
-b, =:
e(al') e(b)
A/, _e- bA/a v ,
81'v'
0, =:
O.
(4.2)
We conclude that the Eq. (4.1), (4.2) define the Hopf algebra - -- K- Weyl group WK' If we forget for a moment about our general theory, w.e can chcck by explicit calculation that our structure is self consistent (Jacobi identities, the relations [L1a, L1bJ = L1[a, bJ) iff goo =: 0 01' b = O. For exam pic:
It is easy to sec that the
W~
has a right-left bicrossproduct structure:
wherc etA) is the standard algebra of functions defined ovcr group A. Thc group A consists of the pairs (A, eb ), wherc A is a matrix of the Lorcntz group and b E R, and with the composition law:
T* is defincd by the rclations:
S(a")
_all ,
e(a")
O.
To sce this it is sufficicut to define the structurc maps as follows:
fJ(al')
b
e JlI'" Q9 a" , b
_!..( (e A"o - bl'o)Jle" K
+ (A o" -
b
e go" )g"e) ,
O. This right-left bicrossproduct structure of the K- Weyl group allows us to define the Weyl algcbra WK as a left-right bicrossproduct structurc:
K-
ThE'
K-
49
Weyl group and its algebra
where T is dual to T* and U(A) is the universal enveloping algebra of the Lie algebra A of the group A. The duality T* ¢=? T is defined by:
< aI', Pv >= i8l'v. The duality between the group A and the algebra
A is defined in
the standard way:
i(g"1'8 13v - g131'8"vl ,
< Al'v' M"13 > < b,M"f3 >
0,
< A"v' D > < bn , D >
0,
i8n ,l.
The structure maps are defined by the following duality relations:
< I, M"f3
< f3(I), M"13 ® P-, >,
P-, >
<1,D I,M"13> < f r> I,D>
, ;
[Mij,PoJ =
[Mij,Pk] = [MiO,PoJ
[MiO,PkJ
=
[D,PoJ
[D, P;] b) the coproducts:
+ I ex> D -
~) - -gOi 1 M ik 1/9 gOi M iO ® ( 1 - e-·
11D
D· I6i I
11 Po
11Pk 11M i }
+ Po ® /, + I 1/9 Pk , Mij 1/9 I + I 1/9 Mij ,
11M iO
II/9M' +M' l/ge-· --M'}I/9PJ'
K
/0 Po
Pk Q9 e-!! ·0
·0
~
1
K
f)k,
50 5.
P. Kosinski and P. MaSlanka Conclusions
We have constructed the K-Poincare group resulting from Poincare group formulated in an arbitrary basis. The quantization is unambiguous due to the absence of ordering problems. The resulting quantum group has a bicrossproduct structure. Using this and the methods developed in [I5] we were able to construct the relevant K-Poincare algebra. The Schouten bracket of the classicalr-matrix we have used appeared to be proportional to the 900 component of the metric tensor. Therefore in the reference frame chosen in such a way that 900 = 0 the relevant Poisson structure can be extended to any group containing Poincare group as a subgroup. This was used to define the Poisson structure on the Weyl group, which allowed us to construct K-deformation of this group. Again we obtained a bicrossproduct structure which allowed us to construct the relevant K- Weyl algebra. The above construction seems to us to be a proper introductory step toward the definition of K-deformed conformal group. One can attempt to quantize the Poisson structure on conformal group resulting from the same r-matrix we used in the case of the Weyl group, hoping that the ordering problems could be overcome in some way (as for example in the case of K-Poincare supergroup [2-6]). As a next step one tries to construct the relevant algebra. This might be more difficult as the bicrossproduct structure is lacking in the case of conformal group. An alternating way of attacking the problem would be to try to incorporate on the quantum level the property of conformal group that it can be obtained from Weyl group by adding (in a special way) the operation of inversion.
REFERENCES [I] .1. Lukierski, A. Xowicki, H. Ruegg, Phys. Lett. B293 (1993) 419. [2] 1'. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, J. Phys. A27 (1994) 6827. [3] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, J. Phys. A28 (1995) 2255. [4] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, Mod. Phys. Lett. AIO (1995) 2599. [5] P. Kosinski, J. Lukierski, P. Maslanka, J. Sobczyk, preprint 1FT UWr887/95, q-alg. 95-01010, to appear in J. Math. Phys. [0] P. Kosillski, J. Lukierski, P. Maslanka, J. Sobczyk, "Quantum deformation of the The Poincare supergroup", in: "Quantum Groups. Formalism and applications". Proceedings of XXX Karpacz Winter School of Theoretical Physics, eds. J. Lukierski, Z. Popowicz, J. Sobczyk, PWN 1995, p. 353. [7]
~1.
Klimek, J. Lukierski, Acta Phys. Pol. B26 (1995) 1209.
[8] J. Lukierski, P. Minnaert, M. Mozrzymas, preprint CPTMB/PT/95-0. [9] A. I.lallesteros, FoOl. Ilerranz, M.A. del Olmo, M. Santander, preprint March 1995.
The
K- Weyl
group and its algebra
[10] S. Zakrzewski, J. Phys. A27 (1994) 2075. [11] P. Zaugg, preprint MIT-CTP, (1994). [12] S. Majid, H. Ruegg, Phys. Lett. B334 (1994) 348. [1:1] 1'. Kosinski, P. :vIaslanka, preprint IMUL3/94. [14] J. Lukierski, H. Ruegg, Phys. Lett. B329 (1994) 189. [15] 1'. Maslanka, J. Phys. A26 (1993) L1251.
51
From Field Theory to Quantum Groups ----_._- ------- - - - - - - - -
RATIONAL SOLUTIONS OF YANG-BAXTER EQUATION AND DEFORMATION OF YANGIANS S.M. KHOROSHKIN
Institute of Theoretical and Experimental Physics 1.17259 Moscow, Russia (e-mail: khoroshkinatvxitep.itep.ru)
A.A.
STOLIN
Department of Mathematics, Royal Institute of Technology S-10044 Stockholm, Sweden (e-mail: astolinatmath.kth.se) AND V.N. TOLSTOY
Institute of Nuclear Physics, Moscow State University 119899 Moscow, Russia (e-mail: tolstoyatanna19.npi.msu.su)
Abstract: A quantization scheme of a certain class of nonstandard rational solutions of CYBE for any simple Lie algebra 9 is given explicitly. We obtain this quantization using a twisting of the usual Yangian Y(g). Explicit computations are performed for the case of S12' The corresponding quantum object (deformed Yangian Y~,«SI2)) is a two-parametric deformation of the universal enveloping algebra U(SI2[UJ) of the positive current algebra sI 2[u]. We consider the quantum double DY~,«sI2)' its the universal R-matrix and also the RTT-realization of Y~,«SI2)'
Introduction This paper is a survey of results obtained by the authors in [1] and [2]. It is devoted to the problem of a quantization of a non-standard rational solution of classical YangBaxter equation (CYUE) for any simple Lie algebra g. We propose here a general scheme of deformation of the Yangians as a solution of this problem for a certain class of such non-standard rational solutions. Yangian of Y(g) of a simple Lie algebra 9 was introduced by V. Drinfeld [3,4] as a deformation of the universal enveloping algebra U(g[u]) of the current algebra g[u]. It
54
S.Y1. KhoroshkiiJ, A.A. Stolin and V.N. Tolstoy
has a lot of applications in the mathematical physics and in the representation theory ([5]-[15]). In particular, the Yangian symmetry Y(si n) was shown for the following one-dimensional N-body integrable models: the Hubbard model [5,6], the classical sin Euler-Calogero-Moser model confined in an external harmonic potential [7], the quantum sin Calogero model confined in the harmonic potential [8], and the quantum Sutherland model [9,10]. The Yangians are employed in [13,14] for an explicit description of the center of the universal enveloping algebra U(g), where 9 is a simple Lie algebra of A-, B-, C-, D-series. A connection between the Yangian Y(gln) and the classical construction of the Gelfand-Zctlin basis for the Lie algebra gin was established in [15]. Tensor products of finite dimensional representations of the Yangian Y(g) produce rational solutions of the quantum Yang-Baxter equation (QYRE). For instance, for 9 = sin these solutions can be obtained by the fusion procedure applied to the Yang solution R( u) = 1 + plu, where p is the permutation of factors in en (9 en. However, there exist other rational solutions of the classical Yang-Baxter equation (CYRE). These solutions were studied in [17]. Every rational solution of CYBE provides a bialgebra structure on g[u]. These structures have not been quantized yet except the case l' = cdu, when the Yangian is exactly the quantization. We present here a quantization scheme for rational solutions of the form cdu + constant. We obtain this quantization using a twisting of the comultiplication in the usual Yangian Y(g). The existence of such element was proved by V. Drinfcld [3]. In Appendix we reproduce from ([18]) another general construction of the twisting element. We realize explicitly the twisting for the simplest non-standard rational r-matrix for S12, namely r = c21u + h", 1\ C"" Here C±a, h", is the standard Chevalley basis for S12' We note that the additional term h", 1\ c'" leads to a deformation of the co-algebra structure of Y( sI2 ). We perform this deformation by means of the twisting of U(SI2) 129 U(SI2) by some special two-tensor of U(SI2) (9 U(SI2) which (appeared in [19]. Moreover, this two-tensor enables us to write down a quantum Rmatrix corresponding to the classical r-matrix r = c2Iu+h",l\e_",. On the other hand, writing down this R-matrix explicitly in the fundamental representation, we develop RTT-formalism (see [20]) to get another presentation of the deformed Yangian. We discuss also properties of the corresponding quantum determinant, and the realization of the deformed Yangian Y,I.{( S12) in terms of generating functions ("field" realization). 1.
Rational solutions of classical Yang-Baxter equation
Let 9 be a simple Lie algebra over C, {Ii} be an orthogonal basis of 9 with respect to the Killing form. Let C2
P(u,v)= --+r(u,v) u-v be a function from C 2 to 9 0 g, where definitions.
C2
= Li Ii (9 l;.
(I.l ) We introduce the following
Rational solutions of Yang-Baxter equation
55
Definition 1.1 We say that P( u, v) is a rational solutions of the classical YangBaxter equation (CYBE) if the function P( u, v) satisfies the following conditions: 12
21
(i) it is skew-symmetric, i.e. P (u,v) = - P (v,u), (ii) the function 1'( u, v) is a polynomial in u, v, i.e. it has the form 1'( u, v) = Lij Pij (u, v) Ii 161 Ij , where Pij (u, v) are usual polynomials of two variables u and v. (iii) the function P( u, v) satisfies the CYBE
13
Here l' (U1' U3) means 13
P (U1' U3) =
L (hAUl, U3){Ii 010 Ij ) ,
( 1.3)
i,j
if P(u,v) has the form P(u, v) = Lij rPij(U, v)Ii0Ij , where rPij(U, v) are usual functions , 12 23 of two variables u and v (rPij : C 2 -> C). The clements P (Ul,U2) and P (U2,U3) are defined analogously. The commutators in the left side of (1.2) are considered as commutators in U(g)8 3 •
Definition 1.2 We say that two rational solution P1(u,v) and P2(U,V) are gauge equivalent if there exists an automorphism 7>. of algebra g[u] such that (7). 0 7>.)PI (u,v) = P2(U,V). It turns out that the degree of the polynomial part r(u,v) for rational solutions of CYBE can be estimated. More exactly, the following result was proved in [17]: Theorem 1.1 Let P( u, v) = ~ + 1'( u, v) be a rational solution. Then there exists a rational solution p( u, v) being gauge equivalent to 1'( u, v) and of the form -
C2
P(u, v) = - u-v where roo,
1'10, 1'01, I'll
+ 1'00 + rlOu + 1'01V + rlluv
,
(1.4 )
E g 119 g.
We will be dealing in the present paper with the case C2
P(u,v) = - u-v
+ 1'0,
(1.5)
where 1'0 := 1'00 E 9 (>9 g. In what follows, we need the following result. Let g((u- 1)) := g 0 C((u- I )). One can define the non-degenerate ad-invariant inner product on g( (u- 1 )): (x, y) = Resu=o tr( ad x . ad y). The following theorem is valid.
56
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
Theorem 1.2 1) There is a 1-1 correspondence between the set of rational solutions of CYBE of the form (l.5) and subalgebras W C g[[u-llJsuch that: (i) u- 2 g[[U- l ]] eWe g[[u- l ]], (ii) w.L = W with respect to the form (.,.) introduced above, (iii) WCDg[u]=g((u- l )). 2) Any W satisfying conditions (i-iii) above defines a subalgebra Leg and a nondegenerate 2-cocycle B on L. In other words the element B is skew-symmetric and satisfies the equation
B([x, y], z)
+ B([z, x], y) + B([y, z]x) = 0
( 1.6)
for any x,y,z E £. Moreover, the element r·o E L 1\ f and it is non-degenerate, l = B E C 1\ C. 3) Conversely, any pair (f, B) such that f is a subalgebra of 9 and B is a nondegenerate 2-cocycle on f, defines a rational solution of the form (l. 5).
ra
r'o
ro
l l Here the symbol means the following. Since ro E f ® f '2:' Hom(C, f), E Hom(f, C) '2:' C (l) C. It should be noted that a Lie algebra with a non-degenerate 2-cocycle is called quasi-Frobenius Lie algebra. Proof (see also [18], [17]). Let P(u,v) be a rational solution of the form (1.5), where ro is given by the following general expression ro = Li,j rij Ii 129 I j. Let us define a linear functional on u-lg[[U-llJ by the formula
pta) =
'L rij(1i,a)Ij
(1.7)
i,j
for any a E u-1g[[U- I ]] and let us define a subspace W C g[[u-llJ consisting of the elements of the form a + pta), where a E u-lg[[U-1lJ. Since P(u,v) is a rational solution, therefore we have that rij = -rji. It is easy to see that this is equivalent to the following condition: W = W.L, which proves property (ii). Now we prove that W is a subalgebra of g[[u-llJ. Taking into account that (ii) is already proved, it is sufficient to prove that ([x,y],z) = 0 for all x,y,z E W. Since the elements x, y, z have the form x = a + pta), y = b + p(b), z = e + pte) for some a, b, e E u-lg[[U-llJ, we have to consider ([a
+ pta), b + p(b)], e + pte))
= ([a, p(b)], pte))
+ ([p(a), b], pte)) + ([p(a), p(b)], e)
Since the inner product is ad-invariant, we have:
([a, p( b)], p( e))
= (a, [p( b), p( e)]) = 'L
rijrkle'ji(1i, b)( h, e)(1m, a) ,
ijklm
where we take into account that [Ii, Ij] ="Lk e7/k' On the other hand
.
Rational solutions of Yang-Baxter equation
57
and the fact that 1'0 satisfies CYBE (this is the same as P( u, v) satisfies CYBE) is equivalent to that ([x,y],z) = 0 and hence to that W is a subalgebra. It is clear that the properties (i) and (iii) are fulfilled because of the construction of Wand therefore every rational solution of the form (1.5) determines W, Conversely, let W satisfy conditions (i) - (iii). Since we have another decomposition u-lg[[U-llJEBg[u] = g[[u- I ]] we infer that W = {a+p(a) : a E u-lg[[U- I ]]}, where p: u-lg[[U-llJ ---> 9 is a linear map. Let fin := Iiu- n- I be a basis of u-lg[[U- 1]] and fin := Iiv n be a basis of g[v]. Then the element L:in(ein is) (!in + p(fin))) is a rational solution of CYBE of the form (1.5). The first statement is completely proved. Let us prove the second statement. Define a projection 1r : g[[u-llJ ---> 9 as follows: 1r( aD + al u- I + ...) = aD. Clearly 1r(W) = L is a subalgebra of g. Moreover, W c .c EB u-lg[[U-1lJ C g[[u- 1 ]]. Then the condition (ii) implies that u-1.c.L EB u- 2g[[u- 1lJ C W where .c.L egis the orthogonal complement to .c with respect to the Klling form. It is easily seen that the subalgebra N = u-1.c.L EBu- 2g[[u- 1 lJ is an ideal in the algebra M = .cEBu-1g[[U- I ]]. The quotient algebra MIN is isomorphic to the semidirect product of .c and commutative algebra £*, where £* is the dual space to .c on which .c acts by the coadjoint representation. We can write MIN = .c + (£*, where (2 = O. Then MIN is equipped with the non-degenerate ad-invariant symmetric inner product coming from the inner product on Meg is) C((u- 1 )). One can check that (a,b) = 0 if a, b E .c: (ex,ey) = 0 and (a,ex) = x(a). Denote by ¢ the natural projection M ---> N. Then ¢(W) = S is a subalgebra of MIN, which coincides with its orthogonal complement. Thus, dim(S) = dirn(.c). Recalling the construction of .c we deduce that S is projecting onto .c under the natural projection .c + e£* --->.c. Therefore S is of the form S = {I + (f(l): I E.c} for some f E Hom(.c,£*). Let x: Hom(.c,£*) ---> £* is) £* be the natural isomorphism. Then we have that B = x(f). One can check that the equality S = S.L implies that B is skew-symmetric and the fact that S is a subalgebra implies that B is a 2-cocycle. The non-degeneracy of B comes from the property (iii), which implies that S n .c = (0) in MIN. The second statement is completely proved. Clearly, all the considerations above can be converted, which proves the third statement. The theorem is proved completely. It should be noted that in the case of 9 = sIn all the rational solutions can be described in a similar way [i 7]. Let
dk = diag( 1, ... ,1 t, ... ,t) E GL(n,C((CI))).
---.,.........k
(1.8)
Then every rational solution of CYRE defines some Lagrangian subalgebra W contained in d};l . sl( n, C[[t- 1]]) • dk for some k. The corresponding combinatorial data are: (i) subalgebra .c c sl(n, C) such that .c + Pk = sl(n, C), where P k is the maximal parabolic subalgcbra of sl(n, C) not containing the root vector ea. of the simple root Ok;
,'58
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
(ii) 2-cocycle B OIl L which is non-degenerate on L n P k . In the case of slz, one has just two non-standard rational r-matrices (up to the gauge equivalence): (1.9) and
cz
Pz(u, v) = - u-v
+ h", iZI e_",u -
e_", iZI h",v .
(1.10)
The corresponding Lagrangian subalgebras are
and
Wz = C:l(slz[[t- 1 ) EB C(e",t- 1 - h",t) ED C{e_",C1)ED Ell C(h",C 1 )
(IJ
C{ C"'C Z )
e C{e_",C Z ) Ell C{h",C Z + 2c",) .
( 1.12)
Now we are going to quantize the rational r-matrix (1.5).
2.
Deformation of Yangians
In this section we describe a procedure for quantization of any rational solution of the form (l.Ei). The output is a deformed Yangian. The Yangiall V(g) as a quantum deformation of the universal enveloping algebra U(g[t]), where g[t] is an algebra of polynomial currents over a simple Lie algebra g, was introduced by Drinfeld [3] firstly in terms of generators which actually are not associate to choice of a concrete basis in g. Later Drinfeld gave in [3,4,21] another realization of the Yangians in terms of generators connected with Cartan- Weyl basis in g. In this paper we use this second Drinfeld realization. We remind the definition of this realization and the basic properties of the Yangian. Let 9 be a simple Lie algebra with a standard Cartan matrix A = {aij li,j=o, a system of simple roots // := {Ol" .. ,02} and a system of positive roots .1+(9). Let ei := e a " hi := ha,. fi := fe>, := C_"'" (i = 1•... ,1'), be Chevalley generators and {e.."f..,}. hE .1+(g), be a basis Cartan-Weyl in g, normalized so that (ea,fa) = I.
Definition 2.1 The Yangian Y := Y(g) associated to g is the Hopf algebra over C generated (as an associative algebra) by the elements eik .- e""k, h ik := h aik , fik := f""k, (i = 1, ... , r; k = 0,1,2, ... ), with the relations:
(2.1) [hiO.ejd = (Oi.Oj)Cj/,
[hio.h] = -(ui,uj)fjl. 1
[hi,k+l,eJtl- [hik,cj,l+d = 1]2"(oi,oj){hik ,Cjd ,
(2.2)
(2.3)
Rational solutions of Yang-Baxter equation
59
1 [hi,k+l,!Jd - [hik,!j,l+d = -1)2(Oi, Qj){h ik , !jt} ,
(2.4)
1
(2.5)
[ei,k+l, eJd - [eik' ei,l+ll = 1)2(0i, OJ){ eik, Cjd , [!i,k+l,!il] - [Jik,fJ,I+1]
1
= -1)"2(Oi,Oj){Jik,!jd
Sym{k} [ei,k, [ei,k 2 . , . [Ci,k n ,) , Cjd ' .. J] = 0 { Sym{k} [fi,kl [Ji,k2 ... [Ji,k n .) , lid·· .J] = 0
,
(2.6)
lor i -=f- j ,
(2.7)
where {a, b} := ab+ ba, nij := 1 - A ij , the symbol "Sym"{k} denotes a symmetrization on k 1 , k 2 , • •• , k;J' The co-multiplication map of Y is given for basic generators eik, hik , lik, (k = 0, I), by Ll(x)=xlSil+llSix,
(2.8)
x E9 ,
L
Ll (eid = Cil (8)1 + 1 lSi eit + hiO lSi eiO - 1) .
I'Y lSi [e"" e'Y]
,
(2.9)
'YELl +(g) Ll (fid = fit lSi 1 + 1 (8) fil + fio lSi hiO +.,.,.
L
[f"" l'Yl lSi
c-y ,
(2.10)
-YELl +(g) Ll(hitl=hit09I+llSihil+hiO®hiO-1)'
L
(Oi,')!'Y®C'Y'
(2.11)
'YELl +(g) We explicitly introduce in the relations (2.3)-(2.6) and (2.9)-(2.11) the Yangian deformation parameter 1) E C\ {O} 1 and therefore we shall use further the notation Y~(g) for the Yangian Y(g). Remarks. (i) The universal enveloping algebra U(g) is generated by the elements eiO, hiO, Jio and it is embedded in Y~(g): U(g) ~ Y~(g). (ii) One can show that Y~(g) is generated only by the elements eiO, hiO, liO, fil, (i = I, .. . ,1'), and therefore we can obtain the co-multiplication map for all generators cik, fib h ik , !ik, (i = 1, ... ,7'; k E Z). (iii) If we replace the right parts of (2.3)-(2.6) by zeros, we obtain the algebra isomorphic to U(g[t]). For any ,\ E C define an automorphism 7>. of Y~(g) by the formulas:
xEg,
(2.12) (2.13)
This "translation" automorphism is compatible with the comultiplication, i.e. (2.14) 1 All Yangians with different that TJ = 1.
11
are isomorphic one to another and therefore one usually assumes
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
60
The Yangian is a pseudotriangular Hopf algebra [3]. It means that there exists an unique element R(>.) = 1 + Lk:1 >.-kRk, where Rk E Y~(g) 0 Y~(g), such that
(7), 0
~)R(II)
= R(1I + >. -
(7), 0 id)L1'(x)
p.) , 13
12
23
(,10 id)R(>') =R (A) R (A) ,
= R(A)((J;, 0 21
R (>') R (->.)
id)L1(x))R(A)-1 , (2.15)
= 10 1 ,
(2.16)
where 7>, is the translation automorphism and, as usually, we denote by ,1' the opposite comultiplication. Now we return to the rational solution of CYBE in the form (1.5). We recall that this solution P( u, v) can be characterized by availability of the following combinatorial data: (i) subalgebra £ of g; (ii) non-degenerate 2-cocycle B on .c. Our approach to quantization of a rational solution of CYBE is based on the following result borrowed from [3]. Theorem 2.1 There exists an element FE (U(£)[[7]]])18I 2 of the form f' =
1
1 + "27]ro
~ 7] kf'k, + L.
(2.18)
k=2
o
where r 1 = B (in sense of the Theorem 1.2), and this element satisfies the following condition: 12
23
F (,1 0 id) F = F (id 0 ,1 ) F ,
(2.19)
where ,1 is the trivial (eocommutative) comultiplieation in U(g). Remark. Another construction for F is described in [18] and we show in Appendix an approach, which enables one to consider both cases simultaneously. Let Y~(g) = Y(g)[[7]Jl. For any F satisfying the conditions of the Theorem 2.1 we define a new algebra y~(F)(g), which has the same multiplication as Y~(g) but its comultiplication is defined as follows (2.20) The following theorem is valid. Theorem 2.2 The algebra Y~(g) is the pseudotriangular Hopf algebra with ,1 (F) and R(F)(u) =F R(u)F- 1.
Rational solutions of Yang-Baxter equation
61
Proof. First of all it is not difficult to verify that the comultiplication (2.19) is a coassociative operation. Now we prove that the element R(F) (u) satisfies the relations (2.15)-(2.17) with .1 (F). The first relation from (2.15) and the second relation from (2.16) are obvious for .1 (F) and R(F)(u). It is easy to see that the following chain of relations takes place
(T" is! id).1 (F)' (x) = (T" Q9 id)
= (j.!
F.1'(x)r =F R(>.)((T" is! id).1 (x))(R(>.))-1 (}\)-I l
R(A)P-I)((T" is! id).1(F)(x)) (
FR(>.)r't
l
=
R(F)(>.) ((T" 0 id).1 (F)(x))(R(F)(Ajt' . This proves the second relation from (2.15). It remains to prove the first relation of (2.16) and (2.17). The first of these relation is a consequence of (2.15) and the following equalities (which are equivalent to (2.19)): 12
21
13
F (.1 is! id) F=F 0-(.1 is! id)1" , l
(U(.1 ® id)1") ((id is! .1 )Ft =
d;~)-I ~ ,
where u(a 0 b 121 c) = b 0 c 121 a. At last, We deduce (2.17) using (2.15) and (2.16). The proof is complete.
3.
Non-Standard Quantization of U(51 2 )
Let e±c" he. be the Chevalley basis for the universal enveloping algebra U(51 2 ) of the Lie algebra 51 2 with the standard defining relations: (3.1 ) We put here and anywhere (0:,0:) = 2. Let U( L) E U(51 2 ) be the universal enveloping algebra of the Borel subalgebra L of 51 2 , generated by the elements he. and E-e.. Let us introduce the following two-tensor (a formal series) F of some extension of U(b_) 121 U(b_):
where ~ E C is some parameter. We have borrowed the element F from [19]. It is not difficult to verify that the following series
is an inverse element to F, i.e. 1"1"-1 valid.
=
F- 1F
=
1. The following proposition is
62
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
Proposition 3.1 The element F (as formal series) satisfies the following relation (3.4 )
where L1 is the usual eomultiplication in U(b_), i.e. L1(a) = a 0 1 + 1 0 a for any a E b_. Proof By direct calculations. 0 As consequence of this proposition we have Corollary 3.1 Let A be an arbitrary !fopf algebra containing the l/opf algebra U(b_) and let A~Fl be the algebra A[[~lJ (i.e. the algebra A over C[[~lJ) with co-multiplication map L1(Fl given by the formula
(3.5) Then Ar) is a l/opf algebra. Proof Coassociativity of L1(Fl follows from the formula (3.4). Existence of the antipode is proved in [21].0 We introduce now the following notations for some elements of utJ(L): 1'.:, := 1 -
~e_o
,
(3.6)
Proposition 3.2 The elements ho , 1':1 satisfy the following relations:
1'cJ,,-1
= 7';:11'0
, [h o , To]
L1(Fl(h o ) = h o 091';;1+10h o ,
= 2(1
- To) ,[h o , I~-I]
L1(FJ(1'.:,)
8(h o ) = -hoI'.:, ,
= 1'.:,01'.:"
= 2(1';:1
L1(Fl(1~-I)
= 1;1 , 8(1';:1) = 1'.:, e(I'.:,) = e(1';;I) = 1 .
5(1'.:,)
e(h o ) = 0,
_1~~2) ,
= 1';;1 09 1';;\
,
The algebra Utl(b_) generated by ho' 1':1 is a Hopf subalgebra of Dtl(L). Proof By direct calculations with (3.1), (3.4) and (3.5).0 The Hopf algebra Ut\b_) is triangular with the universal R-rnatrix R
=
p21
F- 1 = 1 + ~e_o 1\ h o
+ ... =
(3.7)
(3.8) (3.9) (3.10)
Rational solutions of Yang-Baxter equation
63 (3.12)
This algebra is a quantization of the Lie bialgebra b_ defined by the classical r-matrix h",. According to Corollary 3.1 we can extend the twisting by F to 0t)(.gI2)' Then we have
r = e_" 1\
Proposition 3.3 Let ut)(sI2) be an algebra generated the elements h"" e" and 1';;1 with the defining relations
l' 1'-1 0' Q
[h", e,,] L1(F)(h",)
= 1'-11' n
Q,
= 2e"
,
[h a, l' J = 2(1 - 1') [h 0', 1'-1] Q
Q
[1'", e",J
= h"091';I+I09h,,,
= 2~h",
,
Q
= 2(1'-1 a
- 1'-2) Q'
(3.13)
= -2~T;1 haT; I , (3.14) ,1(1")(7:') = 1',,0'1:', L1(F)(T;I) = 1';1 0 1';1, ,
[1';1, e,,]
(3.15)
L1(F)(e,,)
= e,,09T;1 + 10ge"-~h"@T;lh" 5(h,,)
= -h,,7:'
,
5(1',,)
5(e,,) = -e"T" c:(h,,)
ih",(h" -2)@T;I-i h,,(h,,+2)@1';2 ,
= T;l
,
5(1';1)
~h"(h,, + 2)7:'(1'", -
= c(e,,) = 0,
c:(T,,)
= 1'"
2) ,
= c:(T;I) = 1 .
,
(3.16) (3.17) (3.18) (3.19)
Then ut)(sI2) is a Hopf subalgebra of ut)(sI2) and it is a triangular deformation of U(sI2) in the direction of the classical r-matrix r = L " 1\ h". Proof By direct calculations with (3.1) and (3.5).0 Remark 1. In every finite dimensional representation of sl2 the element Tu = 1 - 2~L", is always invertible since L" is nilpotent. Therefore, the theory of finite dimensional representations of Ut l (sI2) is the same as the theory for .gI2' Remark 2. Similar computations for sl2 with another twisting element F were carried out in [24]. However, using Theorem 2 from [22] one can prove the following result. Theorem 3.1 There exists an invertible element l' E U(gI2)[[~ll such that F (7' 09 7')P-1 ,1(7'-1), c(7') = 1 and PI(T) = 1 E GL 2, where PI is the two-dimensional representation of g12' In other words /lopf algebras obtained from U(sI2)[[~ll by twisting by F- I and F are isomorphic as /lopf algebras and the isomorphism is given by conjugation by T.
t
Proof The matrix (PI @ p.)((p 2I l p) was computed in [24J. We can calculate the matrix (PI @ Pl)( p 21 F- I ) and it turns out that we obtain the same matrix (see Lemma 6.1 further). Then Theorem 2 from [22J implies all the statements.O
S.M. Khoroshkin, A.A. Stalin and V.N. Tolstoy
64
4.
Twisting of Yangian Y(sI2)
One can show that the Yangian Y,,(slz) (as a Hopf algebra) can be defined by Chevalley generators h"" e±,,, co-" with the defining relations [16]:
[e""e_,,] == h" , [h,,,c±,,] == ±2e±" ,
(4.1)
[h", eo-oj == eo-" , [e_o, eo-oj == TJe_" ,
(4.2)
[c"" [e", [eo, eo-,,]]] == 6TJe; , [[[e"" eo-",], eo-a], eo-oj == 6TJeL", .
(4.3)
,1( ho )
== h" ® 1 + 1 0 ho ,
,1( e±o)
== e±o ® 1 + 1 0 e±o ,
,1(e o_,,) == eo_", 0:9 1 + 1 0 eo-", 8(h",) == -h" ,
S(e±o) == -e±ci ,
+ Tie o 0
ho
(4.4)
(4.5)
,
8(c o_,,) == -e o- o - TJc",h" ,
c(h o ) == e(e±o) == e(e o-,,) == 0,
(4.6)
e(l) == 1 .
(4.7)
Since Yry(slz) contains U(slz) as a Hopf subalgebra the Corollary 3.1 implies that the algebra
Yry(2
isomorphic to Yry[[~JJ with the cOIIlultiplication (3.5) is a Hopf algebra.
Proposition 4.1 The elcments ho ,e" ,1~±1 (sec (3.6)) and e6-", satisfy the relations:
1'0 T,,-I == 1;;IT", ,
[h,,, 7'c,] == 2(1 - T",) , [T"" e,,] == 2~h" ,
[h", e",] == 2c", ,
[1'c"
eo_",]
== - 2: (T; - 21::' + 1),
[e"" [c"" [e"" e6-,,]]] == 6TJe; , ,1(F)(h,,) == h",0T;;I+I®h""
rho, T;;I] == 2(7~-1 - T;;Z) , [T;;l, e",] == -2~T",-1 hS;;1 ,
[1:;-1, eo-a] == - 2: (T;;Z - 21:;-1 [[[e", e6-",], e6-",], eo_",] == 6TJeL", .
,1(F)(T",) == T",®T", ,
,1(F)(1~-I)
(4.8) (4.9)
+ I) , (4.10) (4.11)
== T,,-1 0 T;;I, (4.12)
,1 (F) ( e,,)
== e" 0 7;;1 + 10e" -~h" 0T;;1 h", -
,1(F)(eo_",) ==
e",
0 7::'
+ 1 ® CO-'" + ~h",
5(h",) == -h",1'c, ,
~h"(h,,, ® T;;I
8(To ) == T;;I ,
5(e",) == -e",T", -
2) 01:;-1- ~ho(ho + 2) 01~-z ,
+ 2:h", 0
(I -7::') ,
5(T;;I) == 1'c, ,
~h",(h", + 2)T",(1'c, -
1) ,
5(co- o ) == -eo_",T;:1 - {h",To + TJ.h",T;;1 - TJ. h", , ~
TJ
e(h",) == c(c",) == e(e6_",) == 0,
2~
2~
c(T",) == e(1:;-I) == 1 .
(4.13) (4.14)
(4.15) (4.16 )
(4.17)
(4.18)
The algebra Yry ..dsI2) generated by the elements h"" e"" 7;=1, e6_", is a Hopf subalgebra - (F) of Yry,e (sI 2)'
Rational solutions of Yang-Baxter equation
65
Proof By direct calculations.D One can see that the Hopf algebra Yry(2(sI2) is a quantization of the Lie bialgebra sI2[u] corresponding to the rational solution (1.5). 5.
Deformed Yangian double DYry(sI2)
Since Yry(.~12) C Yry,~(sI2) C Yry(2(sI2) (as associative algebras) and since T>. acts identically on U(sI2)[[~]] and since Yry,~(sI2) differs from Yry(sI2) by elements from U(sI2)[[~]] therefore the automorphisms T>. extended to Yry<2(sI2) preserve Yry,~(sI2)' Thus we have the following result. Proposition 5.1 The Hopf algebra Yry,~(sI2) is pseudotriangular with R(F)P..) :=
1"21 R(),,)F- 1. In particular, R(F)(>\) is a rational solution of QYBE.
Let us recall that the Yangian double DYry(sI2) (see [16]) is a quasi triangular Hopf algebra with an universal R-matrix R which lies in some extension of DYry(sI2) ® DYry(sI2)' Since U(sI2) C DYry(sI2) we can twist the Yangian double DYry(.~12) by F. Using formal algebraic arguments similar to that of Section 2 we get as result the following proposition. Proposition 5.2 The deformed Yangian double DYry,dsI2) is a quasitriangular Hop!
algebra with the universal R-matrix R(Fl = F 21 RF-l.
In what follows we need the realization of the Yangian double DYry(sI2) given in [16]. In this realization the Yangian double DYry(sI2) is generated by the elements h ko , CkH", (k E Z)2, which are composed into generating functions h+( u) = 1 + r"'Lk>O hkou- k- I , (h o := h,,), 4,,( u) = Lk>O CkH" U- k- 1 , and h- (u) = 1 - TJ Lk
(5.2) (5.3)
(5.4) (5.5) (5.6) 2These notations of the generators are connected with the notations in [16] as follows: hkb := h k , !k (k E Z).
Ckb+" := Ck, Ckb-n :=
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
66
(5.7) where {a,b} = ab+ba. It turns out that Yry(s12) is generated by h+(u), etc,(u), while the dual to Yry(s12) algebra YryO(s12) is generated by h-(u), e;;;,,(u). The universal R-matrix R found in [16] can be factorized as follows: (5.8) where
-
-
k?O
k?O
R+ = IIexp(-I/C(k+l)b-"c;;:ekS+")' R_ = II exp(-17e_(k+l)S+"° ekb-,,) , (5.9) Ro =
II exp(17Resu=v(17-1In h-(v + 2n + I) 1/9 dd 17-
1
In h+(u))) .
(5.10)
u
n?O
Here Resu=v(f(u) I8ig(v)) = Lkfk I8ig-k-1 if f(u) = LfkU-k-l, g(v) = Lgkv - k- I .
Corollary 5.1 The element R(F) = FRF- 1 satisfies QYBE, where F is the same as in (3.2) and R is defined by (5.8)-(5.10). 6.
RTT-realization of the deformed Yangian Yry.dsI2)
We develop in this section so called RTT-formalism (see [20]), i.e. we obtain the RTT-realization or, in other words, the realization in terms of the L-operator. Let p(1) be the two-dimensional representation of s12 in C 2 with the basis II) and 1- I). It is well-known that p(1) is extended to a representation p~1) (u E C) of the Yangian v,/( 051 2 ) by means of
pS1)(h±(w)) II) = (1
pS1)(h±(w)) 1- I)
+ _17_) II) , w-u
pSl)(e~(w)) II) pSI)(e:,,(w))
II)
p!/)(e~(w)) 1- I)
0,
=
= _1_
w-u
1_ I)
,
=
= (I -
_17_)
w-u
_I_II) , w-u
pS1)(e:,,(w))
1- I)
1- I) . (6.1)
= 0,
We find with the help of these formulas
(pSI)
(9
pSI))(R)
=
v)( I
+ ~)=
v)R (u - v),
(6.2)
where
2
2
1
R(u - v) L (u) L (v) =L (v) L (u)R(u - v) , I
2
(6.3)
where L (u) = L( 11) 09 id, L (v) = id 09 L( v). The matrix L( u) is a generating function for }'~(gI2) and Y(s12) ~ Yry(gI2)j(qdetL(u) - I). More exactly, we can formulate the following result.
Rational solutions of Yang-Baxter equation
67
Proposition 6.1 Let L( u) be a 2 x 2-matrix with non-commuting entries, such that 1
2
2
1
(i) R(u - v) 10 (u) 10 (v) =L (v) 10 (u)R(u - v) , [)O) L( i) IP) (l!..) L( U ) = I + --;;+ --;;2 + ... --;;;.+ ... , (iii) qdet L(u) = 11l(u)122(u-l)-lzl(u)llz(u-l) = lzz(u)ln(u-l)-itz(u)lzl(u-l) = I. Then the matrix coefficients of L(u) generate a Hopf algebra isomorphic to The comultiplication L1 and the antipode 8 are given by the formulas
Y~(sI2)'
(6.4)
L1(lij{U)) = 'Llki(u) 01jk(u) , k
8(L(u))
= L-1(u).
(6.5)
The deformed Yangian Y,J.dslz) admits a similar representation. We start with the lemma. Lemma 6.1 In the representation p~i)Q9p~i) the universal R-matrix 'RP') = p 21 nF- 1 has the form R~.((u - v) := (p~1) 0 p~l))(n(F)) =
= (1 +~p~I)(e_,,)0P~1)(h,,))(I-TJuP~2J(1 ~~p~I)(h,,)0P~i)(e_,,)) 1
=
(
-=-~~v ~ _~~v ~
--~
e
~ )
1
~~v
~
0
1-
=
(6.6)
u~v
Proof. By direct calculation.D Let us consider an algebra A of matrix elements of 1.,( u) satisfying the relation 1
2
Z
1
R~.((u - v) L (u) L (v) =L (v) I., (u)R'J.du - v) .
(6.7)
It follows that L(11) = (p~i) 1/9 id)n(F) satisfies (6.7). Algebra A together with the comultiplication (6.4) and antipode (6.5) constitutes a Hopf algebra. The following lemma takes place. Lemma 6.2 (i) The matrix R~.((TJ) is the project07' onto the one-dimensional subspace C(ll) G 1- 1) -1- 1) ® 11) - ~I- 1) 01- 1)) up to a scalar factor. (ii) The following relations hold
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
68 qdetry,~
= 111(u)122(u -1)) -121(u)ln(u - "1) - ~111(u)112(u In(u)ll1(u - "1) -112(u)121(u -1)) + ~112(u)ll1(u - "1) ,
L(u) =
"1)
= (6.9)
t1(F)(qdetry,~ L(u)) = qdetry,~ L(u) 119 qdetry,~ L(u) ,
where the quantum detenT/inant
qdetry,~L(u)
(6.10)
is an element of the Hopf algebra A.
Proof The part (i) is verified by direct calculations. The proof of the second part is standard (see [13]).0 qdetry,~
Lemma 6.3 The quantum determinant
L( u) is a central element of the alge-
bra A. Proof The formula for the quantum determinant qdetry,~ and the quantum YangBaxter equation for Rry,~( u) provide the following equality:
R;~~("1)R~~~(u)R~~~(u + "1)R;~~("1) 1(v) (qdetry,~ L(u))R;~~("1)
= (qdet'I,~ L(u))R'I,~("1) 23
1
23
12
13
L (v)Rry,~(Tl)Rry,~(u)Rry.~(u
23
=
+ "1)Rry,~("1)
,
1
where we use the standard notations: L (u) := L( u) 119 I 119 1, and R~~~ = Li ai 119 bi 119 I and so on if Rry,~ = Li ai 119 bi . Direct calculations show that
( )RI2( )R 13 ( )R23() R 23 ry,~ "1 ry,~ u ry,~ u + "1 ry,~ "1
23 = 2(u-"1)R "1 ry,~
.
Therefore R;~~("1)L 1 (v)( qdetry,~ L( u) )R;~~("1)
and
qdet'l~
= R;~~("1)( qdetry,~ L( u))L 1 (v )R;~~("1))
,
L( u) commutes with L( v), i.e. [qdetry,~
L(u), L(v)] = O.
The proof is complete.O At last we have the following theorem. Theorem 6.1 Let A be a Hopf algebra generated by matrix elements of the generating function L( u) satisfying the following conditions: R'I.~(U
1
2
2
1
- v)· L (u) L (v) =L (v) L (u)·
L(u) =
'1'1 ( ~h"l:f
0) + -
L(o)
I
I
'I';;'
qdetry,~
t1(l;j{u)) =
u
Rry,~(u -
L(l) + -, + 2
v) ,
(6.11 ) (6.12)
u
L( u) = I
L lki(u) 119ljk(u) ,
(6.13) (6.14)
k
S(L(u)) = L -I(U) , where T,,:= I -
2~e_".
Then A is isomorphic to Yry,dsI2)'
(6.15)
69
Rational solutions of Yang-Baxter equation
Proof The proof is analogous to that of Proposition 6.1. Taking into account that
we see that there exists a homomorphism from A to Y~,{(sI2)' To notice that this is an epimorphism, one can use the formula (7.2). To prove that this homomorphism is a monomorphism one can use the same arguments as for non-deformed Yangians (see for instance [23]).0 Remark. To find the constant term in the decomposition of L( u) into the series in u- 1 , one should find
(Z-I 0) z '
o) 1
where Z
= 1 + ~e_"
3!!
+ -21. ~
It is not difficult to see that
Z2
2 2
e_"
+ ... +
(2n - 1)!!
,
n.
~
n
0
n
e_"
+ ...
(6.16)
(6.17)
= 1';;1, hence (6.18)
7.
Realization of the deformed Yangian functions
Y~.{(sI2)
in terms of generating
The realization of the usual Yangian Y~(sI2) in terms of the generating functions ("fields" realization) ht(u) and 4,,(u) (see Section 5) can be obtained from the Gauss decomposition of the L-operator:
L(u) == (p(1)(u) 0 id)(R+Ro1L)
== (
1 0 ) ( kl(u) -et(u) 1 0
0 ) (1 k 2 (u) 0
-e~,,(u))
(7.1 )
1
where k.(u)k 2(u - 1) = 1, k 2(u)k j l(U) = h+(u) (see [16] and [23]). The same procedure can be applied to R(F). We have
L(u) = (p(1)(u)
==
Q9
id)(F 2I R+RoR_r l ) ==
(~~" ~) (-e~(u) ~) (kl~U) k2~U)) (~ -e~t(U))
(1'!
T~~) == (7.2)
=(
1
~h" - et(u)
0 ) ( kl (u)71 1 0
1) (10
0 k 2 (u)T;>
-T;;~e~,,(U)T;t) . 1
(7.3)
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
70
Remark. Strictly speaking, the dccomposition (7.3) is valid in the algebra A defined by the relation (6.7), but simultaneously the formula (7.3) shows that all the generators {h", c,,, 1;'1, co_a} of the deformed Yangian y,1.{($!2) can be expressed in terms of the I.-operator (6.:J), what proves that the homomorphism constructed in Theorem 6.1 is an epimorphism. The Gauss decomposition (7.3) provides the following choice of generators for Yry.{("!2): ht(u)
= 1;,-t ht(u)T,-;t
. et(u)
= et(u) - ~hcx , e~cx(u) = T:te~cx(u)1~t.
(7.4)
Using relations (5.1 )-( 5. 7) one can obtain thc following relations betwecn -
±I
hcx(u), e±cx(u) and 1;'
2:
T:th-:;(u)T: t = 1;;l et(u)T: 1 = et(u) =F 217~
(I ±17~e~a(u)r1i!t(u)(1
±17~e~cx(U)rJ,
(Vi)
± 217~(1 ± 17~e~cx(u)tl ht(u)(1 ± 17~e~cx(u)rl,
(7.6)
T:te_cx(u)T: t = e_cx(u)(1 ±17~e_cx(u)r1.
(7.7)
With the help of the relations (5.1)-(5.7) and (7.5)-(7.7), we can prove the following thcorem. Theorem 7.1 Defining relations for the "field,," i!t(u), e!,,(u) have the form: Hdu)h~(v) = H1(V)i!~(u) ,
(7.8)
r 217~(et(
[et(u), et( v)] = 1/ (et(v) - et(u) + u) - et( v)) , v-u (u - v + 17)91(u)e~cx(v) - (u - v -17)gl(v)e~cx(u) ==
(7.9)
17(gl(u)e~cx(u)) + gl(v)e~cx(v)) ,
(7.10)
(U-1J-17)Hl(u)e~cx(v)-(u-V+1/)gl(V)ht(u)
== 17Hl(v)e~cx(u)+17g1(U)i!cx(v),
( e~(u) - 217~ + 2TI~H2(U))92(V) - 92(v)et(u)
= -
1/(H2(u) - H 2(v)) u-v
(7.11)
+ 217~92(v)
,
(7.12)
(u - v
+ 17)(e~(v) =
217~ + 217~H2(v))H2(u) - (u - V -17)H2(u)
17H2(v)e~(u)
where H 1(u) =
=
+ 17(e~(u) - 21/~ + 2TI~H2(U))H2(U)
,
(7.13)
(I - 217~e~cx(u)rlht(u)(1 - 217~e~,,(u)rl
,
(7.14)
H 2(u) = (I -17~e~a(U)r\t(u)(1 -17~e~,,(U)rl , 91(U)
= (I - 21IU~a(u)r1e~,,(u)
•
92(U)
= (I -17~e~cx(u)rl e~cx(u).
(7.15)
(7.16)
Rational solutions of Yang-Baxter equation
71
Proof. Let us prove for instance the formula (7.10). From (5.6) we have
Substituting c_et(u)
1]~C~,,(u)rl (see 8.
=
1
1
1
I
TJc+et(u)TJ and using that T;;2e~et(u)TJ
=
C~et(u)(l -
(7.7)), we obtain the formula (7.10).0
Appendix. An integral associative formula and quantization of Poisson brackets
Let M be a smooth manifold equipped with a Poisson bracket. To quantize this bracket means to find a family of new operations * on C=(M) depending on a formal parameter h ("Planck constant") such that: (i) f * 9 = f 9 + ~ih{f, g} + L~2 hkFkU, g), where Fk are bidifferential operators on C=(M); (ii) the operation * is associative; (iii) it is a distributive operation with respect to addition; (iv) for any a constant real function a E R is valid the relation a * f = f * a = af. We are interested in the following case: M = £*, where £* is the dual space to a real Lie algebra £, and M is equipped with a Poisson bracket {".} B, where B is a 2-cocycle on £. Let us defiue the following bracket {".} B:
{f,g}B(X)
= (J,g](x) + BU,g)
,
(8.1 )
where f and 9 are two linear functions on £* and we extend the definition using the Leibnitz rule. We do not claim that B is non-degenerate. The following lemma is valid.
Lemma 8.1 The bracket (8.1) is a Poisson bracket.
Proof. The Leibnitz rule {fg, h}H = f{g, h}B + g{f, h}B follows from definitions immediately. Further, it is sufficient to check the .Jacobi identity for linear functions. The latter is straightforward since BU, g) = const and hence {BU, g), h} B = O. Thus, the ,Jacobi identity for {.,.} B follows from the Jacobi identity for £ and the 2-cocycle property for B. Let Ch(x, y) be the Campbell-Hausdorf series, uniquely defined by the relation chx . chy = ehCh(x,y), where x and yare elements of the Lie algebra £. The function Ch(x,y) has a form Cd:r, y) = x
h
h2
+ y + "2[x, y] + 12([~' [C 1]]] + [1], [1],~]]) + ...
(8.2)
and satisfit's tlw following associativity condition (8.3)
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
72
It is well- known that G" (x, y) is a commutator series, namely
where [XI'" Xn] == ~[ .•• [Xl, X2]X3,'" xn]. Define a form B" (x, y) with values in R as follows
h2
h
B,,(x,y)== 2B(x,y)+ 12(B(x,[x,y])+B(y,[y,x]))+...
(8.5)
In other words, we substitute ~ B([[Xh X2] ... Xn-I], xn) instead of [XI, ... ,x n] in the Campbell-Hausdorf series. The following proposition takes place. Proposition 8.1 The form (8.5) satisfies the relation
B,,(C,,(x,y),z)
+ B,,(x,y) == B,,(x,C,,(y,z)) + B,,(y,z)
(8.6)
for any x, y, z E L. Proof Consider the central extension LB of the Lie algebra.c. As a vector space 2" L CD R and admits the following commutator
LB
(8.7)
[(a,A), (b,JI)] == ([a,b],B(a,b)).
It is easy to check that (8.8) Then
A :==
(exp"(x,O) . exp"(Y,O))
. exp"(z, 0) ==
exp("C\(x,y),O) . exp"(z,O)
. exp(O,Rh(X,Y))
(8.9)
,
i.e., exp(O,Bh(a,b)) is a central element of the group G(LB) and hence the clement A has the form A == (exp("Ch(Ch(X,y),z), expBh(Ch(X,y),z)+Bh(X,y)) . (8.10) The statement of the Proposition follows from the associativity in the group As a consequence of this Proposition we have the following result.
G(LB ).0
Corollary 8.1 Let f,9 E COOte), then the following convolution
f *9
==
Jj(Og(
1]) exp'<X,Ch(C~»
.
exp'Bh(e,~) d~d1]
(8.11 )
is an associative operation on COOte), which quantizes the bracket {', .} B. Here is the Fourier transform of the function f.
i(O
Rational solutions of Yang-Baxter equation
73
Proof It is not difficult to see that the operation (8.11) is an associative operation. Indeed, we have
Further It is obvious that
Therefore, we have
(f * g) * h
=
J
j().)g(Jl )h( 1])' expi<X,Ch(Ch(A,/l),ij)> . expiBh(Ch(A,/l),ij) . expiBh(A,/l) d>.dJId1] (8.14)
and Proposition 8.1 implies associativity. Further, if we decompose f * 9 in the series in h, the constant term is
Jj(<)g(1])expi<x,~+ij> d~d1]
=
f(x)' g(x)
(8.15)
and the coefficient at the linear term is equal to
~
J
j(<)g(1])( < x, =
z
[~, 1]] > +B(~, 1])) expi<x,e+ij> d~d1] =
2"( < x, [\7 f, \7g] > +B(\7 f, \7g) .
(8.16)
Clearly, it is nothing but ~([J,g](x) + B(f,g)) if f,g are linear functions. The proof is completed.D Example. Let.c be arbitrary and B = O. Then this formula provides a quantization of the Lie-Kirillov-/<,:ostant bracket (see [IB}). Now we define an action of C(.c) on C, which depends on B satisfying the cocycle condition of Theorem 1.2. Consider £B as in Proposition 8.1. Then £*a contains a hyperplane .ci = {h E £*a : h(O,I) = I}. Clearly.ci is parallel to .c~ = {h E £*a : h(O,I) = O} and .c~ is nothing but C. It is easy to see that .c'B = .c~ EB R and let p be the projection from .c~ onto .ci along R. Then define PB(g) E Diff(.c~) as follows:
Here Diff(.c o) is the set of diffeomorphisms of .co; 9 = (9, A) E G(£B), where A is an arbitrary element of R', "y E G(.c) and the action docs not depend on A. Moreover, PB is well-defined since (0,1) E Center(£B)'
Lemma 8.2 (8.17)
74
S.M. Khoroshkin, A.A. Stolin and V.N. Tolstoy
The Proof is straightforward. Remark. The action PR is non-linear unless B the coadjoint representation of G(£).
= O.
If B
= 0,
po is nothing but
Theorem 8.1 ([22]). (i) Let B be non-degenerate. Then PH acts locally transitively at the point (0,0) E and hence, defines an equivariant local diffeomorphism between G(£) and £'0 (equivariancy should be undel'stood with respect to right shifts on G(£) and PB on £0)' This local diffeomorphism enables one to carry the *-operation from £0 ~ L* to G(£). (ii) Such obtained *-operation on C00( G(£)) uniquely defines
Co
PI E (U(£)[[h]])0 2
and f*g = m(Fd-I(f®g), where f,g E COO(G(L)), 12 23 (iii) (.::1 09 I)(Fd- 1 . (F)-1 = (I QS:.::1 )(Fd- 1(F)-I.
m is the usual multiplication.
l'\ow we assume that £ is a Frobenius Lie algebra, which means that there exists such that 1[( x, y]) is a non-degenerate 2-form on £.
J E L*
Theorem 8.2 ([18]). Let £ be a Frobenius Lie algebra with J E L* and B = O. Then Po acts locally transitively at the point J E L* and uniquely defines r2 satisfying conditions 2) and 3) oj Theorem 8.1. Remark. In both cases F was computed (partially)
Acknowledgments
The authors are thankful to the Swedish Academy of Science for the support of the visit of the first and third authors to the Royal Institute of Technology (Stockholm), during which visit the paper was completed. The first and third authors would like also to thank the Russian Foundation for Fundamental Research, grant 95-01-00814A, and the ISF, grant MH1300, for the financial support.
Rational solutions of Yang-Baxter equation
75
REFERENCES [1] A. Stolin, TRITA-MATH 1994 0042 preprint, Stockholm, 1994. [2J S.M. Khoroshkin, A. Stolin and V.N. Tolstoy, TRITA-MATH 1995-MA-17 preprint, Stockholm, 1994; TRlTA-MATH 1994 0042 preprint, Stockholm, 1994; Lett. Math. Phys. (1995) (to appear). [3] V.G. Drinfeld, Soviet Math. Dokl. 32 (1985) 254. [4] V.G. Drinfeld, Soviet Math. Dokl. 36 (1985) 212-218. [5J D. Uglov and V. Korepin, Phys. Lett. A 190 (1994) 238. [6] D. Uglov, preprint of State University of New York at Stony Brook, (Sept. 1994). [7] K. Hikami, Phys. Lett. A 197 (1995) 393. [8] K. Hikami, J. Phys. A: Math. Gen. 28 (1995) L131. [9] F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard and V. Pasquier, Phys. Rev. Lett. 69 (1992) 2021. [10J D. Bernard, M. Gaudin, F.D.M. Haldane, and Y. Pasquier, J. Phys. A: Math. Gen. 26 (1993) 5219. [11] T. Nakanishi, Nucl. Phys. B439 (1995) 441. [12] A. LeClair and F. Smirnov, Int. J. Mod. Phys. A7 (1992) 2997. [13J A. Molev, M. Nazarov and G. Olshanski, preprint of Australian National University, Canberra, 1994. [\4] A. Molev, J. Math. Phys. 36 (1995) 923. [15J M. Nazarov and V. Tarasov, Pub/. RIMS. Kyoto Univ. 30 (1994) 459. [16] S.M. Khoroshkin and V.N. Tolstoy, Lett. Math. Phys. (1995) (to appear). [17] A. Stolin, Math. Scand. 69 (1991) 57. [18J M. Gekhtman, A.Stolin Proceedings of the workshop in algebra "Moscow-Tainan" (1996) (to appear). [19] M. Gestenhaber, A. Giantino and S.D. Schack, in Lecture Notes in Math. on Quantum Symmetry, 1510 (1992) 9. [20J L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, AIgebm Analysis, 1 (1989) 178. [21] V.G. Drinfeld, Leningmd Math. J. 1 (1989) 1459. [22] V.G. Drinfeld, Soviet Math. Dokl. 28 (1983) 667. [23] J. Ding and LB. Frenkel, Commun. Math. Phys. 156 (1993) 277. [24J Ch. Ohn, Let. Math. Phys. 25 (1992) 85.
Chapter II. QUANTUM GROUPS: APPLICATIONS
From Field Theory to Quantum Groups
LARGE N MATRIX MODELS AND Q-DEFORMED QUANTUM FIELD THEORIES LYA.AREF'EVA
Steklov Mathematical Institute, Vavi/ov 42, eSP-I, 117966, Moscow, Russia e-mail: arefevaatarevol.mian.su
Abstract: Recently it was shown that an asymptotic behaviour of SU(N) gauge theory for large N is described by q-dcformed quantum field. The master fields for large N theories satisfy standard equations
of relativistic field theory but fields satisfy q-deformed commutation relations with q = O. These commutation relations are realized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a nonAbelian gauge symmetry. The gauge master field for a subclass of planar diagrams, so called half-planar diagrams, is also considered. A recursive set of master fields summing up a matreoshka of 2-particles reducible planar diagrams is briefly described.
1.
Introduction
With great pleasure I dedicate my paper to Jurek Lukierski. He made a great contribution in quantum field theory. I hope that this paper will be interesting for him sincc his last interests are relatcd with q-deformed algebras [1,2]. In thc last years there were considerations of models of quantum field theory based on quantum groups and q-deformed commutation relations [1]-[20]. The main problem with these approaches was how to find a physical justification for q-dcformed quantum field theory. Recently, it was shown that there is a remarkable physical justification for q-deformcd quantum field thcory at least for q = O. In [21] it was shown that such theory describes an asymptotic behaviour of SU(N) gauge theory for N ---> oc.
80
LYa. Arcf'eva
The large N limit in QeD where N is the number of colours enables us to understand qualitatively certain striking phenomenological features of strong interactions [22]-[25]. To perform an analytical investigation one needs to compute the sum of all planar diagrams. Summation of planar diagrams has been performed only in low dimensional space-time [26,27]. It was suggested [24] that there exists a master field which dominates the large N limit. There was an old problem in quantum field theory how to construct the master field for the large N limit in QCD. This problem has been discussed in many works, see for example [29]-[35]. More recently the problem has been reconsidered [36][46] by using methods of non-commutative (quantum) probability theory [41,47,48]. Gopakumar and Gross [36] and Douglas [37] have described the master field using a knowledge of all correlation functions of a model. Finally the problem of construction of the master field has been solved in [21]. It was shown that the master field satisfies to standard equations of relativistic field theory but it is quantized according to q-deformed (q = 0) relations
a(k)a*(k l ) = 8(k - kl ), where a(k) and a*(k) are annihilation and creation operators. These operators have a realization in the free (Boltzmannian) Fock space. Therefore, to study the large N limit of QCD it seems reasonable to develop methods of treatment of theory in the Boltzmann space. Let us note that the fact that the master field satisfies the same equations as usual relativistic fields makes actual a development of non-perturbative methods of investigation of these equations. Quantum field theory in Boltzmannian Fock space has been considered in [43][46]. Some special form of this theory realizes the master field for a subset of planar diagrams, for the so called half-planar (HP) diagrams and gives an analytical summations of HP diagrams [4:l]-[46]. In this paper a realization for the master field for lIP diagrams of gauge theories will be given. Then some recursive set of master fields summing up a more rich subclass of planar diagrams will be also sketched. This subclass of diagrams contains a matreoshka of 2-particles reducible planar diagrams. In Section 2 consider the master field for matrix models including the master field for gauge field will be considered. In Section 3 Boltzmann quantum field theory and the lIP master field for scalar and gauge theories will be considered.
2. 2.1.
Master Field for Planar Diagrams Zero-dimensional Master Field
The master field <1> for the Gaussian matrix model in zero dimensional space-time is defined by the relation (2.1 )
Large N Matrix Models for k = 1,2... , where the action S(M) = ~ tr M matrix. The operator !/J = a + a'
2
81
and M is an Hermitian N x N
acts in the free or Holtzmannian Fock space over C with vacuum vector aiD
>= D
10 > (2.2)
and creation and annihilation operators satisfying the relation aa' = 1
(2.3)
Let us recall that the free (or Boltzmannian) Fock space F(ll) over the Hilbert space H is the tensor algebra over H
Creation and annihilation operators are defined as
u'(f)II Q9 ... Q9 In = I Q9 II
(9 ...
(0 In
a(f)II 0 ... 0 In =< I, II > 0h 0 ... 0 In where < 1,9 > is the inner product in H. One has
a(f)a'(9) =< 1,9> . Here we consider the simplest case H = C. The relation (2.1) has been obtained in physical [29] and mathematical [47] works. It can be interpreted as a central limit theorem in non-commutative (quantum) probability theory, for a review see [41]. The basic notion of non-commutative probability theory is an algebraic probability space, i.e a pair (A, h) where A is an algebra and h is a positive linear functional on A. An example of the algebraic probability space is given by the algebra of random matrix with (2.1) being non-commutative central limit theorem. As another example one can consider quantum groups. Theory of quantum groups have received in the last years a lot of attention [49]-[52]. In this case A is the Hopf algebra of functions on the quantum group and h is the quantum I1aar measure. The relations of theory of the master field in the Boltzmannian Fock space with quantum groups was discussed in [42].· There we defined the master field algebra and showed that this algebra is isomorphic to the algebra of functions on the quantum semigroup SUq (2) for q = D. In fact the master field algebra coincides with the algebra of the so called central elements of the quantum group Hopf algebra. Let us
82
I.Ya. Aref'eva
repeat the main steps of these observations. In the Boltzmannian Fock space F( C) we have (2.4) a'a == 1 -10 >< 01. Let us define an operator
F == c'IO >< 01, where r/J is an arbitrary real number. following relations:
(2 ..5)
Then from (2.2), (2.4), (2.3) one has the
aF == 0, aF' == 0, F F' aa'=I, a'a+FP'=1.
= p' F,
(2.6)
We call the algebra (2.6) the master field algebra. From equations (2.6) we get
and the operator P F' is an orthogonal projector. Now let us recall the definition of algebra of functions A q = Fun(SlJq (2)) on the quantum group ,c..;Uq (2). The algebra Aq is the Hopf algebra with generators a, a', c, c' satisfying the relations
°
ac' = qc'a, ac = qca, cc' = c'c, a'a + cc' = 1, aa' + q2cc' = 1
(2.7)
°
where < Iql < 1. Taking q = in (2.7) one gets the relations (2.6) if F == c. Therefore the master field algebra (2.6) is isomorphic to the algebra A o of functions on the quantum (semi) group SUq (2) for q 0. A q is a Hopf algebra with the standard coproduct,
=
,1 :
Aq
->
Aq
~
Aq
,1(gj) = L gl ® gJ. k=O,l
The Boltzman field 1> is a central element of the Hopf algebra A g • The bosonization of the quantum group SUq (2) [20] gives bosonization for the master field. If band b' are the standard creation and annihilation operators in the Bosonic Fock space, [bY] == I, blO >= 0, then 1a
==
q2(N+l)
N
+1
b, c =
satisfies the relations (2.7). Here N = b'b, r/J is a real number. If q (2.8)
a=
1
~b,
vN+ 1
c= e
i,p
(2.8)
e'qN
10 >< 01.
---->
°one gets from (2.9)
Large N Matrix Models
83
Therefore the master field takes the form
1> = b'
1
+
1
IN+T IN+T
b.
(2.10)
The operator N can be also written in terms of creation and annihilation operators a+, a, N = L::;;"=l(a+)k(al.
2.2.
Master Field as a Classical Matrix
Let us consider UrN) invariant correlation functions for a model of selfinteracting Hermitian scalar matrix field M(x) = (Mij(x)), i,j = 1, ... , N in the D-dimensional Euclidean space-time (2.11 ) where S(M) is the action
and M is an Hermitian N x N matrix, (2.12) Witten suggested [24] that there exists a master field which dominates in the large
N limit of invariant correlation functions of a matrix field, i.e.
where M is some 00 x 00 matrix. Since 00 x 00 matrix can be considered as an operator acting in an infinite dimension space one can interpret the RHS of (2.13) as an expectation value of the product of some operators 1>(x;) (2.14) This interpretation gives an alternative definition of the master field 1>(x) as a scalar operator which realizes the following relation (2.15) where < . > means some expectation value. Therefore the problem is in constructing of a scalar field 1> acting in some space so that the expectation value of this field reproduces the large N a'iymptotic of UrN) invariant correlation functions of given matrix field.
I. Va. Aref'eva
84 2.3.
Free Master Field
To construct master field for the free matrix field let us calculate the expectation value for free matrix field in the Euclidean space-time (2.16)
where the action
We have 1
N3 < ir (M(xIlM(X2)M(X3)M(X4)) D(xI - x4)D(X2 - X3)
D(XI - X2)[)(X3 - X4)+
>(0)=
1
+ ND(xI
(2.17)
- x3)D(X2 - X4),
Here we use
< Mij(x)Mj'i'(Y)
>(0)=
bii,bjj,D(x - y),
(2.18)
D is an Euclidean propagator, dD k
D(x - y) =
eik(x-y)
J(21l')D k2 +
m2
(2.19)
Let (2.20) be the Holzmann field with creation and annihilation operators satisfying the relations rp-(x)¢i+(y) = D(x - y),
(2.21 )
It is easy to check that
where [0 > is a vacuum ,p-(x)IO >= 0 =< OI4>+(x). The similar relation is true for an arbitrary n-point correlation function. This consideration proves that the Euclidean Holtzmann field is a maste~ field for the Euclidean free matrix model. Moreover, if we assume the relations (2.23) then we get (2.24)
Large N Matrix Models
D(XI - X2)f)(.T3 - X4)
+ D(x]
- x4)D(X2 - X3)
85
+ qD(xI
- x3)D(X2 - X4),
i.e. relation (2.24) reproduces (2.17) if we identify 1
(2.25)
N = q.
We can also consider the Minkowski space time. To avoid misunderstanding we use a notation M(in) for the free Minkowski matrix field. One has (2.26) where
/F(J:) =
('2~)3
J
o eikxO( _k )b(k 2
-
2
m )dk.
onsider the free scalar Boltzmannian field .p(in)(x) given by
.p(in)(x) = _1_
(21r )3/2 where w( k) = y'P
+ m 2•
J~(a'(k)eikX + J2W( k)
a(k)e- ikx ),
(2.27)
It satisfies Klein-Gordon equation
(0
+ m 2 ).p(in)(x) =
0
and it is an operator in the Boltzmannian Fock space with relations
a(k)a*(k') = 8(3)(k - k')
(2.28)
and vacuum 1.00), a( k) 1.00) = O. A systematical consideration of the Wightman formalism for Boltzmannian fields is presented in [44]. a(k) and a*(k) act in the Boltzmannian Fock space T(H) over 11 == U(R3 ) and one uses notations such as aU) == J a( k )f( k )dk. An n-particle state is created from the vacuum lila) == 1 by the usual formula
but it is not symmetric under permutation of ki . The the following basic relation takes place lim _1_. < 01/1' ((M(in)(y!lJPl ... (M(in)(Yr)yTJIO > NI+,
(2.29)
N-oo
= (Jlol( .p(in) (yd )PI ... (.p(in)( X T lJPT lila)
where k = PI +... + pro To prove (2.29) one uses the Wick theorem for the Wightman functions and 't Hooft's graphs with double lines. According to the Wick theorem we represent the vacuum expectation value in the L.II.S. of (2.29) as a sum of't lIooft's graphs with the propagators ('2.26). Then in the limit N --+ 00 only non-crossing (rainbow) graphs are nonvanished. We get the same expression if we compute the R.H.S. of (2.29) by USiIlg the relations (2.28), i.e. by using the Boltzmannian Wick theorem.
I. Ya. Aref'eva
86
2.4.
Master Field for Interacting Matrix Scalar Field
To construct the master field for interacting qualltum field theory [21] we have to work in Minkowski space-time and use the Yang-Feldman formalism [53]-[56]. Let us consider a model of an Hermitian scalar matrix field M (x) == (M i ) (x)), i, j == 1, ... , N in the 4-dimensional Minkowski space-time with the field equations
(0
+ m 2 )M(x) == J(x)
(2.30)
We take the current J(x) equal to (2.31) where 9 is the coupling constant but one can take a more general polynomial over M(:r). One integrates eq (2.30) to get the Yang-Feldman equation [5:1,54]
M(x) = M(in)(x)
+
J
f)ret(x - y)J(y)dy
(2.32)
where Drct(x) is the retarded Green function for the Klein-Gordon equation,
and MUnl(x) is a free Bose field. The U(N)-invariant Wightman functions are defined as
W(XI,···,Xk) =
1
-k
NJ+'j
< Oltr(M(xd···M(Xk))IO >
(2.:13)
where 10 > is the Fock vacuum for the free field M{in) (x). We will show that the limit of functions (2.33) when N -+ 00 can be expressed in terms of a quantum field ¢>(x) (the master field) which is a solution of the equation
¢>(x) == ¢>(in)(x)
+
J
nret(x - y)j(y)dy
(2.34)
where (2.35) The master field 4>( x) does not have matri x indexes. The following theorem is true. Theorem 1. At every order of perturbation theory in the coupling constant one has the following relation (2.36)
where thf field M(J;) is dffined by (2.:12) and ¢>(x) is defined by (2 . .'14). The proof of the theorem can be found in [21].
Large N Matrix Models 2.5.
87
Gauge field
In this section we construct the master field for gauge field theory. Let us consider the Lagrangian I 2 I 2' L = l7' { -';/"V - 2a(o"A,,) +co"V'"c} (2.37) where A" is the gauge field for the SU(N) group, c and c are the Faddeev-Popov ghost fields and Q is a gauge fixing parameter. The fields A", c and c take values in the adjoint representation. Here
9 is the coupling constant. Equations of motion have the form
(2.38)
One writes these equations in the form
(2.:m) Dc
where
= J,
Dc
= J,
J V = -~o,,[AI" A v]- ~[AJ.l' }'"v] - ~Ovcc - ~efM.,
N2
N2
J =
-~o"[A,,,c], J N2
N2
=
N2
-~[A",o"c] N2
From (2.:39) one gets the Yang-Feldman equations (2.40)
c(x) = c(in1(x)
+
J
Dret(x - y)J(y)dy, c(x) = c(in)(x)
where
D::t(x) = (g"v - (I _
+
J
Q)o~V)lyet(x),
and 9"v is the Minkowski metric. Free in-fields satisfy
Dret(x - y)J(y)dy,
I. Ya. Aref'eva
88
and they are quantized in the Fock space with vacuum 10 >. The vector field A~n) is a Bose field and the ghost fields e(in), e(in) are Fermi fields. Actually one assumes a gauge Q = 1. In a different gauge one has to introduce additional ghost fields. We introduce the notation 1j.'i = (A", c, c) for the multiplet of gauge and ghost fields. The U(N)-invariant Wightman functions are defined as (2.41) We will show that the limit of functions (2.41) when N --> OC! can be expressed in terms of the master fields. The master field for the gauge field A,,(x) we denote BJ1.(x) and the master fields for the ghost fields c(x),c(x) will be denoted ry(x),ry(x) . The master fields satisfy to equations
(2.42) where (2.43) These equations have the form of the Yang-Mills equations (2.39) however the master fields B", ry, ry do not have matrix indexes and they do not take values in a finite dimensional Lie algebra. The gauge group for the field B" is an infinite dimensional group of unitary operators in the I30ltzmannian Fock space. Equations (2.42) in terms of currents read (2.44)
Dry=j,Di/=j, where
jv = -ga"[B,,, Bv]- g[13", F"v]- gavi/ry - gryovi/, j
= -go"[B,,, ry], J = -g[B", O"i/].
We define the master fields by using the Yang-Feldman equations
B,,(x) = T/(:r)
= ry(in)(x) +
J
13~m)(x) + D::t(x - y)jv(y)dy,
JDTet(x - y)j(y)dy,
ry(x)
= ii(in)(x) +
(2.45)
JDTet(x - Y)J(y)dy,
The in-master fields are quantized in the I30ltzmannian Fock space. For the master gauge field wp have (2.46 )
Large N Matrix Models
89
where f~.\l( k) are polarization vectors and annihilation and creation operators satisfy (2.47) The expression (2.46) for the field BI'(x) looks like an expression for the photon field. However because of relations (2.47) the commutator [HI'(x), B,,(x)] does not vanish and it permits us to develop a gauge theory for the field BI'(x) with a non-Abelian gauge symmetry. We quantize the master ghost fields in the Boltzmannian Pock space with indefinite metric 3 (inl( ) _ 1 d k ( *(k) ikx 4(k) -ikx) (2.48) 11 x - (271-)3/2 e + fJ e , 7](in l (x)
= _1_
(271" )3/2
JJ2TkT' J~((3*(k)eikX + J2TkT
,(k)e- ikx ),
where creation and annihilation operators satisfy
,(kh*(k')
= 5(3 l (k -
IJ(k)(3*(k') =
_t5(3 l (k
k'), - k').
(2.49)
We also assume that the prodnct of any annihilation operator with a creation operator of a different type always is equal to zero, i.e.
,(k){r(k') a(.\l(k){r(k')
= (3(kh*(k') = a(.\)(kh*(k) = 0, = ,(k)a(.\)*(k') = (3(k)a(.\)*(k') = 0.
(2.50)
The Boltzmannian Pock vacuum satisfies
(2.51) Let us denote Xi = (HI" 1], Ti) the multiplet of the master fields. The following theorem is true. Theorem 2. At evcry order of perturbation theory in the coupling constant one has the following relation
whlTc the fields AI'(x),c(x) and c(x) are defined by (2.40) and HI'(x),1](x) and Ti(x) are defined by (2.45). The proof of Theorem 2 is analogous to the proof of Theorem 1. We get relations (2.49) for master fields by taking into account the opposite statistics of the ghost fields.
I. Va. Aref'eva
90
Master Field for HP Diagrams
3. 3.1.
Half-Planar Approximation for the One Matrix Model
A free n-point Grccn's function is defincd as the vacuum expectation of n-th power of mastcr field
(3.1) As it is well-known, the Green's function (2.4) is given by a n-th moment of Wigner's distribution [26,47]
dO)
=
2"
j2 d)' ),2"~ = n!(n+I)!' (2n)! -2211'
This rcpresentation can be also obtaincd as a solution of the Schwinger-Dyson equations ,(O) GJ2n
-
-
~
L...J
G(O)
2m-2
G(O)
2n-2m'
m=l
Interacting Boltzmann correlation functions are defincd by the formula [43] (3.2) In contrast to the ordinary quantum field theory where onc dcals with the exponcntial function of an interaction, here we deal with thc rational function of an interaction. In [45] it was shown that under natural assumptions the form (5.1) is unique one which admits Schwinger- Dyson-like equations. For the case of quartic interaction Si"t = 9<jJ4 the Boltzmannian Schwinger-Dyson equations have the form k-l
G
n
G " = """G(O) ~ k-/-l ..I/+n-k-l + /=1
-g[G n -
k G k+ 2
""" G(O) G ~ /-k-l ..I n +k-I-I
(3.3)
/=k+1
+
G,,-k+l Ch+l
+ Gn - k+ 2 G k + G"-k+3(h-d·
For 2- and 4-point correlation functions we have
(3.4) 3.2.
Boltzmann Correlation Functions for D-DimensionaI Space-Time
Hcre we present the Schwinger-Dyson equations for Boltzmann correlation functions in f)-dimcnsional Euclidean space. To avoid problems with tadpoles let us following [46] Consider the two-field formulation. We adopt the following notations.
Large N Matrix Models
91
Let V,(x) = Jj>+(x)+V'-(J:), ¢(x) =¢+(x)+¢-(x) be the Holzmann fields with creation and annihilation operators satisfying the relations
where D( x, y) is V-dimensional Euclidean propagator. The n-point Green's function is defined by
We define an one-particle irreducible (IPI) 4-point function J'4(x,y,z,l) as
where :F4 is a connected part of F4
r4(x,y,Z,t) = :F4 (x,y,z,t)
+ f 2(x,t)D(y,z).
Kote that in the contrast to the usual case in the RHS of (:J.6) we multiply :F4 only on two full 2-point Green functions while in the usual case to get an I PI Green function one multiplies an n-point Green function on n full 2-point functions. Let us write down the Schwinger-Dyson equations for the two- and four-point correlation functions. We have
r 4 (p, k, r) = -g - 9 .1'2(P) = 9
J
dk'/'2(P + k - k')D(k')/'4(p + k - k', k', 7')
JdkdqFi(k)D(q)D(p - k - q)r (p, k, q). 4
(3.7)
(3.8)
where ,
h
1
=
2
P
j~ +m 2 + ~2
Equation (3.7) is the Bethe-Salpeter-like equation with the kernel which contains an unknown function Fl.. Equation (3.8) is similar to the usual relation between the self-energy function 1.,'2 and the 4-point vertex function for
I. Va. Aref'eva
92
3.3.
A Matreoshka of 2-particles Reducible Diagrams
In [43] has been shown that equations sum up HI' diagrams of planar theory. Let us remind the definition of HI' diagrams. Sometimes they are called the rainbow diagrams. The free rainbow diagrams are dual to tree diagrams and they have been summed up in the zero dimensions [57]. The half-planar diagrams for < ir' (M(x n ), ... M(x n )) > are defined as a part of planar non-vacuum diagrams which are topologicaly equivalent to the graphs with all vertexes lying on some plane line in the left of generalized vertex represented tr (M(x n ), ... M(x n )) and all propagators lying in the half plane, We can use F"2 and F4 to construct correlations functions which correspond the sum of more complicate diagrams. Let us consider the following correlations functions (3.9)
J
i::::4
(1
+ II dDx;l4(x2' X3, X4, XI )lj;1 (xd
: r/J(X2)r/J(X3) : lj;1( X4)tIIO),
t=)
here
(3.10) The Schwinger-Dyson equations for the 2- and 4-point correlation functions rjl) and FP) satisfy t.o equations similar to equations (3.7) and (3.8). The obtained FJ!) and 1,~1) may be used to define the next approximation to planar diagrams. One can see that such procedure sums up special type of 2-particles reducible diagrams. These diagrams are specified by the property that they contain two lines so that after removing these lines from the given diagrams one reminds with two disconnected parts and each of these disconnected part is itself a connected 2-particle reducible diagram. It is natural to call this set as an matreoshka of 2-particle reducible diagrams.
3.4.
Master Field for HP Gauge Theory
In the case of gauge theory the set of the HI' master field is given by the field satisfying the following relations (3.11 )
1 c-(x)c+(y) = (27r)D
1 c-(x)c+ (Y)=-(21f)D A;(x)c+(y)
J J
dDk PI cxpik(x - y),
1 . y ), dD kpexpzk(x-
= A;(x)c+(y) =
°
(3.12)
(3.13) (3.14)
Large N Matrix Models
93
A unrcnormalized interacting Lagrangian which assumed to enter in the correlation functions as is (3.5) (a generalization to two-fields formalism is evident) has the form
4g2{2AvAI'AvAI' - AvAI'AI'Av - AI'AI'AvA v } - g{ Ol'cAl'c
+ cOl'cAI' + Al'col'c -
ol'ccAI' - Al'ol'cc - cAl'ol'c}
(a.15)
Divcrgences in onc-loop correlation functions may be removed by the following renorlIlalizations
2
+~ z1/f'(Zf P
r {2AvAI'AvAI' 1
AvAI'AI'Av - A,.AI'AvA v }-
fLZllP(ZllP)-I OI'C-A I'C - C-A I' 0I'C-) 8 1 2 (3.16) where 2
ZllP = I 1
Z.
HI'
2
ZHP 1
= 1
+ -g-(-O') In A 2 161r
g2 13 +(- 161r2 3
O')ln A
= 1' 2 ZHP = 1 + L(~ 161r2 2
- ~)lnA 2
(3.17) (3.18)
(3.19)
Z2 factors enter in the nonstandard way since only two legs (the first and the last) may bring the wave function renormalization. So we have a modification in the definition of beta function (3.20) Therefore we have
(3.21) Recall that the IIsual beta function is /3 = -~¥-. Note that we get good results: beta remains negative and in this approximation it does not depend on the gauge fixing parameter 0', i.e. it is gauge invariant.
94 4.
LYa. Aref'eva Concluding remarks
In conclusion, models of quantum field theory with interaction in the Holtzmannian Fock space have been considered. To define the master field for large N matrix models we used the Yang-Feldman equation with a free field quantized in the Boltzmannian Fock space. The master field for gauge theory does not take values in a finite-dimensional Lie algebra however there is a non-Abelian gaugf' symmetry. For the construction of the master field it was essential to work ill Minkowski space-time and to use the Wightman correlation functions. The fact that the master field satisfies the same equations as usual relativistic fields push us to develop a non-perturbative methods of investigation of these equations. :\'ote in this context that in all previous attempts of approximated treatment of planar theory were used some non-perturbative approximation [59 61]. To sum up a part of planar diagrams we have used the new interaction representation with a rational function of the interaction Lagrangian instead of the exponential function in the standard interaction representation. The Schwinger-Dyson equations for the 2- amI -i-point correlation functions for this theory form a closed system of equations. The solntions of thf'se equations may be used to sum up a more rich class of planar diagrams. This is a subject of further investigations. Acknowledgments The author is grateful to P. Medvedev, L Volovich and A. Zubarev for useful discussions.
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K. "q-Dcformation of Poincare Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure" J. Phy... A28 (1995) 2255; M. Klimek and .1. Lukierski "K.-deformed realisation of D=4 conformal algebra" Acta Phy... Polan. B26 (199.1) 1209.
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[9] S. Watamura, "Hicovariant Differential Calculus and q-deformation of Gauge Theory", preprint II D-Til EP-92-45. [10] T. Brzezinski and S. Majid, Phy". Lett. B298 (1993) 339. [Ii] D.D. Coon, S. Yu and M.M. Backer, Phy". Rev. D5 (1972)1429. [12J V.V. Kurishkin, Ann. Found. L. de Broglie 5 (1980) 111. [1:3J 1. Va. Aref'eva and 1. V. Volovich, Phys. Lett. B264 (1991) 62; L Aref'eva, "Quantum Group Gauge theories", in Proc. oJ the First Sakharov Memorial ConJ. (1991); L Aref'eva and L Volovich, Phy". Lett. B286 (1991) 179; LV. Volovich, "Quantum Group Shear', hep-th/9404110. [14] O.W. Greenberg, Phys. Rev. D43 (1991)4111. [15] J. Rembielillski and K.A. Smolinski, Mod. Phys. Lett. A8(1993) 3335; T. Brzezinski, J. Rembielinski, K.A. Smolinski, Mod. Phys. Lett. A8 (1993) 409; J. Remhielinski, preprint, lIEP-TH/9302019 [16] M. Chriaco, Mod. Phys. Lett. A8 (1993) 2213. [17] LYa. Aref'eva and G.L Arlltyunov, Geometry and Physics, 11 (1993) 409-423. [18] A.P. Isaev and Z. Popowicz, Phys. Lett. B307 (1993) 353. [19] A.P. Isaev and P.N. Pyatov, Phys. Lett. A179 (1993) 81. [20] LYa. Aref'eva, K. Parthasaraty, K. Viswanathan and LV. Volovich, Mod. Phys. Lett. A9 (1994) 689. [21] L Aref'eva and L Volovich, hep-th/9510210 (to be published in Nuel. Phys. B). [22] G. t'Hooft, Nuel. Phys. B72 (1974) 461. [23] G. Veneziano, Nuel. Phys. B160 (1979) 247. [24] E. Witten, in: Recent Developments in Gauge Theories, eds. G. t'Hooft et a!., Plenum Press, New York and London 1980. [25] LYa. Aref'eva and A.A. Slavnov, Lectures in the XIV International School oj Young Scicntists, DlIbna, 1980; A.A. Slavnov, Acta Physiea Austriaca, Suppl. 25 (198:3) :3.')7. [26J E. Brezin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm. Math. Phys. 59 (1978) 35. [27J A. Migdal, .n'Tp, 69 (197.')810; B. Rusakov, Mod. Phys. Lett. A5 (1990)695. [28] YII. Makeenko and A.A. Migdal, Nucl. Phys. B188 (1981) 269. [29] O. Haan, Z. Physik C6 (1980) :345. [30] M.B. Halpern and C. Schwartz, Phys. Rev. D24 (1981) 2146.
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[31] P. Cvetanovic, Phys. Lett. 99B (1981) 49; P. Cvetanovic, P.G. Lauwers and P.N. Scharbach, Nucl. Phys. B203 (1982) 385. [32] LVa. Aref'eva, Phys. Lett. 104B (1981) 453. [33] .1. Greensite and M.B. Halpern. Nucl. Phys. B211 (1983) 343. [34] L.G. Yaffe, Rev. Mod. Phys. 54 (1982) 407. [35] A. Migdal, "Second Quantization of Wilson Loop", hep-th/9411100. [36] R. Gopakumar and D. Gross, Nucl. Phys. B451 (1995) 379. [37] M.R. Douglas, "Stochastic master field", RU-94-81, hep-th/9411025; [38] L Singer, Talk at the Congress of Mathematical Physics, Paris (1994). [39] M.B. Halpern and C. Schwartz, Phys. Rev. D24 (1981) 2146. [40J M. Douglas and M. Li, Phys. Lett. 348B (1995) 360. [41] L. Accardi, Y.Lu, L Volovich, hep-th/9412246 [42] L. Accardi, LYa. Aref'eva, S.V. Kozyrev and LV. Volovich, Mod. Phys. Lett. A10 (1995) 2323. [43] L. Accardi, LYa. Aref'eva and LV. Volovich, preprint CVV-201-95, SMI-05-95, hepth/9502092. [44] L. Accardi, L Aref'eva and L Volovich, "The Master Field in Euclidean and Minkowski Formulation and Free Non-Commutative Random Variables", in preparation. [45] L Aref'eva and A. Zubarev, preprint SMI-35-95. [46] L Aref'eva and A. Zubarev, preprint SMI-36-95, hep-th/9512166. [47] D. Voiculescu, K..J. Dykema and A. Nica, "Free random variables", CRM Monograph Series, Vol. 1, American Math. Soc. (1992); D.V. Voiculescu, Invent. Math., 104 (1991) 201. [48] 1. Accardi, Rev. in Math. Phys. 2 (1990) 127. [49J V. Drinfeld, in Proc. ICM Bcr'keley CA, Providence, RI: AMS, ed. A.M. Gleason (1986) 798. [50] 1.D. Faddeev, N. Reshetikhin and L.A. Tachtadjan, Alg. Anal. 1 (1988) 129. [51] S. Woronowicz, Commun. Math. Phys. 111 (1986) 613; Publ. RIMS: Kyoto Univ. 23 (1986) 117. [52] M. Jimbo, Lett. Math. Phys. 11 (1986) 242. [53] C.:". Yang and D. Feldman, Phys. Rev., 79 (19,')0) 972.
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[.54] J.D. Rjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 196.5. [.55] FoOl. Dyson, Phys. Rev. 82 (19,')1) 428. [,')6] G. Kallen, Arkivf. Phys.2 (19.51) 371. [,')7] D.V. BouJatov, V.A. Kazakov, I.K. Kostov and A.A. Migdal, Nucl. Phys. B275 (1986) 641. [58] K. Rothe, Nucl. Phys. B104 (1976) 344. [,')9] A.A. Slavnov, Phys. Lett., 112B (1982) 1.54; Theor. Math. Phys., 54 (1983) 46. [60] I.Ya. Aref'eva, Phys. Lett. 124B (1983) 221; Theor. Math. Phys., 54 (1983) 1.54. [61] G. Ferretti, Nucl. Phys. B450 (199.5) 713.
FroIll Field TlJeory to QuantuIll Groups
QUANTUM GROUP COVARIANT SYSTEMS M.
CHAICIIIANt AND
P.P.
KULISHt
t IIigh
Energy Physics Laboratory, Department of Physics, Research Institute for High Encrgy Physics P.O. Box 9 (Siltavuorcnpenger 20C), University of Helsinki Helsinki SF-OOOI4, Finland e-mail: chaichianatphcu.helsinki.fi +St.Petersburg Branch of Steklov Mathematical Institutc Fontanka 27, St.Petersburg, 191011, Russia e-mail; kulislJ atpdmi.ras.ru
Abstract: The meaning of quantum group transformation properties is discussed when instead of the Lie grouJl the corresponding quantum group is used while the algebraic (matrix) form of the transformation looks the same. Different algebras are considered which are covariant with respect to the quantum (super) groups 811q('2), SUq(l, I), SlJq(ljI), SUq(n), SUq(mln), OSpq(112) as well as deformed Minkowski space-time algebras. 1.
Introduction
The transformation properties of physical systems related to the Lie groups are of great importance for the understanding of Nature. As a result, applications of the Lie group theory take place in quite different branches of physics and the corresponding formalism is very well developed. The quantum groups and quantum algebras extracted from the quantum inverse scattering method (QISM) happen to be quite similar or even richf'r mathematical objects as compared to Lie groups and Lie algebras. In this paper we shall point out some peculiarities of the quantum group interpretations when the formal transformations of the physical quantities winside with the usual ones whi Ie the coefficients (elements) of these transformations started to be lIOn-commutative quantities belonging to a quantum group (QG) or a quantum algebra.
M. Chaichian and 1'.1'. Kulish
100
These objects (QG and q-algebras) are described using the language of Hopf algebras. In the general situation of Lie group theory one has the Lie algebra Lie( G) (or better to say the corresponding universal enveloping algebra) with non-commutative multiplication and symmetric coproduct .:1, and the commutative algebra of functions F on the Lie group manifold G with non-symmetric coproduct .:1 : F ---+ F @ F. After a q-deformation (or" quantization") the corresponding objects Lic q ( G) and Fq start to be much more similar in between as the Hopf algebras with non-commutative multiplications and non-symmetric coproducts in both cases. Hence, it looks natural to have the same physical interpretation for transformations including both of them. Putting aside the complicated integrable models solved by the quantum inverse scattering method and the quantum conformal field theory we consider systems with finite degrees of freedom such as a set of q-oscillators A covariant under the coaction 'P of the quantum (super-) group Fq • When the coaction 'P : A ---+ Fq @ A does not preserved the physical observables such as Hamiltonian, momenta, etc, the standard problem of the tensor product decomposition of 11.£0' (g/HA emerges, where 11.£0' and HA are the state spaces of the corresponding algebras. If instead of f~ we have the quantUIIl algebra Lie q ( G) then its representations are almost identical to the undeformed algebra (let us omit plenty of technicalities related with the case when q is a root of unity). However, the deformed algebra of functions F q is a new algebraic object with its own representation theory and the corresponding decomposition problem of 11.£0' !XI HA or 11.£0' ® 11.£0' is new as well.
2.
Covariant systems
2.1. A q as su q (2)-covariant algebra. Let us start with a simple covariant system. The q-oscillator algebra A q has three generators with commutation relations (for q real one has A = qN/2a = qNa )
(1 ) (2) (3) The third set can be obtained with fixed q from the quantum algebra contraction procedureS) (,\ = q _ q-l) 0= lirn)..1/2X+/qs,
S
---+ 00,
N
=s-
s1L q (2)
by a
J.
From this contraction one could find that although there is no natural coproduct for A q , the formulas for the su q (2) coproduct that survive under the contraction procedure could be interpreted as covariance of the algebra A q with respect to the
Quantum Group Covariant Systems
101
quantum algebra 8u q (2) (with generators X+, X _, J and well known commutation relations and coproduct .1). The corresponding map is 1/J : A q -> A q 181 8u q (2), such that
1/J(N) = N - J,
(4)
1/J(at)
= atq-J
+ /).q-N X_.
It is easy to check the following consistency properties for this coaction: (1/Jl8lid)o1/J = (id 181.1) o1/J and (id 181 c) 0 1/J = id as well as that 1/J preserves the defining relations of A q . However, the central element z of the algebra A q
(5) is not invariant under this coaction: 1/J( z) # z. If we choose the Hamiltonian of the q-oscillator as H = at 0' and restrict ourselves to the irreducible representation 'H F of A q with the vacuum state (for q E (0,1) there are other irreps5,8)) then z = and
°
(6) The spectrum of If and its eigenstates are obvious
8pee H
= {[n;q-2J,
n = 0,1,2,., .},
After the coaction the changed Hamiltonian describes an "interaction" of the qoscillator with the q-spin
It acts iu the space of physical states 'H ph = 'HF 181 Vj, where Vj is an irreducible finite dimensional representation of the su q (2) of dimension 2j + 1. This space is decomposed into the direct sum of 2j + 1 irreducible representations of A q : 'Hph = L\ 'H~) with corresponding vacuum states 10 >k, k = 0, 1, ... ,2j k
10 >k=
L
1m> I8Ilj - k + m;j > e(Tn, k) ,
m=O
where 1m >E 'HF , I/;j >E Vj , JI/;j >= lll;j >,
(8)
102
M. Chaichian and P.P. Kulish
m-l
c(m, k) = (_v'1qj-k+l)m([m;q-2]!rl/2
IT
N+(j - k + I, j) .
1=0
The spectrum of HI coincidcs with that of H up to the multiplicative factor q2(J-k) in cach subspace 1t~), but it has the multiplicity 2j + I
spec II -- { q2(j-k) [n,. q-2] ,n -- 0, I , 2,... }. The central element w(z) has 2j + I eigenvalues -[k _ jj q-2]. It is interesting to point out that this coaction has no classical (non-deformed) countcrpart in the limit q --> I in the quantum thcory. Such a limit exists in the Poisson-Lie theory (see e.g. [22]). The connection of the q-oscillator algebra A q with su q (2) through the contraction proccdure gives rise also to a more complicated coaction of thc quantum group SUq (2) on A q (scc Subsec. 2.5 and 2.6).
2.2. A q as SUq(l, I)-covariant algebra. Lct us consider the second set (2) of the q-oscillator algebra A q generators with relation
AAt = qAtA
+ 1.
redenoting q2 by q and putting q E (0, I). This relation reminds of the quantum plane with xy = qyx a central extcnsion. Using the two componcnt column Xl = (A, At) it can be rewrittcn in the R-matrix form 7.10)
(9) whcre if = P R is the R-matrix of su q (2) and J is thc four component column j! = (0, I, -q, 0) obviously relatcd to the well-known q-metric 2 x 2 matrix (q. This relation is prcserved under the transformation W: X ---> w(X) = T X, with I' the 2 x 2 matrix of the quantum group SU q(l, 1) gencrators
1'=(ab* a*b) which satisfies the FRT-relation fiT ® T = l' ® TR [1]. The invariance of the inhomogeneous term is just another form of the q-metric relation
T(qT t
= (qdeiqT,
T I8i T J
= 11l 2) = delqT J
,
(10)
provided that the quantum dcterminant of Tis 1: deiqT = aa* -qbb* = a*a-b*b/q = 1 (a defining condition for ,'iUq ( 1,1)). (One can consider central extension of the real
Quantum Group Covariant Systems
103
quantum plane as well with Iql = I covariant with respect to SL q (2, R). Then the reality condition will fix the phase of the constant term.) The map ljJ : A q --> A q 0 SUq(l, I) or in terms of the generators
(11 ) satisfies all properties of a coaction. Its form is reminicent of the famous Bogoliubov transformation. However, now the" coefficients" are non-commuting. The q-oscillator Hamiltonian acting in the same space 'H F as (6) /I
= AtA = [N; q] = (qN - I)j(q - I)
( 12)
(it differs from the previous one by qN factor and renotation of q2) is also not invariant under the coaction ljJ
(13) The commutation relations of the SUq(1, 1) generators (as well as those of SUq(2))
abO = qb*a,
ab = qba
[a, a*] = >"b*b,
bb* = bOb ,
>..=(q-ljq),
themselves rellli nd of the q-oscillator algebra (a "- at, a* "- a, b "- b* "- q-N) with the additional condition due to delq'l' = 1
aa* =
1+ qb*b,
a*a =
1+ b*bjq.
The deformation parameter q being less than 1 forces us to consider the irreducible representation of SUq(1, I) in the Hilbert space 12(Z) with basis 1m >,m = ... , - 2, -1,0, 1,2.... which consists of the eigenstates of commuting operators b, b* for which a acts as a creation (shift) operator
c~
=
1+ q-2m-l.
lienee the transformed Hamiltonian is defined in the space 'HF (9 12( Z). It has the same spectrum {[n; q], n = 0, 1, ... } with infinite multiplicity. The corresponding vacuum states are
10 >~~)=
f) -1)i xi I2j > 01k - j >,
i=O
(14)
104
M. Chaichian and P.P. Kulish
where the second vector in the tensor product belongs to the SUq(l, I) irrep space /2(Z). As in the case of the representation theory the invariant subspaces of the QG F (coaction) corepresentation V can be defined as W C V such that c/J : W -> F @ W . The invariant elements of the F-corepresentation V do not change at all: c/J( v) = v (or better to write IF ® v for one has the possibility in the corresponding representation of the dual Hopf algebra (F)* to contract a dual element X E (F)* with IF to get a number X(IF)). The extensions of the previous examples to higher rank quantum groups give rise to covariant algebras corresponding to different quantum homogeneous spaces 23 ), systems of (super) q-oscillators 4-6,15) and examples of non-commutative geometry.
2.3. The covariant super-q-oscillator algebra s-A q [4] refers to the quantum supergroup S'Uq(lII), with the T-matrix of the gencrators
and the commutation relations
a(i dfJ fJ, = -,fJ,
= qfJa, = qfJd,
a, d,
(i2 =,2 = 0,
= na, = nd, [a, d] = (q - l/qJifJ.
Fixing the central element (super-determinant) sdetqT = (a - fJh)/d = I one gets a simple relation betwccn the even generators d and a
d = a - fJ,/qd
=a-
fJ,/qa
due to the nil potency of the odd generators. The involution (*-operation) is introduced as follows: d = I/a*, , = dfJ*d = afJ*a . This involution leads to TTt = I and it is consistent with the Z2-grading. One has for the generators u, a*, fJ, (3*
u"fJ = (3a*/q,
[a,a*] = (I-l)fJ*fJ,
aa" = I
+ (3(3*,
fJ(3* = _q2fJ"fJ,
a"a = I - (3* (3 .
Introducing a' = a(1 + (i*fJ/2) = (I - (3fJ"/2)a, one gets a'a'" = a'"a' = I and the factorization of the T-matrix (a' = I/a'" )
Quantum Group Covariant Systems
T
= (a' o
105
0 ) ( (I -/3"/3/2) /3/qa' ) /3" /a'" (I + /3"/3/2)
I/a'"
with unit super-determinant. One concludes that the q-deformation (quantization) of the SU(III) super-group is realized by the unitary scaling operator A, A" = A-I acting on the Grassmann variablies "I and "I" , which are not quantized /3 = ATf ,
A/3
=
q/3A,
A/3"
=
q/3*A,
Al = I
and a' == cxp( i
AAt- q2 AtA=I, AB = qBA,
BBt+StS=I+(l-l)AtA, ABt = qBt A,
B 2 = Bt 2 = 0 .
Using the R-matrix formalism similar to (9) it is not difficult to show that these relations as well as the Hamiltonian H = AtA+BtB are invariant w.r.t. the coaction
However, one can consider Hamiltonians which are not invariant w.r.t. the QG transformation. The latter one will extend the initial system after the transformation. In particular, one can consider different versions of the q-dcformed N = 2-SUSYalgebra taken as the super-charges
where I and It are free fermions I = q-N B commuting with A , At [4]. In all the cases the coaction does not extend the space of states for the representation theory of SUq ( 111) is rather poor and only additional Grassmann parameters appear.
2.4 SlJq(n)- and SUq(mln)-covariant (super) algebras. Let I1S introduce 2n generating elements of the SUq(n)-covariant oscillator algebra A q ( n )5), writing them as n-component column and row
Their commutation relations in the R-matrix form (a spectral parameter independent Zamolodchikov - Faddeev algebrafl
M. Chaichian and 1'.1'. Kulish
106
demonstrate easily that, due to the FRT-relation [R,TI T 2 ] = 0 and TTt = 1 the coaction ¢(A) = l' A, ¢(At) = AtTt satisfies all the requirements. One can rewrite these relations in the form (14) using the 2n-component vector X = (AI, ... , An; At, ..., and the corresponding 2n x2n matrix R, which happens to be the R-matrix of the quantum group Spq(2n). Then the inhomogeneous term (2ncomponent vector J) is expressed using the invariant matrix G of Spq(2n): TIGT =
AD
GI,IO).
The invariant Hamiltonian w.r.t. the SUq(n)-coaction is
which in the Fock space can be written in terms of the mode number operators N k ,k = 1,2, ...
It was already pointed out that the SUq (l, 1) quantum group is related to the qoscillator algebra. The same is true for the SUq(2): its defining relations coincide with (3) up to notations and some factors. A realization of the SUq(n) requires n(n - 1)/2 q-oscillators and n - 1 phase factors I4 ). Hence, transforming the algebra A q(n) -+ S Uq(n) 09 A q(n) one jumps from the n degrees of freedom to the n( n + 1) /2 ones. The situation for the quantum super-group SUq(mln) is similar. The covariant super-algebra s-Aq(mln) has Tn boson and n fermion mutually non-commuting qoscillators 4 ) or vice versa. The realization of the SUq(rnln) requires m(m - 1)/2 + n( n - 1) /2 q-oscillators and rn x n Grassmann paramclers as well as some phase factors. 2.5 The next example of a covariant system is related to the reflection equation algebra K (or the q-Minkowski space-time algebra, or the quantum sphere algebra) (see e.g. [18]). Its quantum group covariance depends on the set of R-matrices in the defining equation (a reflection equation) R(2)/' R (I)/' 12 "I 12 "2
=
I ' 1",(3)/' R(4) "2 "12
"1
(15)
12'
with the coaction cp(]() = ](' = TKS where RU), j = 1, ... ,4 define the commutation relations of the I' and S entries I8). For the simple SUq (2) covariant case one has
R\1) = R\~) = R R\;) = R\~) = R matrix Rand l' = st = S-I. The algebra K 12 ,
with the slq(2)
21
has four generators:
Quantum Group Covariant Systems
0', 13, "
{i
107
with relations
0'/3 = q-2/30', [0, /3] = q-I >'0'13, = q2,O', [0,,] = _q-I>.,O', [,8,,] = q-I >'(0 - 0')0', [0',0] = 0,
a,
and two central elements
One has the covariance of JC with respect to the quantum group SUq (2) with the coaction <.p : JC ---+ SlJq (2) 0 JC which is easy to write using the matrix form
<.p(J<) where
J(
= f(' = UJ
and U = (Ut)-I are the following 2 x 2 matrices of generators
J<_(O' ,
(3) {i ,
a
qb) .
U = ( -b t at
Due to the fact that the q-determinant is equal to one
thc quantum group SlJq ('2) has cssentially one unitary irreduciblc rcpresentation H F 23 ) with vacuum state 10 >: ala >= 0, bla >= 1,
The algebra JC with the *-opcration J< = J
a
/(' = ( -bt
qb) (0' ,t) (at -b) at, 0 qbt a '
hence e.g. Let us consider the very simple (onc-dimensional) irreducible reprcscntation of thc algebra JC : 0' = 0 = 0, , E R. The factor HI is one-dimensional and H F has to be decomposcd into the irreducible representations of )C. To reach this aim one has to find eigenvalues>. and the corrcsponding eigenvectors I>. > of 0" = <.p(O') = qf(ba t + ab t ) in HF. Those of them
M. Chaichian and P.P. Kulish
108
related by IAn+1 >~ 'P(TlIA n >, A,,+I = q2 An give rise to the invariant subspace of H F w.r.t. Je'. The Hermitian operator (bat + ab t ) is a Jacobian matrix with the entries qn c" on the sub-diagonal. Hence the problem of the non-trivial deficiency indecies could take place I6 ). 2.6 A more complicated covariant system is related to the quantum super group
OSPq( 112)2). According to the general arguments of the Introduction, the coaction map gives rise to the extension of the dynamical system and to the representation of the covariantly transformed system in the tensor product with one of the factor being an irreducible representation of the corresponding QG. To find an irreducible unitary representation of the quantum super-group OSPq( 112) one has to introduce a *-operation and to analyse the commutation relations among the generators Tij , i, j = 1,2,3. The matrix T of the OSpq{l12) generators is even and has the dimension 3 in the fundamental representation and the grading (0, 1, 0). The compact form of the quadratic relations among the generators is given by the Z2-graded FRT-relation ( Z2-graded tensor product [2, 3])
ilTiSl1'=T®TR. The osp( 112)- ii-matrix
R has the spectral
R=
qPs - q-I p.,
decomposition [2]
-
q-2 PI ,
where the projector indices refer to the dimension 4s + 1 of the subspaces corresponding to spin 1, 1/2, (see their explicit expresions in [9]). Due to the structure of the R-matrix and the orthosymplectic condition Ts1Cq T = IC q [2], there are only three independent generators among nine entries of T. One can easily see this from the Gauss decompositionl 7) of the matrix l'
°
where TL , Tu are lower and upper triangular matrices with unities on their diagonal and the diagonal factor 1'0 = diag(A, B, C). Among the Gauss decomposition generators one finds three independent ones: A, (1Lb and (Tv h2' while the element B is centraP7). Introducing the nine elements of the Gauss decomposition: Tf) = diag(A,B,C),(liJ21,31,32 = (x,y,z) and (l'uh2,13,23 = (u,v,w) one finds from the FRT-relation 2l : A = U = T12 / qA,
Tn,
x = T2I/A,
v = u 2/w,
y = x 2/w,
w = _ql/2U,
z = X/ql/2,
B = 1'22 - 721 (Tn )-1 TI2 ,
where the elements B = AC = CA are central and w = ql/2 _ q-I/2. Due to the commutativity of 1'13 and T31 which are conjugated to each other 1':l1 = -1'/3 /q according to the *-operation from 2), these elements are diagonal in the
Quantum Group Covariant Systems
109
°
Zrgraded Fock representation rtF with the vacuum: T21 10 >= and the element 1'12 as a creation operator. From the structure of the quadratic relations among the generators Ti/l it follows that the four clements 1'\2, 1'32, 1'\3, 1'31 form a subalgebra of the OSPq( 112)
Hence, the irreducible representation in the Fock space rtF is given by 1'\2 as creation operator and 1':12 as annihilation operator, while
with 7'13 and 7 31 diagonal in the basis In >c::= (1'd n lO >. Let us now define the quantum OSp-plane, which is an associative super algebra A with three generators a, ~, b and the Z2-grading p(a) = p(b) = 0, p(~) = 1. Taking into account the similarity of the quantum super-group OSpq(112) to the symplectic group case [I, 7] the defining relations of A can be written in the R-matrix form with a central extension (16)
where
it is the ospq(112)
R-matrix, X = (a,~,b)t and the nine component vector
J
= (0, _q-I/2,0,0, 1,0,0,ql/2,0)
is the rewritten invariant matrix Cq 2). relations
One has for the generators the quadratic
( 17) where /1 = ql/2 + q-I/2 = >../w. The vector J is the eigenvector of the rank one projector PI which gives rise to the centrality of the element
This central element is invariant under the OSPq( 112) coaction: X ---t T X. The algebra A was identified in [9] as a twisted q-super-oscillator. Although A has the same irreps as (3) with b = at the coaction is more complicated with respect to (11) of the Subsec.2.2 for it includes the number or scaling operator ~ = TJqN as well.
110 3.
M. Chaichian and P.P. Kulish
Conclusion
The problem of the quantum group coaction interpretation and the corresponding tensor product decomposition are especially interesting in the framework of the Poincare group deformation ll - 13). The corresponding quantum group has many generators and rather complicated quadratic relations among them. Even in the very simple (trivial ?) case when the deformation of the Poincare group is given by the twisting 12 ) there are two Weyl generators defining the representation. The Hamiltonian of the relativistic system being only covariant under the group transformation law will get extra degrees of freedom after the quantum group coaction 13). Another kinematical group: the q-Galilei algebra G q , was connected with the X X Z-model dispersion relation due to the eqivalence of the trigonometric function addition law and a non-commutative coproduct 20 ). Realizing the generators of G q in terms of the local spin operators onc can obtain by the duality the quantum group coaction. Further interesting possibilities for the representation theory refer to the case when coproduct or coaction maps the original algebra into a tensor product with non-commutative factors (see e.g. [18,21]). Acknowledgments The authors thank P. Presnajder and M. Scheunert for useful discussions. PPK appreciate the influence and support of the :"Jon-perturbative quantum field theory workshop of the Australian National University and Laboratoire de Physique Theorique et Haute Energie associe au C.i\ .R.S. REFERENCES [I] L. D. Faddecv, ~. Yu. Reshetikhin and L. A. Takhtajan, Alg. Analiz 1 (1989) 178; (Leningrad Math. J. 1 (1990) 193). [2] P. P. Kulish and N. Yu. Reshetikhin, Lett. Math. Phys. 18 (1989) 143.
[3] M. Chaichian and 1'. P. Kulish, Phys. Lett. B233 (1990) 72. [4] M. Chaichian, P. P. Kulish and J. Lukierski, Phys. Lett. B262 (1991) 43.
[5] W. Pusz and S. L. Worollowicz, Rep. Math. Phys. 27 (1989) 231. [6] V. Rittenberg and M. Scheunert, J. Math. Phys. 33 (1992) 436. [7] P. P. Kulish, Phys. Lett. A161 (1991) .50. [8] P. P. Kulish, Theor. Math. Phys. 85 ( 1991) 157; ibidem 94 (1993) 193.
[9] F. Thuillier and J. C. Wallet, Phys. Lett. B323 (1994) 153. [10] P. P. Kulish, Proc. of Varenna School on Quantum Groups: Theory and Applications, to be published by Plenum Press, 1995.
Quantum Group Covariant Systems
III
[II] J. Lukierski, A. Nowicki, 11. Ruegg and V. Tolstoy, Phys. Lett. B264 (1991) 331. [12] M. Chaichian and A. Demichev, J. Math. Phys. 36 (1995) 39R. [13] J. A. de Azcarraga, P. P. Kulish and F. R6denas, Phys. Lett. B351 (1995) 123; hepth/9405161,Fortschr. Phys. (1995). [14] M. Arik, in: Symmetries in science VI, Plenum Press, 199:1, p. 47. [15] M. Chaichian, H. Grosse and P. Presnajder, J. Phys. A27 (1994) 2045. [16] I. M. Burban and A. C. Klimyk, Lett. Math. Phys. 29 (1993) 13. [17] 1::. V. Damaskinsky, P. P. Kulish and M. A. Sokolov, Zap. Nauch. Semin. POMI, 224 (1995) 155; £SI-95-217; q-alg/ 9505001. [18] P. P. Kulish and R. Sasaki, Prog. Theor. Phys. 89 (1993) 741. [19] P. P. Kulish, Alg. Analiz, 6 (1994) 195. [20] F. Boneehi, 1::. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Phys. A25 (1992) 1.939; Phys. Rev. B46 (1992) .5727. [21] T. 11. Koornwinder, in: Orthogonal polynomials: theory and practice, ed. P.1'\evai, :'\ATO AS Serie 1!J91, p. 257. [22] O. Babelon and D. Bernard, Commun. Math. Phys. 149 (1992) 279. [23] L. Vaksman and Va. Soibelman, Func. Analiz Pril. 22:3 (1988) 1; Va. Soibelman, Int. J. Mod. Phys. 7 Supp. 18 (1992) 859.
From Field Theory to Quantum Groups
THE RENORMALIZATION GROUP METHOD AND QUANTUM GROUPS: THE POSTMAN ALWAYS RINGS TWICE MIGl:EL
A. MARTIN-DELGADOt AND GERMAN SIERRA
t
t Departamento de Fisica Te6rica I Universidad Complutense. 28040-Madrid, Spain +lnstituto de Matermiticas y Ffsica Fundamental. C.S.I.C. Serrano 12.1, 28006-Madrid, Spain
Abstract: We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearance of a quantum group anomalous term which vanishes in the classical case. This is called the quantum group anomaly. We derive the new qRG equations for the XXZ model and show that the RG-f1ow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. In the ITF model the qRGflow equations coincide with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. 1.
Introduction
The Renormalization Group method has become one of the basic concepts in Physics, ranging from areas such as Quantum Field Theory and Statistical Mechanics to Condensed Matter Physics. The many interesting and relevant models encountered in these fields are usually not exactly solvable except for some privileged cases in one dimension. It is then when we resort to the RG method to retrieve the essential features of those systems in order to have a qualitative understanding of what the physics of the model is all about. This understanding is usually recast in the form
114
M.A.
~artin-Oelgado
and G. Sierra
of a RG-flow diagram were the different possible behaviours of the model leap to the eyes. Many authors in the past have contributed significantly to the idea of renormalization and it is out of the scope of this paper to give a detailed account on this issue here. We shall be dealing with the the version of the RG as introduced by Wilson [1] and Anderson [2] in their treatment of the Kondo problem, and subsequent developments of these ideas carried out by Orell et al. at the SLAC group [3] and 1'feuty et al. [1]. It was Wilson in the late sixties and early seventies who set up the framework of the method in its more thorough and complete version. According to his own words, he did so in his search for a better understanding of what a quantum field theory is. To this end he also introduced another tool, the Operator Product Expansion (OPE) for the field operators. Both the RG method and the OPE have become two cornerstones in modern Quantum Field Theory. It has been long known that the physics of (1 + 1)-dimensional quantum many-body systems and 21) statistical field theories were very special in many aspects when compared to their higher dimensional generalizations. It was after the seminal work by Belavin, Polyakov and Zamolodchikov (B1'Z) [5] when the special role of conformal symmetry in two dimensions was brought about in connection to those many special properties exhibited by 20 systems such as, criticality, integrability, etc ... The basic tool employed by BPZ to develop their conformal program was precisely one of the tools introduced by Wilson, the OPE, which with the help of two-dimensional conformal symmetry is powerful enough so as to classify the local fields forming the local algebra according to the irreducible representations of the Virasoro algebra and to determine the correlation functions of those fields. Among all the Conformal Field Theories studied by BPZ they singled out what they called Minimal Models as those for which the conformal program is more successful in determining their properties the best. The minimal models, also called Rational Conformal Field Theories (RCFT), are those CFT which contain finite number of primary fields in the operator algebra. For these models the anomalous dimensions of the operators, or equivalently the critical exponents, are known exactly and moreover, their correlations functions can be computed as solutions of special systems of linear differential equations. The underlying phenomenon responsible for such remarkable features is the truncation of the operator algebra, that is, the primary fields form a closed operator algebra. After the introduction of Quantum Groups by Orinfeld [6] in the mid-eighties, some CFT theorists realized the relationship between these new algebraic structures and those appearing in the RCFT. It was shown in reference [7] that the representation theory of SL(2, q) with q a root of unity provides solutions to the polynomial equations of RCFT, as well as a rather efficient way to compute the duality matrices. The rationality condition is met by requiring q to be a root of unity. In this case
The Renormalization Group Method
115
the representation theory of the quantum group is significantly different from the classical one. The first surprise is that when qN = I there are only N - 1 distinct finite dimensional irreps with spins j = 0, 1/2, I, ... ,( N - I )/2, this last one irrep being singular. This truncation of the representation theory of quantum groups was put in correspondence [7] with the previous truncation of the operator algebra found by BPZ in the minimal models. With this historical perspective in mind, what we have suggested by introducing what we call the Renormalization Quantum Group Method (qRG for short) [8] is to establish the connection of the truncation of states characteristic of the Real-Space RG methods (the other tool introduced by Wilson) with the special features of the (1+I)-dimensional systems exemplified by the RCFT. The connection we have found can be cast into the following squematical form, qRG-truncation ...... RCFT whose precise content will be explained in the following sections with several examples. Physicists working in field theory and condensed matter generalized the Real Space Renormalization Group methods introduced by Wilson [I] to other problems by using the Kadanoff's concept of block [3,4]. The Block method (BRG) has the advantage of being conceptually and technically simple, but it lacks of numerical accuracy or may even produce wrong results. For this reason the analytical BRG methods were largely abandoned in the 80's in favor of numerical methods such as the Quantum Monte Carlo approaches. In the last few years there has been new developments in the numerical RG methods motivated by a better understanding of the errors introduced by the splitting of the lattice into disconnected blocks. A first step was put forward in [9] where a combination of different periodic boundary conditions applied to every block lead to the correct energy levels of a simple tightbinding model. This method however has not been generalized to models describing interactions. Recently, we have clarified the role played by the boundary conditions in the real-space renormalization group method [10] by constructing a new analytical BRG-method which is able to give the exact ground state of the model and the correct 1/N 2 -law for the energy of the first excited state in the large N(size)-limit. A further step in the generalization to interacting models was undertaken by White in [II] where a Density Matrix algorithm (DMRG) is developed. The main idea is to take into account the connection of every block with the rest of the system when choosing the states which survive the truncation procedure. The standard prescription is to choose the lowest energy states of the block Hamiltonian. Instead, in the DMRG method one replaces the block Hamiltonian by a block density matrix and chooses the eigenstates of this matrix with the highest eigenvalues. The density matrix is constructed out of the ground state of a superblock which contains the desired block. In these notes we want to explain another RG method, the qRG method, which
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M.A. Martin-Delgado and G. Sierra
uses the concept of quantum groups. This mathematical notion emerged in the study of integrable systems and it has been applied to conformal field theory, invariants of knots and manifolds, etc. [6,12]. The new application of quantum groups that we envisage has been partially motivated by the aforementioned work of White, Noack and collaborators and it is probably related to it. This relation is suggested by the fact that quantum groups describe symmetries in the presence of non-trivial boundary conditions. The typical example to understand this property of quantum groups is given by the lD Heisenberg-Ising model with anisotropic parameter,1. The isotropic model ,1 = ± 1 is invariant under the rotation group SU(2), but as long as /,11 i- 1 this symmetry is broken down to the rotation group U(1) around the z-axis. One can "restore" this full rotation symmetry by adding appropriate boundary operators to the Hamiltonian of the open chain. The classical group SU(2) becomes then the quantum group SUq (2), where the quantum parameter is related to the anisotropy by ,1 = q+r' [J:I,14]. The "restoration" of a classical symmetry into a q-symmetry is achieved at the price of deforming the algebra and the corresponding addition rule of angular momentum. The q-sum rule, which is called the comultiplication, becomes non local and violates parity. The total raising (lowering) operators acting on the whole chain are a sum of the raising ( lowering) operators acting at every single site times a non-local term involving all the remaining sites which appear in an asymmctric way: sites located to the left or to the right at a given site contribute differently. These features of q-groups made them specially well-suited to implement a RG method which takes into account the correlation betwecn neighboring blocks. In the forthcoming sections we show how this can be done explicitly in two examples in 1D: the Heisenberg-Ising model and the Ising model in a transverse ficld (ITF). This paper is organized as follows. In Sect.2 we present a brief introduction to the Block Renormalization Group methods based upon the concept of the intertwiner operator T. This will allow us to see the truncation procedure inherent to the BRG method as tensoring representations of the Hamiltonian symmetry algebra, the intertwiner operator T being a Clebsch-Gordan operator. In Sect.3 we use the HeisenbcrgIsing model to show how the truncation process pertaining to the rcal-space RG is nothing but a tensor product decomposition of irreps of the symmetry algebra. The properties of the model are qualitative and quantitative well described by the BRG equations in the massive region ,1 > 1, while in the massless region one predicts the massless spectrum but not criticality at each value oj ,1 > 1. This latter fact is rather subtle and elusive. In Sect. 4 we set up the foundations of the qRG method using the Heisenberg-Ising model as an example. We derive the new qRG equations and show that the RG-f1ow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. Moreover, the qRG equations for the renormalized spin operators show the appearance of a novel feature: the quantum group anomaly. In Sect. 5 we apply the qRG method to another model, the Ising
The Renormalization Group Method
117
model in a transverse field (ITF model). Here, the qRG-How equations coincide with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. Seet.6 is devoted to conclusions and perspectives.
2.
A Brief Review of Block Renormalization Group Methods (BRG)
In this sectiou the block renormalization group method is revisited and we preseut a new and unified reformulation of it based on the idea of the intertwiner operator T to be discussed below. For a more extensive account on this method we refer to [15] and chapter 11 of reference [16] and references therein. The block RG-method is a real-space RG-method introduced and developed by the SLAC group [:1]. Let us recall that Wilson developed his numerical real-space renormalization group procedure to solve the Kondo problem [1]. It was clear from the beginning that one could not hope to achieve the accuracy Wilson obtained for the Kondo problem wheu dealiug with more complicated many-body quantum Hamiltonians such as Heisenberg, Hubbard, etc. ... The key difference is that in the Kondo model there exists a recursion relation for Hamiltonians at each step of the RG-elimination of degrees of freedom. The existence of such recursion relation facilitates enormously the work, but as it happens it is specific of impurity problems. From the numerical poiut of view, the Block Renormalization Group procedure proved to be uot fully reliable in the past particularly in comparison with other numerical approaches, such as the Quantum MonteCarlo method which were being developed at the same time. This was one of the reasons why the BRG methods remained undeveloped during the '80's until the beginning of the '90's when they are making a comeback as one of the most powerful numerical tools when dealing with zero temperature properties of many-body systems. Let us first summarize the maiu features of the real-space RG. The problem that one faces generically is that of diagonalizing a quantum lattice Hamiltonian H, i.e.,
HI1/J >= EI1/J >
(2.1 )
where 11/J > is a state in the Hilbert space H. If the lattice has N sites and there are k possible states per site then the dimension of H is simply (2.2) As a matter of illustratiou we cite the following examples: k = '1 (Hubbard model), k =:l (t-.) model), k = 2 (Heisenberg model) etc. When N is large enough the eigenvalue problem (2.1) is out of the capability of any human or computer means unless the model turns out to be integrable which only happens in some instances in d = 1. These facts open the door to a variety of approximate methods among which the RG-approach is one of the most relevant. The main idea of the RG-method is the
M.A. Martin-Delgado and G. Sierra
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mode elimination or thinning of the degrees of freedom followed by an iteration which reduces the number of variables step by step until a more manageable situation is reached. These intuitive ideas give rise to a well defined mathematical description of the RG-approach to the low lying spectrum of quantum lattice hamiltonians. To carry out the RG-program it will be useful to introduce the following objects:
• H : Hilbert space of the original problem.
• H': Hilbert space of the effective degrees of freedom. • H: Hamiltonian acting in H.
• ll': Hamiltonian acting in H' (effective Hamiltonian). • T : embedding operator: H' ---; H
• Tt :truncation operator: H ---; H' The problem now is to relate H, H' and T. The criterium to accomplish this task is that If and H' have in common their low lying spectrum. An exact implementation of this is given by the following equation: (2.3)
HT= I'll'
which imply that if IJi E, is an eigenstate of H' then TlJii, is an eigenstate of fJ with the same eigenvalue (unkss it belongs to the kernel of T: TlJii, = 0), indeed, IfTIJi~, = TH'IJi~., = E'TIJi~,
To avoid the possibility that TIJi ' = condition,
(2.4)
°with IJiI i- 0, we shall impose on I' the (2.5)
such that (2.6) Condition (2.5) thus stablishes a one to one relation between H' and Im(T) in H. Observe that Eq. (2.3) is nothing but the commutativity of the following diagram:
H' /1'
1
.2..
H
1
7-{'.2..H
1I
The Renormalization Group Method
119
Eqs. (2.3) and (2.5) characterize what may be called exact renormalization group method (ERG) in the sense that the whole spectrum of H' is mapped onto a part (usually the bottom part) of the spectrum of H. In practical cases though the exact solution of Eqs. (2.3) and (2.5) is not possible so that one has to resort to approximations (see later on). Considering Eqs. (2.3) and (2.5) we can set up the effective Hamiltonian II' as:
II' = Tt HI'
(2.7)
This equation does not imply that the eigenvectors of H' are mapped onto eigenvectors of II. Notice that Eq.(2.7) together with (2.5) does not imply Eq. (2.3). This happens because the converse of Eq.(2.5), namely TTt =1= 17-( is not true, since otherwise this equation together with (2.5) would imply that the Hilbert spaces 11. and 11.' are isomorphic while on the other hand the truncation inherent to the RG method assumes that dimH' < dimH. The fact that Ttl' =1= hi reflects nothing but the irreversibility of the RGtransformation. Indeed, we go from 11. to 11.' as prescribed by eq. (2.7) but we cannot reverse that equation. What Eq.(2.7) really implies is that the mean energy of H' for the states p' of 11.' coincides with the mean energy of II for those states of 11. obtained through the embedding T, namely,
< p'1H'1tfJ' >=< TP'lHITp' >
(2.8)
In other words Tp' is used as a variational state for the eigenstates of the Hamiltonian H. In particular T should be chosen in such a way that the states truncated in 11. , which go down to 11.', are the ones expected to contribute the most to the ground state of IJ. Thus Eq. (2.7) is the basis of the so called variational renormalization group method (VRG). As a matter of fact, the VRG method was the first one to be proposed. The ERG came afterwards as a perturbative extension of the former (see later on). More generally, any operator 0 acting in 11. can be "pushed down" or renormalized to a new operator 0' which acts in 11.' defined by the formula,
(2.9) ;\fotice that Eq.(2.7) is a particular case of this equation if choose 0 to be the Hamiltonian II. In so far we have not made use of the all important concept of the block, but a practical implementation of the VRG or ERG methods does require it. The central role played by this concept makes all the real-space RG-methods to be block methods. Once we have established the main features of the RG-program, there is quite freedom to implement specifically these fundamentals. We may classify this freedom
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M.A. Martin-Delgado and G. Sierra
in two aspects: • The choice of how to reduce the size of the lattice. • The choice of how many states to be retained in the truncation procedure. We shall address the first aspect now. There are mainly two procedures to reduce the size of the lattice: • by dividing the lattice into blocks with n s sites each. This method introduced by Kadanoff to treat spin lattice systems.
IS
the blocking
• by retrieving site by site of the lattice at each step of the RG-program. This is the procedure used by Wilson in his RG-treatment of the Kondo problem. This method is clearly more suitable when the lattice is one-dimensional. We shall be dealing with the Kadanoff block methods mainly because they are well suited to perform analytical computations and because they are conceptually easy to be extended to higher dimensions. On the contrary, the DMRG method introduced by White [11] works with the Wilsonian numerical RG-procedure what makes it intrinsically one-dimensional and difficult to be generalized to more dimensions. The first step of the BRG method consists in assembling the set of lattice points into disconnected blocks of n s sites each. In this fashion there are a total of N' = N Ins blocks in the whole chain. This partition of the lattice into blocks induces a decomposition of the Hamiltonian (2.1) into an interblock Hamiltonian H B and a interblock Hamiltonian H BB : (2.10)
where>' is a coupling constant which is already present in H or else it can be introduced as a parameter characterizing the interblock coupling and in this latter case one can set it to one at the end of the discussion. Observe that the block Hamiltonian H B is a sum of commuting Hamiltonians each acting on every block. The diagonalization of lis can thus be achieved for small n s either analytically or numerically. Eq. (2.10) suggests that we should search for solutions of the intertwiner equation (2.:3) in the form of a perturbative expansion in the interblock coupling constant parameter >., namely, (2.11 ) }/' = }/~
To zeroth order in
+ >.H; + >.2 H~ +...
>. Eq. (2.3) becomes
(2.12)
The Renormalization Group Method
121
(2.13) Since HB is a sum of disconnected block Hamiltonians defined through the relation
h;f l , j' =
1, ... , N' implicitly
N'
L h;f)
HB =
(2.14)
j'==1
one can search for a solution of To in a factorized form N'
(2.15)
To = IITo,j' j'=1
and an effective Hamiltonian
Hb
which acts only at the site j' of the new chain, N'
' -- '" H0 L..J his') j' - H'8'
(2.16)
j'=1
Observe that H;, is nothing but a site-Hamiltonian for the new chain. Eg. (2.13) becomes for each block: (B)
(s')
•
h j , 7 0 ,j' = To,j,h j ,
(2.17)
The diagonalization of h;~l for j' = I, ... , N' will allow us to write
h;~)
k ns -k'
k'
=
L Ii))'
{i
j,(il
+
i=1
where Ii >j' for j
=
L
la)j' (" j,(al
(2.18)
,,=1
1, ... , k' are the kl-Iowest energy states of h;~). Moreover, we
suppose that h;f l is the same Hamiltonian for each block so that {i does not depend on the block. The truncated Hamiltonian h;~) and the intertwiner operator l'o,j' are then given by: k' " I')' hj'(s') == 'LJ 1- j'
('I'
(i j' Z
(2.19)
Ii)], j,(il'
(2.20)
i=]
k'
TO,j' =
L i=]
Later on we shall show examples of these relations. The final outcome of this analysis is that the effective Hamiltonian HI has a similar structure to the one we started with, namely 11. The operators involved in Ii.:, and H~,s' may by all means differ from those of lis and lIss> but in some cases the only difference shows up as a change in the coupling constants. This is known as
122
M.A. Martin-Delgado and G. Sierra
the T'eno1'Tnalization of the bare coupling constants. When this is the case, one may easily iterate the HG-transformation and study the RG-flows.
3.
Block RG-Approach to the Heisenberg-Ising Model
To exemplify the standard BRG-method we shall study a Id-lattice Hamiltonian, the Heisenberg-Ising model. Prior to considering in detail the Antiferromagnetic (AF) Heisenberg model we shall make some general considerations concerning Hamiltonians which commute with a symmetry group g. Notice that for the AF Heisenberg model 9 is nothing but the rotation group SlJ(2). Let us call 9 an element of the group 9 and JrH(g) a representation of 9 acting on the Hilbert space 'H. We say that 9 is a symmetry group of the Hamiltonian H if
\lg E 9
(3.1 )
Similarly we want the effective Hamiltonian H' to be invariant under the action of 9 acting now on the Hilbert space 'H',
\lg E 9
(3.2)
For this to be the case, the RG-transformation must preserve the symmetry of the original hamiltonian. This call be simply achieved if one chooses T as the intertwiner or Clebchs-Gordan operator. Indeed we may recall that if a representation say fl v, is contained in the tensor product fl v, ® Il v2 then one can define the CG-operator as,
C>12 3
• .
(3.3)
which in fact satisfies the intertwiner condition:
(3.4) This equation expresses the commutativity of the following diagram,
p.5)
Thus from a theoretical point of view, the truncation process pertaining to the realspace HG is nothing but a tensor product decomposition of representations of the group g. To make this point more explicit let us suppose that 'H and 'H' are given as
tensor products as: (3.6)
The Renormalization Group Method
H' = 0~'V'
123
(3.7)
where V and V' are irreducible representation spaces of g. Then the block method of the previous section applied to this case is equivalent to the tensor product decomposition:
V:2I .n.,. :81 V -----+ V'
(3.8)
In Eq. (:l.8) one is establishing that the irrep fl v ' is contained in the tensor product of N copies of the irrep flv. Obviously, the tensor product decomposition usually contains different irreps. The criterion to choose a particular irrep, or a collection of irreps is the one of minimum energy. All the states of a given irrep V' will have the same energy. The summary of the discussion so far is that the intertwiner operator' To can be identified with the Clcbsch-Gordan operator,
(3.9) Let us illustrate this ideas with the AF Heisenberg-Ising model whose Hamiltonian is given by: N-l
/IN
=J
:L (5j8j+l + 5J 8J+l + L15i8i+1)
(3.10)
j=l
where L1 2': 0 is the anisotropic parameter and J > 0 for the antiferromagnetic case. I[ L1 = lone has the AF-Heisenberg model which was solved by Bethe in 1931. I[ L1 = 0 one has the XX-model which can be trivially solved using a Jordan- Wigner transformation which maps it onto a free fermion model. For the remaining values of L1 the model is also solvable by Bcthe ansatz and it is the 10 relative of the 20 statistical mechanical model known as the 6-vertex or XXZ-model. The region L1 > 'I is massive with a doubly degenerate ground state in the thermodynamic limit N -+ 00 characterized by the non-zero value of the staggered magnetization, mst =
(~ :L Si(-I)j).
(3.11 )
J
The region 0 S L1 S I is massless and the ground state is non-degenerate with a zero staggered magnetization. The phase transition between the two phases has an essential singularity. 'INc would like next to show which of these features arc captured by a real-space ltG-analysis. The rule of thumb for the ltG-approach to half-integer spin model or fermion model is to consider blocks with an odd number of sites. This allows in principle, although not necessarily, to obtain effective Hamiltonians with the same
M.A. Martin-Delgado and G. Sierra
124
form as the original ones. Choosing for (3.10) blocks of 3 sites we obtain the block Hamiltonian:
=
~ {['~1 + 52 + '~1]2 -
(51
+ '~3)2 -
3/4}
+ ((5: 5; + 5;5~),
(3.12)
t=:.d-l. If ( = 0 the block Hamiltonian liB is invariant under the 5U(2) group and according to the introduction to this section, we should consider the tensor product decomposition: (3.13)
The particular way of writing H B given in Eg. (3.12) suggests to compose first '~1 and '?3 and then, the resulting spin with .C;2. The result of this compositions is given as follows:
3 3
'2' 2) 3 1
=
I iTT), EB
= J/2,
1
12, 2) = y'3(1 Hi) + llii) + I TTL)), 1 1
'2' 2)1 1 1
12'2)0=
1 = y'2(1
(3.14) EB
Hi) -I liT)), EB
1 J6(2 Iili )-11H)-ITT1)),
= J/2,
(3.15)
= 0,
(3.16)
ER=-J.
(3.17)
lIence for ( = 0 we could choose the spin 1/2 irrep. with basis vectors I~, ~)o and ,~, -~)o in order to define the intertwiner operator To. However, if (-I- 0 the states (3.14) -(3.17) are not eigenstates of (3.12). The full rotation group is broken down to the rotation around the z-axis. The states 1~, &) and I~, ~)o are mixed in the new ground state which is given by:
(:3.18) where x = _ _2-.:(_.d_----,=1) == 8 + .d + 3v'.d 2 + 8
and its energy is
(3.19)
The Renormalization Group Method
J
E B = --[,1 1
+ J LV + 8]
125
(3.20)
along with its 1- ~) partner. This are now the two states retained in the RG method. To be more explicit, wc havc I
1+ -2)
1
=
1- -2) = -
I J6(1+2x
2
1 J6(1+2x
[(2x + 2)1 ili) + (2x - 1)1 iil) + (2x - 1)llij)],
(3.21 )
)
2)
[(2x + 2)lli1) + (2x -1)llLj) + (2x -1)1 ilL)]·
(3.22)
Thc intcrtwincr operator To reads then,
1'0 = I + ~)(i I' + 1- ~)(ll' 2
2
(3.23)
J
where I i)' and 1)' form a basis for the space V' = C2 • The RG-cquations for the spin operators Si (i = 1,3) are thcn givcn by
t
~
~x
(3.24)
ToSfTo=CS'i i= 1,3,
(3.25)
t S~zl' = T.0 10
(Z
c:,z . =
~~)ll
1, (3
(3.26)
where C, etc. are the renormalization factors which depend upon the anisotropy parameter by (X
<,
=
~
y = 2( 1 + x)(1 - 2x) 3( 1 + 2x 2 ) z _
2(1 +X)2 +2x 2 ) '
~ = 3(1
,
(3.27)
(3.28)
Observc thc symmetry bctwccn thc sitcs i = 1 and 3 which is a consequencc of the even parity of the states (3.21) -(3.22). The renormalized Hamiltonian can be easily obtained using Eqs.(3.21)-(3.28) and (3.10), and apart from and additive constant it has the same form as II, namely [17J, (3.29) wherc
M.A. Martin-Delgado and G. Sierra
126
(3.30)
(3.31 ) Iterating these equations we generate a family of Hamiltonians H}:;;~m(.I(m), ,1(m)). The energy density of the ground state of llN in the limit N ---> 00 is thcn given by lim Eo
N
-00
"
i'V
= e BRG = L.. ~ 00
_I_ e (J(m) ,1(m)) 3m +1 B ,
(3.:32)
m=O \
whcrc initially .l(O) = .I, ,1(0) = ,1 and Egs.(3.30) -(3.31) provide the flow of the coupling constants. The analysis of Eq.(3.:11) shows that there are 3 fixed points corresponding to the values ,1 = 0 (isotropic XX-model), ,1 = I (isotropic Heisenbcrg model) and ,1 = 00 (Ising model). The computation of C:;'RG in this case is facilitated by the fact that (3.32) becomes a geometric series at the fixed point. The exact results concerning thc models ,1 = 0 and ,1 = I are extracted from references [18] and [19]. The case with ,1 ---> 00 is exact because the states I ± given in (3.21) - (3.22) tend in that limit to the exact ground state I i 1i) and 11 i 1) of the Ising model. As a matter of fact,
k)
The region 0 < ,1 < I which flows under the RG-transformation to the XX-model is massless since both .1(m) and ,1(m) go to zero. We showed at the beginning of this section that all this region is critical (a line of fixed points) and therefore massless. The RG-equations (3.30) -(3.31) are not able to detect this criticality except at the point ,1 = o. Only the massless ness property is detected. The region ,1 > I which flows to the Ising model is massive and this follows frolIl the fact that thE' product .l(m),1("') gocs in the limit In ---> 00 to a constant quantity .1(=),1(00) which can be computed from Egs. (:UO) -(:l.:ll) and (:1.23) .l(=),1(oo) =
IT ~ (I + X )4 m
m=O
9 (I
+ 2x;,Y
(3.33)
where x'" is given by (:3.19) with ,1 replaced by ,1(m). This quantity gives essentially the mass gap above thc ground state and also the end-to-end or LRO order (Long Rangc Order) given by the expectation valuc [(S(1). k~(N))[ in the limit N ---> 00.
The Renormalization Group Method
127
In summary, the properties of the Heisenberg-Ising model are qualitatively and quantitatively well described in the massive region Ll > 1 while in the massless region o < Ll < lone predicts the massless spectrum but no aiticality at each value of Ll. This latter fact is rather subtle and elusive. One would like to construct a RGformalism such that the Hamiltonian HN(Ll) would be a fixed point Hamiltonian for every value of Ll in the range from -I to 1. Hence we postpone this discussion to next section. The phase transition between the two regimes is correctly predicted to happen at the value Ll = 1. This is a consequence of the rotational symmetry, namely at Ll = 1 the system is SU(2) invariant and the RG transformation has been defined as to preserve this symmetry. When Ll =1= I the SU(2) symmetry is broken and this is reflected later on in the RG-ftow of the coupling constant Ll. The region 0 < Ll < I corresponds to a central charge c = 1, namely, it is realize by a boson compactified in a circle which radius depends on Ll. We may wonder whether the criticality of the region ILlI ::; 1 is due to some non-trivial symmetry underlying the anisotropic Hamiltonian fl.
4.
Quantum Groups and the Block Renormalization Group Method for the Heisenberg-Ising Model
We present in this section a novel treatment of the Block Renormalization Group method for one dimensional quantum Hamiltonians based on the introduction of a quantum group. Our aim is to address the important questions left open in the previous section and to clarify the peculiar role played by the Renormalization group in the one dimensional physics [8]. Let us consider the following open spin chain Hamiltonian,
(4.1 )
where q is an arbitrary quantuIll parameter. This Hamiltonian is known to be integrable [13,20,23,24]. In an interesting paper Pasquier and Saleur [14] established the q-group invariance of (4.1) which served to get a better understanding of the interplay between q-groups and CFT at a discrete or lattice level. We have already talked about the relation between q-groups and CFT in the introduction when reference [7] was mentioned, but this concerned the models in the continuum. What we would like to show now is that the real-space RG method applied to (4.1) may be perhaps the way to link both the discrete and continuous approaches between the relation of q-groups and eFT. As a matter of fact, (4.1) is invariant under the following quantum group generators [14]: S+,8- and SZ
128
Yl.A. Martin-Delgado and G. Sierra
C'Z
"
=
1~ z 2" L.. (jj ,
(4.2)
J=I
N
S± =
L
q-t(u:+"'U;_l)(j;qt
(4.3)
j=1
which satisfies the quantum group algebra:
(4.4 ) (4.5) In the limit q - t lone recovers from Eqs.(4.2) -(4.5) the usual algebra and addition rules of su(2). For q i= 1 this algebra is the quantum universal enveloping algebra Uq (su(2)), or simply the quantum su(2) group denoted by SUq (2). The important property is that the generators (4.2)-(4.3) commute with the Hamiltonian (4.1):
(4.6)
If we set q + q-I
Ll = - 2 -
(4.7)
we observe that the bulk terms of Eq.(4.1) and the one in Eq.(:UO) coincide. The only difference appears in the boundary term of Eq.(4.1) which is essential for the existence of the quantum group symmetry. There are two important cases which we can consider:
q : real and positive
Iql =
1
=> Ll 2:: 1,
=> ILlI s:: 1.
(4.8)
(4.9)
If q is real then HN(q) is Hermitean while if q is a phase then HN(q) is not Hermitean but nevertheless its spectrum is real for H N( q) and HN( q-l) are related by a similarity transformation. This case is the most interesting one. In particular if we write q as a root of unity q = e;~l, then the Hamiltonian (4.1) is a critical Hamiltonian with a Virasoro central algebra given by q=
..
e;;+T
6
=>
c = 1 - j.l(j.l
+ 1)
(4.10)
However for the time being let us keep q as an arbitrary parameter characterizing the anisotropy of the model (4.7).
The Renormalization Group Method
129
Let us try to apply the Block Renorrnalization Group Method to the analysis of the Hamiltonian (4.1). To this end we have to write (4.1) in the following form: N-I HN(q) =
L
hj ,j+l(q,
J),
(4.11 )
j=l
Observe that each boundary term in hj,j+1 cancels one another leaving only those at the end of open chain (i.e. j = 1 and N). The nice feature of the site-site Hamiltonians (1.12) is that all of them commute independently with the q-group generators S± and 8 z ; [hj,j+I,S±]
= [hj,j+t,SZ] = 0
Vj
= 1, ... ,N -1.
(4.13)
Using hj,j+1 we can construct q-group invariant block Hamiltonians. If the block has for example 3 sites then we will have, (4.14) In the isotropic case (i.e. q = 1) we employ the Clebsch-Gordan decomposition (:1.13) in order to find the eigenstates of the block Hamiltonian H B (E = 0) given in (3.12). For quantuIll groups, we can also perform q-CG decompositions. The new feature is that now the q-CG coefficients depend on the value of q. For generic values of !J the analogue of Eqs.(:3.l4) -(3.17) are given by: (4.15)
eB
+
q q-I = .1--- ,
4
(1.16)
q
eB
+ q-I
= .1--4 •
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M.A. Martin-Delgado and G. Sierra
eB =
J
2 _ q _ q-I 4 '
where [:3]q denotes the q-number with the usual definition
[n]q =
q" - q-" 1 n EN. q - q-
(4.20)
The normalization of the states in Eqs. (4.15) - (4.19) is as if q were always a real number, a bra vector (I means transposing a ket vector. If q goes to 1 the vectors in Eqs. (4.15) - (4.19) go over to the vectors in Eqs. (3.11) - (3.17). On the other hand, for q i- 1 there is a certain similarity between the states It,±~)o and the states I ±~) given in Eqs. (3.21) - (3.22). However the main difference is that I~, ±t)o are not parity invariant (1 <--> 3), (2 <-> 2), while states (3.21) - (3.22) are. If we let q go to zero then Ll goes to +00. Let us recall that in this limit the states I ± ~) (3.21) - (3.22) go to the exact ground state of the Hamiltonian (3.12). However in the case of the states I~, ±~)o one does not recover the exact eigenstates in this limit. This shows that the states (1.15) - (1.19) are not appropriate for a discussion of the AF region Ll > 1 (or q real). lIence we shall confine ourselves to the critical region ILlI < 1 where q is a pure phase. As we did already for the isotropic case we shall truncate the basis (1.1.,)) - (1.19) to the states I~, ±t)o and therefore the intertwincr operator To is given by
To =
I~, ~)o(i I' + I~, -~)o(ll'
which satisfies the normalization condition
(4.21)
The Rellormalization Group Method
T~To
=1
131
(4.22)
where the superscript t stands for the transpose of the operator (instead of the adjoint ). This means that we can get the effective Hamiltonian H' through the formula:
H' = T~IITo. The RG-equations for the spin operators t....
...x
ToSiTo = ~S'
.5;
(4.23)
(i = 1,3) are then given by
= 1,3,
(4.24 )
t uV1' - . c:,v. 1, 3 , T,O"j 0 - ~" 1 =
(4.25)
i
(4.26) where ~ is the renormalization factor which depends upon the anisotropy parameter through the q-parameter in the following fashion,
~=
q
+ q-l + 2
2(q+q-l
+ 1)
(4.27)
and
""1=-r/3:=""= 4(q+q-l+l)'
(4.28)
Observe that there are quite a few remarkable differences between this "quantum" renormalization prescription Eqs. (4.24) - (4.27) with respect to the ordinary renormalization expressed in Eqs. (3.24) - (3.28). To begin with, the renormalization constant ~ is common to all the spin operators regardless of its spatial component. This is a reflection of the SUq (2) preservation of the RG method adopted. And last but not least, observe the presence in Eq.(4.26) of an extra term proportional to the identity operator. We may call this term a quantum group anomaly. Using these equations we can compute the renormalization of the block-block Hamiltonian hBB which turns out to be of the same form as the original site-site Hamiltonian (4.12) with the same value of q, namely (4.29) -1
- q -2q with
(O"~z - O"~z+l)
+ eBB( q, J)]
132
M.A. Martin-Delgado and G. Sierra
e
+ 3q-l + 4) . 32(q+q-l+1F
J) = J (q - q-l )2(3q
(
BBq,
(4.30)
Combining Eqs.(4.), (4.:30) and (4.15) - (4.19) we finally arrive at quantum group RG-equations (4.31) with (4.32)
q' = q,
(4.33)
(4.:34) IIcuce we obtain a quite remarkable result we were searching of, namely, that the coupling constaut Ll or alternatively q does not flow under the HG-transformation, while J(m) goes to zero in the limit when m ---+ 00, which in turn implies that theory is massless. The computation of the ground state energy of liN for any value of N is very simple since one has only to compute a geometrical series. The result is
Eo(N) = N
1 - (e/3)M 3_ (eB
e
+ eBB) -
1 -eM 1_ eBB
e
(4.35)
where AI is the number of RG-steps we have to make in order to resolve the chain of N = :3 M sites. As a check of the validity of this expression and because of its own interest as well, we shall consider the case q = e i1f / 3 in (4.35) which yields
E'0 ( N, q = ei1f/3) =
-83 N +"83
(4.36)
This expression coiucides with the exact result obtaiued through Bethe ansatz in reference [20]. Recall that the finite-size corrections to the free-energy for conformally invariant two-dimensional systems behaves as [21,22],
E = eN + C s
7rC
-
1
(4.37)
24 N
where e is the bulk energy per unit length and Cs is the surface energy (which vanishes for periodic boundary conditions). Since the term proportional to 1/ N is absent in (4.36) one observes that this value of q corresponds to a central extension c = 0 of the Virasoro algebra, in agreement with equation (4.10) (p 2).
=
The Renormalization Group Method
133
At first sight it looks surprising that an approximation method such as the Block Renormalization Group yields the cxact rcsult at least in thc case q = ei~/3. The peculiarity of this value of q, and in general when q is a root of unity, has been noticed in various contexts [12]: conformal field theory, quantum groups and in fact they are intimately related. The first thing to be noticed is that at q = ei~/3 the two denominators of the states I~, t) and It, t)l vanishes reflecting the fact that the "norm" of these states is zero, i.e., they are null states and therefore they must be dropped out in a consistent theory. It has been shown in [14] that because of (S±)3 = 0 the 6 states I~, rn) and It, mh do not fortn two irreps of dimensions 4 and 2 but rather a simple indecomposable but not irreducible representation of SUq (2). All these facts motivates that the tensor product decomposition (3.13) for the case q = ei1f / 3 should be really be taken as (4.38) where the irrep 1/2 on the right hand side denotes the one generated by It,m)o. This truncation is mathematically consistent and coincides precisely with the truncation we havc adopted in our Block Renormalization Group approach to the q-group invariant Hamiltonian (4.1). From a physical point of view, the truncation (4.38) means that the states I~, and It, th are not "good excited states" above the "local ground state" given by It, m)o. In other words, above the ground state there are not well behaved excited states. This is why thc central extension is c = 0 which means that the unique state in the theory is actually the ground state. What the BRG method does is to pick up that piece of the ground state which projects onto a given block! In the case of q = e i1f / 3 we have therefore construct for chains with N = 3 M sites the exact ground state of the model through the BRG method. It is worthwhile to point out that this derivation is independent of the Bethe ansatz construction and relics completely on the quantum group symmetry. Another interesting example is provided by q = e i r./4 which corresponds to the critical Ising model (c = 1/2). The RIIS for this q in eg. (4.38) contains two spin-l/2 irreps. According to thc qRG method the truncation of the spin-3/2 irrep shoulcl be a legitimate operation involving no approximation at all. In references [7] the representation thcory of q-groups was put in one-to-one correspondcnce with that of Rational Conformal Field Theories (RCFT). There it was observed that the truncation inhcrent in the construction of the RCFT's has a parallel in the truncation of thc representation theory of q-groups with q a root of unity. The result we have obtaincd in this letter suggests that q-group truncations can be carried over a RG analysis of q-group invariant chains. In other words, using q-groups we can safely truncate states in the block RG method. We may summarize this discussion squematically as we have mentioned in the introduction.
t)
134
5.
M.A. Martin-Delgado and G. Sierra
qRG Treatment of the ITF Model
This simple model has been widely used to test the validity of BRG methods [3,4]. The Hamiltonian of an open chain is given by II = L::7=1 1 hj •j + J where (5.1)
r
The standard choice is p = pi = 12, in which case (5.1) has 4 different eigenvalues. The BRG method with a block with two sites chooses just the 2 lowest ones. However if (p, p') = (r, 0) (or (O,!')) the Hamiltonian (5.1) has two doubly degenerate eigenvalues ±CB (ea = J.J2 + f2). This choice is not parity invariant but it implements the self-duality property of the ITF model, yielding the exact value of the critical fixed point of the lTF which appears at (1'IJ)c = 1 [25]. In the following we shall make the choice (p, pi) = (r, 0). This degeneracy of the spectrum of (5.1) has a q-group or·igin. The relevant quantum group is again SUq (2) with q4 = 1. However the representations involved are not a q-dcformation of the spin 112 irrep. as in the previous example, but rather a new class of irreps. which only exist when q is a root of unity. They are called cyclic irreps. and neither are highest weight nor lowest weight representations as the more familiar regular irreps. If we call E, F and I< the generators of SUq (2), which correspond essentially to S+, S- and q2S in the notation of the previous example, then a cyclic irrep. acting at a single site of the chain is given by: Z
Ej =
Fj
aCTi,
= bCT},
J()
= ACT;
(5.2)
where a = ~~, b = -~~. The parameter..\ is the label of the cyclic irrep. Strictly speaking, we have a particular kind of cyclic irreps. Indeed, the ones which allow the existence of an intertwiner for their tensor product. Using (5.2) and the addition rule of SUq (2) we can get the representation of E, F and I< acting on the whole chain:
E
a
LN
..\j-1 CT z .,. CTj_ICTj z x , 1
(5.3)
j=1 N
F
b
L ..\j-N CT·CT· Y )
z z )+1 ... CTN ,
(5.4)
j=1 N
J(
..\N
II CTJ.
(5.5)
j=1
Now it is a simple exercise to check that these operators commute with (5.1), [h).J+I, I~'] = [hj,)+J,
F]
= [hj,j+l, J(] = 0,
Vj,
(5.6)
The Renormalization Group Method
135
assuming that we choose (5.7)
A = 1'/.1 .
The last of the equalities in (5.6) expresses the well-known Z2-symmetry of the ITFmodel which allows one to split the spectrum of the Hamiltonian into an even and odd subsectors. The other two symmetries are new and explain the degeneracy of the spectrum of hj.j +l . By all means the whole Hamiltonian H = L:j hj.j+1 is also invariant under (5.3). l'\otice that Il differs from the standard ITF simply in a term at one of the ends of the chain. This is the same mechanism as for the XXZ Hamiltonian: one needs properly chosen operators at the boundary in order to achieve quantum group invariance. Similarly as for the XXZ model the RG-analysis of the ITF becomes a problem in representation of quantum groups: blocking is equivalent to tensoring representations. What is the tensor product of cyclic irreps.? Here it is important to realize that all cyclic irreps. of SUq (2) have dimension 2, what distinguishes them is the value of A. The tensor product decomposition of two cyclic irrep. Al and A2 of the type given in (5.2) is given by:
(5.8) where the 2 means that AIA2 appears twice in the tensor product. If we perform a blocking of two sites we will get two cyclic irreps. corresponding to A2 • Then we expect from If-group representation theory that the new effective Hamiltonian hj,j+1 will have the same form as (5.1) but with new renormalized coupling constants .1' and r' satisfying:
A' = T' .1'
= (!-)2 = .1
A2
(5.9)
This is indeed the result obtained in [25]. We arrive therefore at the conclusion that the HG-flow of the ITF Hamiltonian (5.1) is equivalent to the tensor product decomposition of cyclic irT'Cps of SUq (2). This q-group interpretation of the RG-flow is independent of the size of the blocks: for a n-site block the RG-flow would be A ----> An. The fixed point A = I of (5.9) describes the critical regime of the 11'10' Hamiltonian and it corresponds to a singular point in the manifold of cyclic irreps. [27]. At A = I the operators (5.3) are still symmetries of the Hamiltonian (a, b taking any non-zero value) and they recall the Jordan-Wigner map between Pauli matrices and Id-Iattice fermions. Of thc two cquivalent irreps A2 appearing in the tensor product A I8l A we only pick up onc of them. which is the lowest cncrgy. Observe that thc loss of information implied by the R(; method is in the ITF case somehow redundant information from the q-group point of view. The RG-flow in A is in some sense exact and not affected by the R(;-procedure.
136
M.A. Martin-Delgado and G. Sierra
The results we have obtained in the ITF model may perhaps be realized in IIlore complicated models. Namely, that the space of coupling constants or equivalently, the space of Hamiltonians of a given theory which is where the RG takes place, is the Spec manifold of an underlying quantum algebra, so that the RG-ftow is given by the tensor product decomposition of the algebra. For this to work we have to consider quantum groups with a rich and "exotic" representation theory, which is indeed the case when q is a root of unity. In these cases we know from references [26] and [27] that the Spec of these quantum groups is indeed very rich. We have used here the simplest situation. As we said above, it would be interesting to know whether there are more complicated realizations of these ideas. The chiral Potts model is a potential candidate for this realization due to its well known connection to S'Uq (2) with qN = 1.
6.
Conclusions
We have presented in this paper a brief description of the Renormalization Quantum Group Method (qRG) which is specially well-suited to treat 1D quantum lattice Hamiltonians. We have applied real-space RG methods to study two quantum group invariant Hamiltonians, namely, the Heisenberg-Ising model and the Ising model in a transverse field (ITF model). They are defined in an open chain with appropriate boundary terms. The defining feature of this qRG method is that the quantum group symmetry is preserved under the RG transformations except for the appearance of a quantum group anomalous term which vanishes in the classical case. We have called it the quantum group anomaly. As for the Heisenberg-Ising model, with the aide of the qRG equations we have shown that the q parameter describing the anisotropy coupling constant L1 does not flow under the RG-transformation when q is of modulus one, while the coupling constant J goes to zero implying that the theory is massless. In this fashion, the RG-ftow diagram obtained with the qRG method gives the correct line of critical points exhibited by the exact model. In the ITF IIlodel, we have shown that the qRG-flow coincides with the tensor product decomposition of cyclic irreps of SUq (2) with q4 = 1. Cyclic irreps. were used in [28] to derive the Boltzmann weights of the Zwchiral Potts model [29]. In [28] the labels of the cyclic irreps. have the meaning of rapidities rather than coupling constants as in our realization. The Z2 CP-model is nothing but the ITF model. We may wonder whether the general Zwchiral Potts model admits a q-group RG treatment along the lines of this work. This problem will be considered elsewhere. A model that admits a qRG analysis is the XY model with a magnetic field h (XYh). The results will be presented in [30]. It suffices to say here that the q-group underlying the model is SUq (2) with q4 = I and the representation used are the so called nilpotent irrcps. [12], which are also described by a parameter .\ analogue to that in (5.2) and related to the magnetic field h. The XYh model is equivalent to a free fermion with chemical potential.
The Renormalization Group Method
137
The results we are obtaining can be translated into a q-group symmetry between fermions, either free as in the XY or ITF models or interacting as the XXZ model. Another interesting model of interacting fermions is the Hubbard model, which has been studied using RG-methods in [31] . The integrability of the I D Hubbard model [32] suggests that it might be studied using our qRG techniques. All the Hamiltonians analysed in this letter are one dimensional, so the quantum groups are of the type that we know. Despite the fact that the Yang-Baxter equation (the precursor of q-groups) has a higher dimensional analogue called the Zamolodchikov or tetrahedron equation [33], the corresponding high dimensional analogue of quantum groups is not known. This fact represents a barrier to a qRG analysis of Hamiltonians defined in dimensions higher than one. Another possibility, which is suggested by our results, would be to define quantum groups as those which contain symmetries which are anomalous under RG transformations. This definition is independent of the space dimensionality. The quantum anomalous term in equation (4.26), and an analogous term also present in our qRG treatment of the ITF model, gives a discrete realization of this idea. A continuum analogue of this anomaly is given by the Feigin-Fuchs current, which has an anomalous operator product expansion with the energy-momentum tensor [34]. At this point it may be worth to recalling the continuous version of quantum groups in CFT of refcrence [35], which uses the Feigin-Fuchs or free field realization of the latter. Putting all these arguments together, we arrive at the conclusion that quantum groups are indeed defined by symmetries anomalous under RG transformations. This point of view about quantum groups may set up the pathway to new developments in the ficld. Finally, it is somcwhat amusing the way in which the word group enters the title of these notes. It refers both to the Renormalization Group method and to Quantum Groups, but neither of thcm are really groups! Acknowledgments Work partially supported 111 part by CICYT under contracts AEN93-0776 (M.A.M.-D.) and PB92-I092, European Community Grant ERBCHRXCT920069 (G.S.).
REFERENCES [I] K.R. Wilson, Rev. Mod. Phys. 47 (1975) 773. [2] P.W. Anderson, J. Phys. C3 (1970) 2436. [3] S.D. Drell, :vi. Weinstein, S. Yankielowicz, Phys. Rev. D 16 (1977) 1769. [4] R. Juillen, P. Pfeuty, J.N. Fields, S. Doniach, Phys. Rev. B 18 (1978) 3568. [5] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. 8241 ( 1984) 333.
M.A. Martin-Delgado and G. Sierra [6] V.G. Drinfeld, "Quantum Groups" in Proceedings of the 1986 International Congress of Mathematics, cd. A.M. Gleason (Am. Math. Soc., Berkeley); M. .limbo, Lett. Math. Phy••. 10 (19R5) 63, Lett. Math. Phys. 11 (1986) 247. [7] L. Alvarez-Gaume, C. Gomez, G. Sierra, Phys. Lett. 8220 (1989) 142; Nucl. Phys. 8330 (1990) 347. [8J M.A. Martin-Delgado and G. Sierra, UCM-CSIC preprint, 1995, to be published. [9] S.R. White, R.M. Noack, Phys. Rev. Lett. 68 (1992) 3487. [10] M.A. Martin-Delgado and G. Sierra, L"CM-CSIC preprint, 1995, to be published in
Phys. Lett. B. [11] S.R. White, Phys. Rev. Lett. 69 (1992) 286:l; Phys. Rev. B 48 (1993) 10345.
[12] C. Gomez, M. Ruiz-Altaba, G. Sierra, Quantum Groups in Two-Dimensional Physics, Cambridge University Press (to be published). [1:l] E.K. Sklyanin, J. Phys. A 21 (19RR) 2375. [14] V. Pasquier, H. Saleur, Nucl. Phys. B 330 (1990) 523. [15] P. Pfeuty, R . .IulIien and K.A. Penson, in Real-Space Renormalization, editors T.W. Burkhardt and .I.M ..I. van Leeuwen, Series Topics in Current Physics 30, SpringerVerlag 1982. [16] .I. Gonzalez, M.A. Martin-Delgado, G. Sierra, A.II. Vozmediano, Quantum Electron Liquids and Hight-To Superconductivity, Lecture Notes in Physics, Monographs vol. 38, Springer- Verlag 1995. [17] .I.M. Rabin, Phys. Rev. IJ 21 (1980) 2027. [18] E. Lieb, T. Schultz and D. Mattis Ann. Phys. 16 (1961) 407. [19] R. Orbach, Phys. Rev. 112 (19.58) 309. [20] F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R..I. Raxter, G.R.W. Quispel, J. Phys. A 20 (1987) 6397. [21] H.W. Hlote, .I.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742. [22] L Affleck, Phys. Rev. Lett. 56 (1986) 746. [23] M. Gaudin, Phys. Rev. A 4 (1971) 386. [24J LV. Cherednik, LV., Theor. Math. I'hys. 61 (1984) 977. [25] A. Fernandez-Pacheco, Phys. Rev. D 19 (1979) 3173. [26] D. Arnaudon, A. Chakrabarti, Comm. Math. Phys. 139 (1991) 461. [27] C. de Concini, V.G. Kac, P7'Og. Math. 92 (1990) 471.
The Rellormalization Group Method
139
[28] E. Date, M. .limbo, K. Y1iki, T. Miwa, Comm. Math. Phys. 137 (1991) 133; Publ. Res. Inst. Math. Sci. 27 (1991) 639. [29] 11. Au-Yang, H.M. McCoy, .I.II.H. Perk, S. Tan, M.L. Yan, Phys. Lett. A 123 (1987) 219; 11. Au-Yang, R..I. Baxter, .I.H.H. Perk, Phys. Lett. A128 (1988) 138.
[:30] M.A. Martin-Delgado, G. Sierra, paper in preparation. [31J .I.E. Hirsch, Phys. Rev. 822 (1980) 52.59; C. Dasgupta, P. Pfeuty, J. Phys. C 14 (1981) 717; C. Vanderzande, Phys. Lett. A18 (1985) 889;.1. Perez-Conde, P. Pfeuty, Phys. Rev. B 47 (1993) 8.56. [32] E. Lieb, F. Wu, Phys. Rev. Lett. 20 (1968) 1445; B.S. Shastry, Phys. Rev. Lett. 56 (1986) 2453. [33] A.B. Zamolodchikov, Comm. Math. Phys. 79 (1981) 489. [34] V.S. Dotsenko, V.A. Fateev, Nucl. Phys. B (1995). [35J C. Gomez, G. Sierra, Phys. Lett. 8240 (1990) 149; Nucl. Phys. 8352 (1991) 791.
From Field Theory to Quantum Groups
REMARKS ON THE QUASI-BIALGEBRA STRUCTURE IN QUANTUM MECHANICS ANATOL NOWICKI 1
Dipartimcnto di Fisica, Universita di Firenze and INFN - Firenze, Italy
Abstract: It is shown that the description of the time evolution for many-particle quantum mechanical states can be embedded into the quasi-bialgebra structure where the time parameter plays the role of a deformation parameter. Space rotations and translations are also briefly discussed using the notion of the noncoassociative comultiplicatioIl.
1.
Introduction
Recently, several applications of Hopf algebras to physics as quantum groups and deformed quantum algebras are investigated. The deformed space-time symmetry groups and algebras are extensively considered with a hope that these notions turn out to be suitable in the description of high energy phenomena. However, till now the fundamental problems related to a physical interpretation of the Hopf algebra structure are not solved, for instance the notion of the deformed momentum coproduct is not clear [1]. It is hard to find the applications of the Hopf algebra formalism even in the case of simple quantum mechanical systems. Perhaps, this strong mathematical structure can not be realized in quantum mechanics at all. In this note, we relax the notion of Hopf algebra to the notion of quasi-bialgebra introduced by Drinfeld [2], and we present some simple quantum mechanical examples which can be described using a quasi-bialgebra structure. In particular, we deal with the concept of noncoassociative comultiplication obtained by the twisting procedure
[a,4].
142 2.
A. Nowicki
Noncoassociative coproduct and quasi-bialgebra structure
Let us recall the basic notions related with noncoassociative comultiplication and quasi-bialgebras introduced by Drinfeld in [2]. We denote by
A
= (A, L1, E) an algebra with comultiplication L1 and counit
E.
Definition 1. The algebra A = (A, L1, E) is quasi-bialgebra if and only if there exist invertible clement 4> E A ® A ® A satisfying 3-cocycle condition: (104)) ((10 L1 ® 1)4» (4) 0 I) = ((10 1 ® L1)4» ((L10 1 (1)4» and invertible elements
f. 9 E
(I
Q9
(2.1)
A such that for all a E A the following relations hold:
L1)L1(a) = 4> ((L1 ® 1)L1(a)) 4>-1 ,
(2.2a)
(I ® E)L1(a)
jar l
,
(2.2b)
(E 0 1)L1(a)
gag-I,
(2.2c)
(IQ\:E01)c1> = j&;g-I.
(2.2d)
The relation (2.2) tells us that in the case c1> -I- id (c1> is called the Drinfeld associator) we deal with nonassociative coproduct L1(a). This fact is the fundamental difference between quasi-bialgebras and bialgebras for which the coproduct is coassociative.
It is well known, that for a given quasi-bialgebra by the twisting procedure we obtain a new quasi-bialgebra.
Definition 2. The quasi-bialgebras A = (A,L1,E,4» and A 1" = (A,L1 F ,E,4>F) are related by twisting if there exists an invertible clement FE A Q9 A such that (2.6) for all a E A. In this case the following relation holds
(2.7) Element F satisfying (2.6) is called twist junction. In fact, this general definition of a twist function F in the case of quasi-bialgebra gives us the similarity transformation of the coalgebra sector. If one compares this definition of the function F with that one for the Hopf algebras [3], it is seen that the counital and 2-cocycle conditions are relaxed.
Proposition 1. [2] If the algebra A F = (A, L1 1" , E, 4>1") is related to the quasi-bialgebra
Remarks on the Quasi-Bialgebra Structure in Quantum Mechanics
A
= (A, Ll,E, 1'/»
143
by the twist function F then A F is a quasi-bialgebra also.
Let us notice that if A is a bialgebra (i.e. with coassociative coproduct) then A F is not (in general) a bialgebra. The properties of the quasi-bialgebra A F depend on the explicit form of the twist function F. It is easy to prove the following statement:
Proposition 2. If A is a bialgebra and the twist function F is of the form F = J Q9 where 11(f) = J (<: J (i.e. group-like element) then the following relation holds:
J
(2.8) for all a E A.
In this case, the relation (2.8) allows us to replace the noncoassociativity condition (2.2) for the comultiplication I1 F by this one. It appears that we can build coassociative tensor product realization of the quasi-bialgebra A F using the relation (2.8). We shall see in the next Chapter that the conditions of the Proposition 2a. are satisfied in quantum mechanical description of the time evolution and space rotations for noninteracting physical systems. Let A = (A,I1,c:,P = id) be a bialgebra with trivial, non deformed coproduct Ll( a) = a Q9 I + I Q9 a then the following holds:
J Q9 9
Proposition 3. If a twist function F is of the form F =
11(/) =
J (X) J,
EA
(X)
A where:
l1(g) = 9 ® g,
(2.9)
then
FLl(a)p-l = (far l ) Q9/ + I Q9 (gag-I), l I I'/> = (X) Jgrlg- Q9 9
r
(2.10) (2.11 )
and the quasi-bialgebra structure is given by:
(I Cy I1 F )Ll F(a) = I'/>
((d,' ® I)LlF(a)) p- I ,
(I Q9c:)I1 F (a) F
(c: Q9I)I1 (a) (I
r;)
c: Q9 1)1'/> =
(2.12)
Jar l ,
(2.13)
gag-I,
(2.14)
J r;) g-I
.
(2.15)
We finish with our general considerations of the quasi-bialgebra structure the following:
144
A. Nowicki
Remark. If the twist function F is of the form F = f ® f where f is a group-like element then this twist generates an authomorphism (i.e. linear homomorphism) of the dual algebra. In the other cases we obtain a nonlinear homomorphism of the dual algebra.
3.
Quasi-bialgebra structures in quantum mechanics
We shall consider well known simple examples from quantum mechanics where the quasi-bialgebra structure is explicitly realized. Let us consider the standard undeformed Poincare algebra with the commutation relations in the form:
0(3) rotation sector (i,j,m = 1,2,3) [M k , Md = ifkl",M",
(3.1 )
- boost sector: (3.2)
.. four-momentum sector (Il, v
= 0, 1,2,3):
[P", P v ] = 0
[Mi, Pol
0,
[Mk, Pd
iCklm Pm,
(3.3)
with trivial, ulldeformed coproduct:
L1(M k ) = Mk®I L1(.c k ) L1(P,,)
+ I®M k
+ I ® .c k P" ® I + I ® P" .c k ® I
(3.4) (3.5)
(3.6)
Example 1. Time evolution of many-particle states. First, we llotice that the element f = U(t) = eitPo is group-like because of the trivial coproduct (:t6). Therefore, the twist function F(t) = f ® f = U(t) 09 U(t) satisfies the conditions of Proposition 2. and the formula:
L1 t (a) = FL1(a)F- 1 =
U(t)aU+(t) 09 I =
+ I ® U(t)aU+(t)
a(t)09I+I®a(t)
(3.7)
Remarks on the Quasi-Bialgebra Structure in Quantum Mechanics
145
defines the observable a in time t for the two-particle states. It is obvious that .:1 t (a) give us a noncoassociative cornultiplication i.e.
(I
C;;;
.:1 t ) .:1 t (a) =
!p
((.:1 t Q';/).:1 t (a))
(3.8)
!P-l
wit.h the Drinfeld associator of the form (2.11): !P
= U+(t) (9 1(9 U(t)
(3.9)
using the relation (2.8) we can define the observable a in time t for three-particle statc as follows:
which is the standard quantum-mechanical formula for three-particle observable a in time t. This procedure can be immediately generalized for many-particle statcs. Therefore, we see that the time evolution of many-particle states can be put in thc quasi-bialgebra schemc with time t treated as a deformation parameter. One can repeat this procedure for the K-Poincare algebra [5] because in this case the clement f = eitPo is also group-like one and the conditions of Proposition 2. are fulfilled.
Example 2. Three-dimensional rotations of coordinate system
In this case one can choose the twist function F in the form: (3.11 )
where ii describes O(3)-rotations and iiM = (}kMk. It follows from (3.4), that f(ii) is a group-like elemcnt i.e.
f( ii)
Q';
f( ii)
(3.12)
and we obtain the twisted coproduct:
.:1 F (o) (M k)
R k/(ii).1(Md,
(3.13)
.:1f"(O)(Pd
R kl(ii).:1(P1) ,
(3.14)
146
A. Nowicki
L1 F (a)(£k) = Rkl (ci)L1(£t} , L1 F (a)(po) = L1(Po)
(3.15)
(:1.16)
where Rkl(ci) is the three-dimensional rotation matrix. The noncoassociativity of L1F(5) is described by ep (2.11) in the form: (3.17)
Similarly, one can find the quasi-bialgebra structure for the boost transformations with (3.18)
In the case of K-Poincare algebra [5], Proposition 2 holds only for threedimensional rotations because they are not deformed. It does not hold for the boost transformations.
Example S. Translations in space and dual algebra
In this example we express the twist function F by the transformation of shift on a vector r : (3.19)
;-.Ioncoassociative twisted coproduct is given by :
L1 F (M) = L1 F (i) = L1 F (P) = L1 F (po) =
L1(M) + (r x P) 01 L1(i) + rL1(Po) , P 0 1+ 10 P , Po 0 +10 Po
+ 10 (r x
P), (3.20)
where (x denotes vector product). We obtain a quasi-bialgebra structure with the Drinfcld associator:
(3.21) On the other hand, the usual associator has the form:
(10 L1F (1 C9 L1 F (1 C9 L1 F
L1 F «) l)L1 F (M) = 1 Ii) 1 ® (i x P) - (i x P) d'li) l)L1 F (i) = r(1 @ 1 rg; Po - Po Ii) 1 ® l), L1 F (><) l)L1 F (P!'l = 0
on the linear basis of Poincare algebra.
0-)
1 C9 1, (3.22)
Remarks on the Quasi-Rialgebra Structure in Quantum Mechanics
4.
147
Final remarks
The aim of our contribution was to point out the fact that basic mathematical formalism describing timc evolution and spacc symmctry transformations of quantum systcms can bc expresscd by the notion of noncoassociative coproduct in quasibialgebra framework. Thc next stcp is to consider the examples of interacting quantum systems in terms of quasi-bialgebra. This problem is under investigation. Acknowledgments I would like to thank R. Giachetti. E. Sorace, M. Tarlini for interesting discussions on this subject and Dipartimcnto di Fisica and INF'\! in Fircnze for financial support. I would likc to express my gratitude to prof. J. Lukierski - who introduced me to the world of theoretical physics.
REFERENCES [I] J. Lukierski, ll. Ruegg, V.N. Tolstoy, "Quantum ,,-Poincare 1994" in Proceedings of xxx Karpacz Winter School of Theoretical Physics, ed. J. Lukierski, Z. Popowicz, J. Sobczyk, PWN, Warszawa 1995. [2] V.G. Drinfeld, Algebra Anal. 1 (1989) 114 (in Russian); V.G. Drinfeld, Leningr'ad Math. Joum. 1 (1990) 1419. [3J N. Rcshetikhin, Lett. Math. Phys. 20 (1990) 331; B. Bnriquez, Lett. Math. Phys. 25, (1992) Ill. [4] J. Lukierski, ll. Ruegg, V.!\'. Tolstoy and A. Nowicki, J. Phys., A27, (1994) 2389; J. Lukierski, A. Nowicki, V.N. Tolstoy, "Twisted Poincare Algebras" in Proceedings of xxx Karpacz Winter School of Theoretical Physics, ed. J. Lukierski, Z. Popowicz, J. Sobczyk, PW:'\, Warszawa 1995; A. Nowicki, "!\'oncoassociative Twisting; Its Application to ,,-Poincare Algebra" ill Pr'oceedings of the 4th International Colloquium "Quantum Groups and Integrable Systems", Praguf.:, 22-24 June, 1995. [5] J. Lukierski, A. :':owicki, ll. Ruegg and V.N. Tolstoy, Phys. Lett. B 264, (1991) 331; J. Lukierski, A. :':owicki and H. Ruegg, Phys. Lett. B 293, (1992) 344.
Chapter III. SUPERSYMMETRY
From Field Theory to Quantum Groups
LAGRANGIAN MODELS OF PARTICLES WITH SPIN: THE FIRST SEVENTY YEARS ANDRZEJ FRYDRYSZAK
Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Pi. M. Borna 9, Poland
Abstract: We briefly review models of relativistic particles with spin. Departing from the oldest attempts to describe the spin within the lagrangian framework we pass throught various non supersymmetric models. Then the component and superfield formulations of the spinning particle and superparticle models are reviewed. Our focus is mainly on the classical side of the problem, but some quantization questions are mentioned as well. 1.
Introduction
The aim of the present brief review is to indicate some essential aspects of the theory of relativistic point particle with spin. Selected models are presented mostly historically as they were appearing, to show the development of ideas in the period of 70 years - starting from the very begining. The first published work concerning the lagrangian description of the relativistic particle with spin was the paper by Frenkel [i] which appeared in 1926. In that time main considerations go towards the derivation of the equations of spin precession in the external electromagnetic field. Then to its relativistic generalization. The last one was achieved by Bargmann, Michael and Telegdi [3] 33 years after the Frenkel model was constructed. However only the work of Frenkel contains the Lagrangian defining the model of a particle, not only a considerations on the equations of motion level [4]. Then there is a forty years long gap in the activity in constructing such a models. However, in the meantime - in the fifties, the idea of anticommuting coordinates emerges in works of Martin [20], Matthews and Salam [21], and Tobocman [22]. Later it strongly influenced particle models [30,48]. The silence was broken with the work of 13arut [5]. Then in early seventies wider interest in the subject begins with the works of Hanson, Regge [6]; Grassberger [8];
Andrzej Frydryszak
152
Casalbuoni [14]; Rerezin, Marinov [18]. Some of these models involve anticommuting coordinates. Further growth of the interest was stimulated by dynamically developing research in the supersymmetry and supergravity and then superstrings theory with the wide use of Z2-graded structures. This period lasts from the eighties to the present decade with such a new models: Brink-Schwarz [28], Brink-diVecchia-Howe [26], de Azca,rraga-Lukierski [24], Siegel [35,36], Volkov-Soroka-Tkach [71]. Above metioned models fall in principe into different categories. The classification can be made due to the such attributes as mass, algebraical (conventional or anti-commuting) and geometrical character of the internal degrees of freeedom (vectorial, spinorial, twistorial). In the sequel we shall adopt the following naming conventions. Models involving only conventional coordinates will be called the classical models. The models involving anti-commuting coordinates are generally called here pseudoclassical. These with the anti-commuting vectorial degrees of freedom are called the spinning particles and these with the spinorial anti-commuting degrees of freedom are called superparticles. The type of extension of the configuration space of the relativistic particle by the commuting or anti-commuting coordinates determines the symmetry and the behaviour of the model in the external field and upon quantization. Starting demand is to have object which is at least Poincare invariant. Classical vectorial particles and spinnig particles couple properly to the external fields, however only the latter ones can be correctly quantized and do not give undesirable classical selfacceleration and effect of Zitterb('wegung type. Spining particles are in some sense the classical limit of the Dirac particle. After the first quantization these new anti-commuting variables are mapped into the Dirac matrices and they disappear from the theory. This is a general feature of the spinning particle models. On the other hand the extension of the configuration space by the anti-commuting spinorial variables yields the models which are super-Poincare invariant. In contrast to the spinning particle models their first quantization gives theory which still involves the anti-commuting variables. As a result of quantization we get rather not a single quantum particle with spin but a (minimal) supermultiplet. The organization of the review follows the models classification sketched above. We begin with the two principal categories of the classical models and the so called arbit.rary spin particles. Then the pseudoclassical group of models is presented including spinning particles, snper-particles, twistorial and harmonic particles, arbitrary superspin models. f\ext we comment the double supersymmetric models with the spinning superparticle in the component and superficld form. We conclude this brief review recapitulating some new developments including first attempts of q-deformation the relativistic model of the spinnig particle and the K-relativistic model of a particle (exceptionally without spin). 'I'll(' literatur(' on the particle with spin is vast. We include here, only the very selective list of references. Among them are some of the works of Professor Lukierski,
t
LAGRA~GIAN
MODELS OF PARTICLES
153
who has been very active in this area. He has introduced (together with J.A de AZCiirraga) one of the nontrivial models of the superparticle. I am very happy to dedicate this brief review to .Jurek on the occasion of his sixtieth birthday.
2.
Classical models
In this section I shall briefly present the classical models of the relativistic particle with spin. The adjective classical means here not only that a model is not quantum but also that it is described by means of the commuting variables only.
2.1.
Vectorial models
Historically the first model has been introduced by Frenkel. The spin of the particle in his model is described directly by a tensor of spin S"v, which is assumed to be proportional to the tensor of internal magnetic moments M"v' It enters the lagrangian via the "transversality condition"
(I) to reduce the number of independent degrees of freedom. We shall call it the Frenkel condition. Explicit form of the action is as follows
(2) It yields the equations of the motion of the form
S"v - (x"Svp - xvS"p)a P (AX" + S"paP)'
° °
(3) (4)
Some developments of this model were done 33 years later by Barut. To describe internal degrees of freedom he introduces the frame of four fourvectors q(;) i = 0, 1,2,3; such that q(O) is proportional to (x") and the rest is orthogonal to (x"). Csing implicit form of the action (5) s = drL(x",q(;),q(;))
J
and the following definition of the tensor of spin
(6) he gets the following form of the equations of motion
o o
(7) (8)
154
Andrzej Frydryszak
Moreover the Fr<'nkel condition is valid. Historically the next classical model was of different kind. It was an exemplification of the idea that particle is an irreducible representation of the Poincare group. Namely in the model of Hanson and Regge the configuration space has coordinates (x,,, A,w), where A E Ll (Ll - orthochronous Lorentz group). It turns out that dependence of the lagrangian function on the x, A, Ii should be restricted to L(x",O""V), where o",'v = ,1.\" Ii~. Now the tensor of spin is given by the formula
(9) and the equations of motion take the same form as in the Barut model. The demand, that in the non-relativistic limit the particle has only three spin degrees of freedom is realized by the condition
(10) It was originally introduced by Dixon [2.5]. In this model it should be included into the action. Let us note that the Frenkel condition and Dixon condition yield the essential differences in possible motions of particles, even in the free case. Explicit realization of such an action takes the form
(11 ) (12)
o
( 13)
o
(11 )
The last constraint follows from the reparametrization invariance of the action. The mass of the particle is renormalized here by the square of the S"v. This model does not give after the first quantization the Dirac particle. Along similar lines is constructed the BMSS model proposed in Ref. [10]. Here again the particle with spin is directly tied up to the irreducible representations of the Poincare group To this end, as a configuration space one takes (zl', A) E The matrix is decomposed into the momentum and spin tensor, where
pl.
pl.
mAllo,
s
mO
(15)
i,XAO"12 A - 1 .,X E ~
(16)
-i8~~
(17)
This means that
(18)
LAGRANGIAN
~ODELS
155
OF PARTICLES
and by the construction (19)
(20) Therefore the model written on such a space has to be of Dixon category. lagrangian finally defining the model is taken in the following form
L = p"i"
i>..
.
+ "2Tr(a12;1-1;1)
The
(21)
The resulting equations of motion are the same as in the Hanson-Regge model. The four-momenta and four-velocities are related in the standard way. This type of model has been recently reformulated and a correspondence to the psudoclassical model was proposed [11]. ]\'ow let us corne back to the vectorial models. In 1978 Grassberger proposed the description [8] in which the Minkowski space is extended by the two four-vectorial internal degrees of freedom (x,,) >--+ (x",a",b,,). The Poincare invariant action is defined by means of the lagrangian I .2
x ) + x,,((Jb" -
L = 2n!(1 -
au")
I .
+ 2(b"a" -
CL"b")
(22)
The Lagrange multipliers m, a, {3 are introduced to provide the necessary constraints
x·2
I
a"x" b"x."
0 0
(23) (24) (25)
The tensor of spin obtained from the above lagrangian is composed of the new vectorial internal co-ordinates i.e.
(26) and it obeys the Frenkel condition with S"vS"v = const. Now there are some new features in the equations of motion of this model, namely
d
.
-(mx" + 7]")
dT
.S'"v + P[vXv]
o
(27) (28)
This means that if such a particle is coupled to the external electromagnetic field, or has only passed through the bounded area with non-vanishing field, due to the presence of Tlv term the center of mass and the center of the charge need not coincide. Five years later Cognola, Soldati, Vanzo and Zerbini [9] proposed another vectorial
156
Andrzej Frydryszak
model. Its configuration space is the same as for the Grassberger model, but the new lagrangian takes the form
I - -m 2(pJ.l.vX." X.v)l2 j
-
-
a. v bV
(29)
with S'!'v given by eq.(26) and (30)
The constraints are now of the form
(31)
o
(32)
o
(33)
what obviously means that the Dixon condition is fulfilled. The internal degrees of freedom are here othogonal to the momenta, moreover the direction of fourvelocity and fourmomenta can be different. Equations of motion take the standard form. Another branch of the models of the classical spinnig particle is connected with the Souriau's notion of the space of motions [56J and the coadjoint orbit method. As a sample we indicate here the Refs. [56,58,59J and recently [57]. The classical models sketched in this section have one property in common, they do not give after the first quantization the accepted quantum relativistic Dirac particle. On the other side, coupled to the external electromagnetic field, in the limit of the weak homogenous field, they yield the Hargmann-Michel-Tclegdi equations. The models fulfilling the Frenkel condition have helisoidal curves as the solutions of the equations of motion. This can be interpreted as a counterpart of the Zitterbewegung solution for the Dirac particle, however from the classical point of view such a trajectory for a free particle can be hardly accepted. Despite the technical subtleties of different models of this kind the behaviour of the particular type of the particle with spin depends mainly on the type of the "othogonality condition" for the internal degrees of freedom i.e. the Frenkel or the Dixon condition. The latter one seems to be more natural. 2.2.
Spinorial models
Finally let us comment classical models of the particle with spin described by the spinorial coordinates. The presence of the commuting spinoI' not necessarily means that model describes the particle with spin [61,62]. In the twistor-like approach the massless point particle for example, has the action of the form (34)
LAGRANGIAN MODELS OF PARTICLES
157
where Pm is the particle momentum and A" is a commuting spinorial variable, needed to ensure the mass shell condition (another interesting spinorial model has been discussed in Refs.[68] and [69] with the action of the form S = J dT),'mA:i;m). The bove action is a good starting point to supersymmetric generalizations. The model of a point particle with spin described by commuting spinors as dynamical variable has been proposed originally by Barut and Duru [31, :32]. The free part of the lagrangian is given in the form
(35) A is a Dirac spinor, p is a constant. An interesting property -of this model is that it exhibits the classical analog of the Zitterbewegung, which can be seen directly from the solutions of the equations of motion [31]
+ ~",PI' sin TnT)A(O) m
A(T)
(cos mT
).(T)
A(O)( cos mT - -,I'P" sin mT)
-
(36)
z
m
where
i,I'PI'A
(38)
-i)"l'pl'
(39) (40)
A,I' A
XI'
The deep analysis of this model can be found in [:31,32] and some recent comments in [33]. In thE' last decade there were considered other models [63-67]. As an ilustration let us consider two of them. The lagrangian of the first model [63J is closely related to the spinning supcrparicle model [77], and is given in the form
L
= ~(C-li:2 + em~) -
h:i;· j
+ 2iji"
(41 )
where h, rna E ~+ and T! is the Majorana spinor. The current ja = ij,aT! has vanishing square. The resulting equation of motion have the form Ii
hj)
0
(42)
2
0
h:i;,T! -2-;'
0
(43) (44)
-(e-1:i; -
dt
:i;2
+ rnGc
Obviously the conserved angular momentum tensor has a contribution from the "internal" degrees of freedom
(45)
158
Andrzej Frydryszak
where Pa is defined by expression in the first equation of motion given above. The Dirac quantization of this particle, after solving the second class constraints (and hence with the breaking of the Lorentz covariance in the spinorial sector of the phase space) gives the condition on states which singles out arbitrary spin and relates the mass and the spin mJ± =
J+- 1- + ±h 2
J
h 2 (J
1 + mo + -)2 2
(46)
The limit for the massless case can be considered as well and gives the description of particles with arbitrary helicity. The second example, the arbitrary spin particle model comes [64] from the geometrical construction of the model on the six dimensional product space of the Minkowski space M and the two-dimensional sphere 8 2 • The family of Lagrangians of the model involves the joint interval in M and 8 2 , with the metric on 8 2 depending explicitly on fourvelocities. It is parametrized by the mass and spin (m, s) and has the following form
(47) where;; is a complex coordinate on the 8 2 , (z") = (1, z) and ~a = (O"a)""z" z", eJ, 2 e2 are einbein fields associated with the reparametrizations in M and 8 . The.d is an additional ("spherical") mass .d = hmcJs(s momentum tensor has the form AA IVl
ab = J:"J!b - XbP"
+
(
+ I)
(s - spin). The conserved
( ) ,,/3;; Ii J!z ) z-0 (-O"ab ) "iJ Z. iJ pz - Z" O"ab
(48)
with the "internal" part defined by complex spherical coordinates. The Dirac quantization of this model can be performed in the covariant way and gives irreducible representations for arbitrary choice of spin.! Let liS note that analogously to the Barut-Duru particle this model exhibits the Zitterbewegung effect as well. Let us recall that this phenomenon is typical for the vectorial models.
Pseudo classical models
3.
In this section we pass to the models with the internal degrees of freedom"described by the anti-commuting co-ordinates. The origins of the pseudomechanics should be dated back to the 1956, to the work of Martin [20]. However, anti-commuting variables appear firstly in the context of the functional integral for fermions in the works of Matthews, Salam [21] and Tobocman [22]. The extension of a configuration space to superspace enlarges the underlying symmetry group. Depending on the type of I
The author thanks S. L. Lyakhovich for comments to this paragraph
159
LAGRANGIAN Y10DELS OF PARTICLES
the model the extension of the Poincare algebra yields the super-Poincare algebra or some super algebra of the other kind. There are two types of such a models: vectorial and spinorial. In models of vectorial type (the spinning particle models) the extension gives a untypical vectorial superalgebra, with the odd generators having vectorial index. Characteristic feature of such models is conventional character of the first quantized theory. Namely, the odd variables upon quantization are mapped into Dirac matrices and disappear on the quantum level. Such particle can be considered as a pseudoclassical limit of the conventional Dirac quantum particle. In fact, this was the non-achieved goal of the vectorial classical models presented in the previous section. The spinorial models (the superparticle models) have the super-Poincare algebra (or its extension) as a symmetry generators. This models are connected more closely to the relativistic supersymmetry and the superparticles can be viewed upon, as a minima!. irreducible representations of the super-Poiucarc group. However, such an objects contains the whole multiplet of fields with different spin but not only the spin one half component. On the first quantized level one still deals with the anti-commuting variables and instead of the wave functions the the wave super-functions have to be considered. The Dirac equation is not used literary but finds its superspace counterpart. It is worth noting that there exists an equivalence betveen some pseudoclassical and classical models of particles with spin which allows to generalize the notion of Zitterbewegung to the pseudoclassical case [16] (d. as well Ref. [17]). 3.1.
Spinning particles
Twenty years after the anti-commuting variables were introduced into the physical literature for the first time, there was proposed the spinning particle model by Berezin, Marinov [is] and Barducci, Casalbuoni, Lusanna [19]. The configuration space for this model is described by the set of co-ordinates (xl" 01" Os), where the O-variables are anticommuting between themselves; 01' beeing fourvector and Os a scalar. Proposed lagrangians were of the form 1·
I·
-m
1·
1
.
(X" - ;;:OIJ.Os)(x" - ;;:0,,(5) - 2°1'0" - ;,/50S
-rrl\/-x
2
+~
(0,,01'
+ 0505 -
(J~x201J. + Os)'\)
(49)
(50)
Let us focus on the model given by the first of above lagrangians. It is invariant under the supertranslations X JL
0" Os
, !---4' X J1.
0'" O's
xI' -
0" Os
{IJ.A0 5 + {sHO"
+ t" + (s
(51 ) (52)
(53)
160
Andrzej Frydryszak
where A, B arc numerical constants. The algebra of generators of these transformations is defined by the following relations
{Q", Qv}
(54)
{Qs, Qs}
b
(55)
{Q", Qs}
(H-C)P"
(56)
The Q", Q., commutes with P". Performing canonical analysis of the model one gets the first class constraints
P2 -m 2
o
(57)
p"O" - mOs
o
(.'i8)
After the first quantization one obtains precisely the Klein-Gordon and Dirac equations. For the O-sector of the phase space the anti commutation relations
[OI',Ovt
-hgl'v
(59)
[Os, Os] +
n
(60)
Ost
0
(61 )
[0
1"
show that the classical variables originating from the Grassmann algebra arc mapped after quantization to the elements of the appropriate Clifford algebra, here 01, I-> /'f"'lI''''IS, OS I-> /'f"'lS (where II" "'Is - Dirac matrices). Above model can be generalized taking into account the repararnetrization invariance, which yields the supergravity in d= 1 [19] (cf. also J. van Holten's contribution to this volume). For the relativistic point particle one can explicitly achieve time reparametrization invariance by means of einbein field e( T). In the case of the spinning particle it is necessary to introduce its supersymmetric partner 1jJ(T). Resulting lagrangian takes the form
L = e- 1 :i: 2
+ em 2 + i(Oj)1' + OsOs) + i(mOs -
e-1:i:1'01')1jJ
(62)
The action given by this lagrangian is invariant under Poincare transfo\,mations, reparametrizations and the local supersymmetry transformations, what justifies the associacion with the J) = 1 supergravity. Namely,
/jT
-O(T)
(63)
/jxl'
0:( T):i:1'
+ it( T)01' a(T)OI' + (2e)-lf(T)(2:i:1' -
(64)
/jOI' 60 s
6t {)1./J
a(T)Os - mC(T) d T(O:(T)e + i((T)V') (T d dT (O(T)ljJ + 2t(T))
i1jJOI')
(65) (66) (67) (68)
LAGRANGIAN MODELS OF PARTICLES
161
The Euler- Lagrange equations for the e and 'IjJ are of algebraic character and this fields can be easily eliminated what yields the other version of the lagrangian which was considered in [29, :30]. The most general form of the action for the spinning particle with the supergravity multiplet can be given by the lagrangian of the form [39]
L
ige- I ;j;2 - ibcm 2 + goj)1' + &0 505 + mb05'IjJ (ge- I + 2e- 3;j;2 g')'IjJ(;j;I' BI') + 2e- 2g'(;j;I' BI')2
(69) (70)
All the spinning particle models have the property that they are the classical limits of the Dirac field theory and the anticommuting variables are present only in the classical description. The coupling of such models to the external electromagnetic or Yang-Mills fields yields vectorial superspace versions of the Bargmann-Michel-Telegdi or Wong equations [19]. Therefore in some sense the spinning particle models are improved versions of the conventional vectorial models, now with the proper quantum picture. Let us finish this section with the superfield formulation of the spinning particle proposed by Ikernori [49,50]. The first step consists in considering instead of single conventional time parameter a generalized super-time as a (111) dimensional superspace with coordinates (t, 1/), where 1/ is the new anticommuting variable. This means that trajectories of a system will take values in the superspace too. Kamely,
X(t,7]) = x(t)
+ i7]O(t),
X E C OO (t)[7]]o
(71)
Above superfield unifies in one object the even and odd coordinates of the spinning particle. The supersymmetry present in the super-time space is called the little SUSY: (t, 7]) - - t (t + T, aT/, 1/ + Cl'), where T, is an even and Cl' an odd infinitesimal parameter. The algebra of supercharges and covariant derivatives is of the form
017 D = 017 [Q, Q]+
-
7]Ot,
= 20t
o
=
07]'
(72)
(73) (74)
[D, D]+ = -20t
(75)
[Q, IJ]+ = 0
(76)
To introduce the local invariance the d = 1 supergravity multiplet is needed. It enters the super-zweibein field (EX), where aM = (at, a,,) and \7 A = EX aM. The action takes the form
(77)
162
Andrzej Frydryszak
The customary choice of the gauge for the super-zweibein field is the following
FM
-
(
( JA ) -
£ -1
_e-1,1, )
-e-l17
e-1£
-
'"
'
E = e + T/JjJ
(78)
Recently the model of the spinning particle with arbitrary number of supersymmetries on the world-line has been constrncted [51,55]. Such an lV-extended little SCSY in the massive model of the spinning particle, after the field redefinitions in the equations of motion, yields the supersymmetric Lax equation. Moreover it can be used in the study of hyperbolic Kac-Moody algebras.
3.2.
Superparticle models
The extension of the Minkowski space to the superspace with the additional spinorial coordinate is the basic structure of the supersymmetric field theories [18]. The super-Poincare group becomes the fundamental symmetry of the theory. Now, the superparticles are generalizations of the relativistic point particle from the Minkowski space to such a superspace, with still the same requirement at the background - to describe" correctly" the spin. Because they incorporate super-Poincare invariance there is a close connection between superparticles and representation of supersymmetry. Some models provide a natural examples of actions which yield, after the first quantization, the minimal irreducible representations of the given superPoincare superalgebra. The superparticles have reach symmetry, as local as rigid [a1]. In many respects the quantization procedure is diflicult because of the complicated structure of the phase space constraints. This aspect of the superparticle models makes that they are instructive toy models used to understand the superstrings and the variety of their quantization procedures. There are important differences between massless and massive models, ho~ever we will not stress them, aiming only to ilustrate generally the historical development in the construction of the models. The first pseudoclassical relativistic particle model with the spinorial grassmanian coordinates was proposed by Casalbuoni in 1976 [14]. On the configuration superspace (xl" (Jo" iF') the he defined the lagrangian of the form L = -rnJw!,w!'
(79)
dw" = dx" - i(dOu!,(J - (Ju"dO)
(80)
where is the super one-form, invariant under supertranslations
,
X"f-7x!,
x" - i(tu,,(J - Ou"t)
0",
0", Oex
0"
f-7 f-7
0' a
B~
(81 )
+ fa
(S2)
+ (0.
(Sa)
163
LAGRANGIAN MODELS OF PARTICLES
The lagrangian is too poor to give after the first quantization the Dirac equations and some interesting supersymmetric multiplets. The chiral supermultiplet content of the Casalbuoni's G4 model was analysed by Almond [15]. To improve this model the first order fermionic kinetic terms are needed. But they cannot be introduced in a strightforward way, because of the relation
o"dT ~()" = ~(O 0") dT"
(84)
Four years later Volkov and Pashnev [23] tried to cure this drawback using more general super one-form, invariant under super-Poincare transformations. Namely, 2
d8 = dw"dw"
+ adO"'d()" -
a*dO"dO o ,
aEC
(85)
and then the action of the form S=
_rnJ
T
_mJ Jw"w" + aO"O" - a·eo0" T
2,Jd;;i. Tl
=
2
T
(86)
TJ
Now the fermionic kinetic term is present and gives the first class constraints. However, not the one playing upon quantization the role of the Dirac equation. Nevertheless the content of the model is more reach, since the first quantized theory contains some multiplets (two scalar multiplets and one vector multiplet of states with the negative norm). The Brink-Schwarz action for a superparticle of mass Tn in d dimensions uses again invariant super oncform w.The reparametrization invariance is provided by the einbein field what enables to consider a massless superparticle as well. In 1981 they proposed an action of the form [28]
(87) where
w"=:i;n+ior"o,
n=I,2, ... ,d-l
(88)
Specialy massless case is intersting here, because there is an additional invariance present (Siegel [:37]) b"O" O,.:l'H
bKt'
r;:;"fJ KfJ,
r:::;"'fJ =
Wn
(l''')''fJ
(89)
-ibKOr"()
(90)
4ieOK
(91)
whre K is an anticommuting spinoral parameter. This symmetry allows to reduce some of the () - degrees of freedom (here half of them, in general at most half)
[36-38]. The first massive superparticle model which exhibits
K -
symmetry was introduced
164
Andrzej Frydryszak
in 1982 by de Azcarraga and Lukierski [24]. In their model this symmetry was firstly observed but the role of such a gauge invariance in reduction of the degrees of freedom was first pointed out by Siegel [37] and he introduced modified action. To finally overcome the problem with the fermionic kinetic term present in the Casalbuoni's model one has to enlarge the superspace. In the de Azca.rraga-Lukierski model it is done by considering the N-extended Y1inkowski superspace (xl',Oj",Of), i = 1,2, ... , N and introducing central charges to the superalgebra. I1ence the resulting underlying rigid symmetry gets enlarged to N -extended super- Poincare superalgebra. The new "isotopic" structure allows to use internal symplectic metric A ij = -Aji and the expression of the form Oi" A ij 1;oj" now is not a total time derivative and can contribute nontrivially to the action. After obvious modification in the super one form (92) the lagrangian function can be written as
~ + Z'(Oi0 A ij O'jo 1 = -mVwJlwl' J
Oi A ij O'-F') +"
(93)
The fermionic kinetic term is in fact of the Wess-Zumino type and changes under supersymmetry transformations by a total time derivative. Indeed, let Z/J be a symmetric, Lorentz invariant matrix (where I, J could be multi-indices e.g. 1= (Q, i) and Z/J = f"BA ij ), then the simple example of the WZ-term for a supersymmetric particle is of the form (94) What means that one starts from the closed super-twoform h = idOZdO. It;s exact; with b = idOZO one can write h = db. From the invariance of of h under supertranslations it follows that d(b,b) = 0 and at least locally b,b = df for some superfunction f. For the AL-action it means that f-variation yields the total time derivative change in the lagrangian. The AL-model after the first quantization yields the irreducible representations of the N -extended super- Poincare superalgebra. The whole spectrum of supersymmetric multiplcts was found as a result of the first quantization [43,81] not only for the massive case but also for the massless [44]. In the quantization of this model there was firstly applied the supersymmetric generalization of the Gupta- Bleuler [40,41] quantization method [42,43], which later was used in quantization of various systems exhibiting the similar structure of the second class constraints (i.e. hermitean splitting of the set of the second class constraints into the subsets of conjugated, relatively first class constraints). The coupling of this model to the external fields gives interesting results. Comparing to the traditional equations of the spin precession in the external electromagnetic field we obtain that the superspace generalization of the Bargmann-Michel-Telegdi
LAGRANGIAN MODELS OF PARTICLES
165
equations takes the form [45] , L ',,". e, x"
+ 9 s,spin PA "" FPA
(95)
U
9 F,," W "
+ (- e.-
L'"PW W. 9 ). W", p ,,-
(96)
2m e (W' - 2m "w" - W· "w" )F"P'zp,
(97)
-- i2 E: ""PAP P('O AO-) - _i_ h W,,were 2m 2 E: ""PAP "S'PA • Spin an d .JcSpin "" l kO' k. F'or th' c ex t erna I YangMills field we obtain, within the minimal coupling, the generalized Wong equations [45] >
p" i"
91
F"" I"'x" + 92 SSpin 0" FPA I" a PA a
ra A b I C X. " 91} bc " a W"I
92 F ""
a
2m
92) bc
Fb
91 + ( -2m -
91 (W· vWIJ.
--
ra
-
''"
ICS'''" Spin
). F"PW W. la 92 W" a p "
W·) LWp' la J.lW V I'a Zp ,
(98) (99)
(100) (101 )
However, the AL-model is supersymmetric therefore the fully supcrsymmetric coupling to the supersymmetric field is of greater interest. It can be found in Ref. [60]. In the case of the supersymmetric Yang-Mills and supergravity theories it gives in a natural way the conventional sets of constraints for these fields. 3.3.
Twistorial models
The supersymmetric particle models using the twistor-like variables were developed in the eighties, firstly in the component formulation then in the superfield one [61,62,65,66,71,72]. The very important result obtained within this formulation consists in re-expressing upon use of the equations of motion the local world-line supersymmetryas K-transformation [71,72]. General feature of this kind of models is a possibility of manifestly covariant quantization. To merely signal the existence of very reach developments let us recall the superfield version of the model beeing generalization of the following component action [71J
(102) Namely, SI = -i
J
dTd7] Pm (DX m + i81m De),
(103)
where D = a~ + i7]OT and Pm = Pm + i7]Pm, X m = X m + i7]Xm, eo = Oalpha + 7]>'",. In the component version this action contains additional to the S an auxiliary term of the form
(104)
Andrzej Frydryszak
166
The mechanism of trading the twistorial superparticle's K-symmet.ry for world-line supersymmetry is analysed in series of papers [71-73] and recently in Ref. [74]. Relation between the different forms of the superparticle dynamics, involving spinorial coordinates is analysed in Ref. [75]. The possibility of manifestly covariant quantization of the massless particle model was the motivation of development of the model of harmonic superparticle [76]. The action of this model is a generalization of the Siegel model with some new (harmonic) bosonic variables which are parametrising a suitably choosen coset spaces.
3.4.
Arbitrary superspin models
The model of the classical arbitrary spin particle [64] discussed in Sec.2.2. can be generalized to the pseudoclassical model with the I\'-extended super-Poincare symmetry [70]. After the Dirac quantization this model gives the on-shell massive chiral superfields (the central charges can be introduced as well). The extension of the configuration space M x 8 2 is done in the Minkowski sector, it is changed into the N-extended superYrinkowski superspace with coordinates (x",O"IJN) I = 1,2, ... , N, a = 0,1,2,3. On the new configuration space M 4 14N x 8 2 there is defined the Lagrangian of the form
( lOti) where
n"
." '0 10" " 0'-1 - O· I 0" "0 I x+z ( 0" )
'z"ziJ
" ,,(I
mJV(V
+ 1)
(106) ( 107) (108)
The V is a superspin parameter. The central charges analogous to those of the Azcarraga-Lukierski model can be considered here as well [70].
4.
Doubly supersymmetric models
The doubly supersymmetric models were considered firstly by Gates and ;'IIishino [51]. To extend the ~SR string theory they proposed a new class of superstring models which possess both spacetime and world-sheet supersymmetries. Then within this scheme the particle model was considered [52,53]. There are also two other approaches to such particle models: the twistor-like superfield models (commented in previous section) and the spinning particle model (invented firstly in the component form [77, 78]). The spinning particle models are revieved in Ref. [82]. here we shall restrict ourselves to the brief ilustration of the superfield realization [79,80]. In the (supersymmetry)2
LAGRAKGIA;-.J MODELS OF PARTICLES
167
particle models one introduces the supertime space (t, 1/) and the superMinkowski superspace. Therefore the trajectories of a point object are the mappings
(109) where
X"'(t,ll) eA(t,ll)
::= ::=
x"'(t) + 1/A"'(t) OA(t)+1I'pA
(110) (III)
Introducing the covariant object
(112) where V is the covariant superderivative given by VA == E';!iJM (cL eg. (71)) one can write the supersymmetric invariant action in the form
s
~
Jdtd1/8det(E,;!)V7}Y'" . Y", ==
2"1Jdt( e - [ ,w2 _e('P/ m p)2
- 2C -1,/.''''' 'f/A W",
+ A~)
-
2'('''' 1 W
(11:1) -
,1.,,,,)'f/A 'P/mp-
(114) (115)
The superfidd covariant phase space description of this model in the rigid supersymmetry case was given in Ref. [81]. One can say that developments in the spinning and superparticle models has been resumed in their superfield formulation which appeared in the second half of the eighties. It has turned out that all types of the pseudomechanical description can be put together and organized in a joint superfield model (Jet us note that there exists the superfield formulation of a spinning particle alone, but not of the superparticle, which has to coexist in the superfield formulation with the spinning particle).
5.
Recent developments: q-deformed spinning particle and K-relativistic model
Finally let us mention the brand new aspect of the relativistic particle models, namely their deformations. Actually there are not well established deformed models. However, without entering into the question why things have to be (or not to be) deformed we shall recall two examples: the q-dcformed spinning particle and t.he Krelativistic part.icle. The example of t.he q-deformed relativistic spinning particle was considered by Malik [83]. With the use of the first order Lagrangian of the spinning particle and the q-deformed graded commutation relations for (x m , 1/;"', p"', e) in the phase space he introduces" deformed" G Lq (2)-invariant Lagrangian
(116)
168
Andrzej Frydryszak
The model is under investigation and its Dirac" deformed" quantization is still to be performed. The K-rclativistic particle is more "physical". It lives in the K-deformed Minkowski space [86,87] with the mass shell condition modified to the following form
(2K sinh po )2 _ 2K
jJ
= m2
( 117)
This model can bl' described within the formalism with commuting as well as noncommuting space-time coordinates. The interesting properties of the object of this kind are discussed in Refs. [84,85]. However an inclusion of spin to this model is still an open question. Acknowledgements This work is supported in part by KHN Grant
#
2 1'302 087 06.
REFERENCES [I] Frenkel .1., Z. fiir Physik 37 (1926),243 [2] Thomas L.H., I'hil.Magazine 3 (1927), 1 [3] Ilargmann V., Michell,., Telegdi V.L., Phys. Rev. Lett. 2 (1959),435 [4] Corben H.C., "Classical and Quantum Theories of Spinning Particles", Holden-Day, San Francisco 1968 [il] Barut A.a., "Electrodynamics and Classical Theory of Fields and Particles", MacMillan, :\ew York 1964 [6] Hanson A.J., Rcggc T., Ann.Phys. 87 (1974),498 [7] Hanson A.J., Regge T., Teitelboim C., Constrained Hamiltonian Systems", Accad.Naz.dei Lincei, Rome 1976 [8] Grassberger 1'., J.Phys. A: Math Gen. 11 (1978), 1221 [9] Cognola G., Soldati R., Vanzo 1,., Zerbini 5., Phys. Lett. 104 B (1981),67 [10] Balachandran A.P., Marmo G., Stern A., Skagerstam Bo-S., Phys. Lett. 89 B (1980), 199 [II] Cho J-H., Hyun S., Kim J-K., "A Covariant Formulation of Classical Spinning Particle", prcprint YU:vIS-93-09, Seoul 1993 [12] Cog[]ola G., Soldati R., Zerbini 5., preprint CTFn, Univ. di Trento, 1982 [13] Stern A., Skagerstam Bo-S., Physica Scripta 24 (1981),193 [14] Casalbuoni R., Nuovo Cim. 33 A (1976), 389
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[15] Almond P., "The Suprsymmetry Extended Weyl Algebra and Casalbuoni's G4 Model", preprint QMC/02-81, London 1981 [16] Barut A.O.,Pavsic ~1., Phys. Lett. B 216 (1989), 297 [17] Cho J-H., Hyun 5., Kim J-K., "Relation between Classieal and Pseudo-classical Spinnig Particle", preprint YCYlS-93-8, Seoul 1993 [18J Berezin F.A., Y1arinov M.S., Ann. Phys. 104 (1977),
3;~6
[19] Harducci A., Casalbuoni R., Lusanna 1,., Nuovo Cim. 35 A (1976), :H7 [20] Martin J., Proc. Roy. Soc. of London A 251 (1959),536 [21] Matthews P., Salam A., Nuovo Cim. 2 (1955), 120 [22] Tobocman W., :-,'uovo Cim. 3 (1956), 134 [23] Volkov D.V., Pashnev A.T., Teor. Mat. Fiz. 44 (1980),321 [24] de Awirraga J .A., J.ukierski J., Phys. Lett. 113 B (1982), 170 [25] Dixon W.G., Nuovo. Cim. 34 (1964),317 [26] Brink L., Di Vecchia P., Howe P., Nud. Phys. B 118 (1977),76 [27] Brink L., Oeser 5., Zumino B., Di Vecchia 1'., Howe P., Phys. Lett. 64 B (1976),437 [28] Brink L., Schwarz J.H., Phys. Lett. 100 B (1981),310 [29] Galvao C.A.P., Teitelboim C., J. Math. Phys. 21 (1980), 1863 [30] Sundermeyer K., "Constmined Dynamics" Springer, Berlin 1982 [31J Harut A. 0., Duru 1.11., Phys.Rev.Lett. v.52, 23 (1984),2009 [32] Barut A. 0., Duru 1.11., Phys.Rev.Lett. v.53, 25 (1984),2355 [33] Leon J., Martin J.M., "Introducing spin to classical phase space", LAEFF preprint #26/1995, Madrid [34] Townsend P.K., "Spacetime supel'syrnrnetl'ic particles and strings in background fields" in proceedings of the first Torino meeting on Superunification and Extra Dimensions,1985 [35] Siegel W., Class. Quantum Grav. 2 (1985). L95 [36] Siegel W., Phys. Lett. 203 B (1988),79 [37] Siegel W., "Introduction to string field theory", World Scientific, Singapore 1988 [;~8]
Evans J.M., ;\"ucl. Phys. B 331 (1990),711
[;~9]
Harducci 1\., Giachetti R., Gomis J., Sorace E., J. Phys A: Math. Gen. 17 (1984), 3277
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[40] Gupta S., Proc. Roy. Soc. of London 63 A (1950), 6R1 [41] Bleuler K., Helv. Phys. Acta. 23 (1950),567 [42] Lusanna 1.., in "Supcrsymmetry and Supergravity" proceedings of XIX Karpacz Winter School ofThpor. Phys. World Scientific, Singapore 1983 [43] Frydryszak A., Phys. Rev. D 30 (19R4), 2172 [44] Frydryszak A., Phys. Rev. D 35 (1987),2432 [45] Frydryszak A., "Supersymmetric particle with internal symmetries in external electromagnetic and Yang-Mills fields" 1FT/UWr 1982 [46] de Azc:irraga J.A., Lukierski .1., Phys. Rev. D 28 (1982), 1337 [47] Frydryszak A., Lukierski .I., Phys. Lett. 117B (19R2), .51 [48] Gates Jr. S.J .. Grisaru :\1.'1'., Rocek M., Gates W., "Superspace" Benjamin/Cummings, London 198:3 [49J Ikemori 11., "Superjicld formulation of superpar·ticle" preprint, 19RR
#
DPNU-88-03, :'lagoya Univ.
[50] Ikemori H., Z. fUr Physik C: Particles and Fields 44 (1989),625 [51] Gates S.J. Jr., Nishino II., Class. Quantum. Grav. 3 (1986),745 [52] Gates S.J. Jr., Majumdar P., Mod. Phys. Lett. 4A (1989),339 [53] Gates S../ . .I r., in "Functional integration, geometry and strings", proceedings of XXV Karpacz Winter School of Theor. Phys., Hirkhiiuser, Basel 1989 [54] Gates S.J. Jr., Rana L., "A Theory of Spinning Particles for Large N-extended Supersymmmeil'y " hep-th/9504025 [55] Gates S.J. Jr., Rana , "A Theory of Spinning Particles for Large N-extended Supersymmmetry (II)" hep-th/9510151 [56] Souriau .I.-M., "Structure des systemes dynamiques", Dunod, Paris 1970 [57] Zakrzewski S., "Extended phase space for a spinning particle ", hep-th/9412100 [58] Duval Ch., Horvathy P., Ann. Phy,. (:"Y) 142 (1982),10 [59] Duval Ch., Ann. lnst. 11. Poincare A XXV (1976), :l45 [60] Lusanna 1.., Milewski B., :"ucl. Phys. B 247 (1984),396 [61] Bengtsson A.h.II., Bengtsson 1., Cederwall M., Linden N., Phys. Rev. D 36 (1987), 1766 [62] Penrosp R., Mac Callum M.A.H., Phys. Rep. 6 (19n), 109
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[6aJ Hasiewicz Z., Siemioll P., Defever F., Int. J. Mod. Phys. A 17 (1992), :J979 [61] Kuzenko S.M., Lyakhovich S.L., Segal A.Yu., Int. J. Mod. Phys. A 10 (1995), 1529 [65] Penrose R., Rindler W., "Spinal's and space-time", Cambridge Cniv. Press, Cambridge 1986 [66J Gershun V.D., Tkach V.I., JETI' Lett. 29 (1979),320 [67] Howe P.S., Penati S., Pernici M., Townsend 1'., Phys. Lett. B 215 (1988),255 [68] Ferber A., t"ucl. Phys. B 132 (1977),55 [69] Shirafuji T., Prog. Theor. Phys. 70 (1983), 18 [70] Kuzenko S.M., Lyakhovich S.L., Segal A.Yu., Phys. Lett. B 348 (1995),421 [71] Sorokin D.P., Tkach V.I., Volkov D.V., Mod. Phys. Lett. A4 (1989),901 [72] Sorokin D.P., Tkach V.I., Volkov D.V., Zheltukhin A.A., Phys. Lett. 216B (1989),302
[ra] Bandos LA., t"urmagambetov A., Sorokin D.P., Vo1kov D.V., Class. Quantum. Grav. 12 (1995), 1881 [74] Galperin A.S., Howe P.S., Stelle K.S., ~ucl. Phys. B 368 (1992),248 [75] Cederwall M., "A note on the Relation between Different Forms of Superpartiele Dynamics", preprint ITP-9a-33, Goteborg 199a (hep-th/9:310177) [76] Sokatchev Eo, Class. Quantum. Grav. 4 (1987),237 [77] Aoyama S., Kowalski-Glikman .1., Lukierski J., van Holten J .W., Phys. Lett. 217B (1989),95 [78J Kowalski-Glikman J., Lukierski J., Mod. Phys. Lett. A4 (1989), 2437 [79] Kavalov A., Mkrtchyan ILL., "Spinning superparticle", preprint Yer I'hI/1068(31)-88, Yerevan, 1988 (unpublished) [80] Frydryszak A., "Superjield Spinning Superpartiele Model and (Supersymmetryj2 ", preprint ITP UWr 726/89, Wroclaw, 1989 (unpublished) [81J de Azca.rraga J.A., Frydryszak A., Lukierski J., Phys. Lett. B 247 (1990),289 [82] Frydryszak A., Lukierski .1., "Spinning Superparticle Models - Recent Developments", preprint ITP /UWr 752/90, WrocJaw 1990 [83] Malik R.P., "On q-deformed spinning t'elativistic pat·ticle", hep-th/950302 [84] Lukierski J., Ruegg H., Zakrzewski W.J., in "Quantum Groups. Formalism and Applications" proceedings of XXX Karpacz School of Theor. Phys., PWN, Wroclaw 1995 [85J Lukierski J., Ruegg II., Zakrzewski W.J., Ann. Phys. 243 (1995),90
172
Andrzej Frydryszak
[86J Lnkierski .J., :\'owicki A., Ruegg H. Phys. Lett. B 264 (1991), 3:11 [87] Lukierski J., :\'owicki A., Ruegg 11. Phys. Lett. B 313 (1993), 357
From Fidd Theory to Quantum Groups
D = I SUPERGRAVITY AND SPINNING PARTICLES J.W.
VAN HOLTEI\"
NIKHEF!FOM, P.O. Box 41882 1009 DB Amsterdam NL
Abstract: In this paper I review the multiplet calculus of N = I, I local supersymmetry with applications to the construction of models for spinning particles in background fields, and models with space-time supersymmetry. New features include a non-linear realization of the local supersymmetry algebra and the coupling to antisYIllmetric tensor fields of both odd and even rank. The non-linear realization allows the construction of a D = 1 cosmological-constant term, which provides a mass term in the equations of motion. D
1.
=
Worldline supersymmetry
Supersymmctry, as a symmetry between bosons and fermions, was discovered almost 25 years ago [1,3]. Apart from mathematical elegance, supersymmetry has the quality of improving the short-distance behaviour of quantum theories and has therefore been proposed as an ingredient of many models of physical phenomena, most often but not exclusively in the domain of particle physics and quantum gravity. Many of these applications are presently speculative, but it was realized already early that supersymmetric extensions of relativistic particle mechanics describe ordinary Dirac fermions [4,7]. Supersymmetric theories of this type are known as spinning particle models. They are useflll providing low-energy descriptions for fermions in external fields [7,12] and path-integral expressions for perturbative amplitudes in quantum field theories [13,20]. They are also useful in studying aspects of higher-dimensional supersymmetric field theories and superstring models [21]. Like in string theory, in the discussion of supersymmetric point-particle models one has to distinguish between worldline supersymmetry, where the supersymmetry concerns transformatiolls of the worldline parameters of the physical variables (proper Dedicated to Jurek Lukierski on the occasion of his 60th anniversary.
174
J.W. van Holten
time), and space-time supersylllmetry which refers to supersymmetry in the target space of the physical variables. Both types of supersymmctry are encountered in the literature. In fact, in [22J a model was constructed possessin~ both types of supersymmetry simultaneously. In the present paper I review the construction of pseudo-classical spinning particle models with worldline supersymmetry. To this end I present the multiplet calculus for local worldline supersymmetry (N = 1 supergravity in one dimension) and construct general lagrangians for D = 1 supersymmetric linear and non-linear (j-modcls with potentials; many elements of this formalism were developed in [24,25J. The coupling to all kinds of background fields, including scalars, abelian and non-abelian vectors fields, gravity and anti-symmetric tensors is discussed in some detail. I finish with the construction of a model which exhibits space-time supersymmetry as well [22].
2.
D = 1 supermultipIets
Supergravity models in a+-I space-time dimensions describe spinning particles. Indeed, the local supersymmetry and reparametrization invariance generate first-class constraints which after quantization can be identified with the Dirac and Klein-Gordon equations. IIence the quantum states of the model are spinorial wave functions for a fermion in a d-dimensional target space-time. The models I construct below are general N = 1 supergravity actions in f) = 1 with at most 2 proper-time derivatives, and the correponding quantum theories. IIigher-N models have been considered for example in [26,27]. Extended target-space supersymrnetry has been studied in [28,29]. In one dimension supersymmetry is realized off-shelP by a number of different sets of variables, the supermultiplets (superfields): 1. The gauge multiplet (e, X) consisting of the einbein e and its superpartner X, the gravitino; under infinitesimal local worldline super-reparametrizations, generated by the parameter-valued operator D(~,e:) where ~(T) is the commuting parameter of translations and e:( T) the anti-commuting parameter of supersymmetry, the multiplet transforms as
De
d(~e) . ---2ze: v dT '"
d(~X)
de:
~+-dT'
(1)
2. Scalar lIlultiplets (x, 1/'), used to describe the position and spin co-ordinates of particles. The transformation rules are lThe term off shell implies that the supersyrnmetry algebra is realized without using dynamical constraints like the equations of motion.
[) =
1 Sllpergravity and Spinning Particles
175
dx
8x = ~ dr - icV',
(2)
where the sllpercovariant derivative is constructed with the gravitino as the connection:
dx
+ ix1/;·
dr
(3)
3. Fermionic multiplds (TJ,j) with Grassmann-odd TJ and even f. The fcomponent is most often used as an auxiliary variable, without dynamics of itself. The transformation properties under local super-reparametrizations are
U
df r
. 1 '1"l e
(4)
= ~-d - lc-vTTJ·
The supercovariant derivative is formed using the same recipe as before:
(5)
4. A non-linear multiplet consisting of a single fermionic component transformation rules {j 17 =
d17 . 1 '1"l (-+c-zc-17v17 T , r e
~d
17
with the
(6)
with the supercovariant derivative
(7)
On any componcnt of any multiplet thc commutator of two infinitesimal variations with parameters (6.2,101,2) results in an infinitesimal transformation with parameters (6,103) given by
(8)
J.W. van Holten
176 For the non-linear representation variation
(T
the proof requires use of the supersymmctry
(9) Then a simple result is obtained:
(10) It follows that each of these multiplets is a representation of the same local supersymmetry algebra, and this algebra closes off-shell. However, the parameters of the resulting transformation depend on the components of the gauge multiplet (e, X), indicating that the algebra of infinitesimal transformations is a soft commutator algebra, rather than an ordinary super Lie-algebra. Among the representations discussed, the gauge multiplet and the non-linear multiplet have mauifestly non-linear transformation rules. The variations of the other two multiplets are linear in the components of these rnultiplets. For this reasou the scalar and fermionic rnultiplets arc called linear representations of local supersyrnmetry, although some of the coefficients depend on the gauge variables (c, X).
3.
Multiplet calculus
The linear representations (scalar and fermionic) satisfy some simple addition and multiplicatiou rules; this tensor calculus has been developed iu [25]. The rules can also be formulated in tenus of D = I superfields [24]. As concerns addition, any linear multiplets of the same type can be added component by component with arbitrary real or complex coefficients. The linearity of the transformation rules then guarantees the sum to be a multiplet of the same type. The multiplication rules are also simple. There are 3 different product formula's: I. The product of two scalar multiplets E = (x,1/J), E' = (x',1//) is a scalar multiplet
E x~' = 1-'" = (J:.r',J:li,'
+ x'0).
(II)
This rule can be extended to arbitrary powers of scalar multiplets, for example:
(12)
In this way one can define functions of scalar rnultiplets by power series expanSlOns.
D = I Supergravity and Spinning Particles
=
2. The product of a scalar lTlultiplet E (TI, J) is a fermionic multiplet
177
(x, lj;) and a fermionic multiplet P
Ex p = pi = (xTJ,xf - ilj;TJ).
=
(13)
3. The product of two fermionic multiplets is a scalar multiplet:
- ip
X
pi = E ' = (-iTJTJI,fTJ' - fry).
( 14)
:"iext I introduce the operation of derivation of scalar and fermionic multiplets; the super-derivative on linear multiplets is a Grassmann-odd linear operator turning a scalar multiplet into a fermionic multiplet, and vice-versa, with the following components: 'D2.' = pi =
(01,0/, l'D e
T
x) .1
(15) On product multiplcts this super-derivative satisfies the Leibniz rule, with in particular the result 'DE" = n'DE x 2.,,,-1.
(16)
The super-derivative satisfies an operator algebra similar to the supersymmetry algebra:
(17) where 'D T is the supercovariant proper-time derivative on componcnts, encountered before in eqs.(3) and (5).
4.
Invariant actions
1. Invariant actions can bc constructed for the lincar as well as the non-linear multiplcts. As there exists no intrinsic curvature in D = I, there is no invariant action for the gauge multiplet involving the einbein, but there is a very simple action for the gravitino, namely
J
dTiAX,
( 18)
178
J.W. van Holten
where A is a constant. The equation of motion for this action by itself is not consistent (it requires A to vanish); but this is changed if one adds other terms to the action, like the ones discussed below. Also, one can replace X in the action by dxldT, but then the action becomes a total derivative. A cosmological-constant like action can be constructed with the help of the nonlinear multiplet; it reads
(19) This is also the kinetic action for the fermionic r5 variable, which in view of the anticommuting nature of 17 can be only linear in proper-time derivatives. :"Jote, that the non-linear nature of the multiplet allows one to rescale the variable r5 and thereby change the relative co-efficients between the various terms in the action. A rescaling of 17 by a factor lie gives the action
5 nl (r5;e)
=
2' ' JdT (e-"':"'X(J-..!....(J{;-) , 2 c
C
(20)
where I have introduced the dot notation for ordinary proper-time derivatives. Of course, the rescaling also changes the non-linear transformation rule for 17 under supersymmetry to (21) Combining the actions S'A and Snl in such a way as to get standard normalization of the fermion kinetic term for 17 leads to the action (with the dimension of h)
SgTaV
=
m
J
dT (ieAX -
~ e + iCX(J + ~ (J{;-) ,
(22)
where m is a parameter with the dimension of mass and c has the dimension of a velocity, The Euler- Lagrange equations for the fermions X and (J' then give 17
= A,
I d(J X = - - = O. C
dT
(23)
Thus the constant A is like a vacuum expectation value of the fermionic variable 17. However, the variation (21) of (J' is such that for non-zero C it can be gauged away completely by a supersymmctry transformation. Therefore it does not represent a true physical degree of freedom. Of course, this seems to contradict the equation of motion (2:1), but we observe that also the equation for the einbein is inconsistent, requiring the constant c to vanish. Again, these problems are solved by adding further terms to the action. In applications c usually represents the velocity of light, which
D = 1 Supergravity and Spinning Particles
179
can convcnicntly ht> taken as unity (c = I). 2. ;"ext we turn to a formula for the construction of invariant actions for linear multiplets. Given a fcnnionic multiplet !P = (1),!), an action invariant under local supersymmetry transformations is Slin
=
J
dT
(ef - iXT/).
(24)
Note, that in eqs.(19) and (24) the integrand itself is not invariant, but transforms into a total proper-time derivative:
J
5SIin =
dT ddT
(-in/) ,
(25)
and the same for (iSnl with T/ -> u. Eq.(25) shows, that (is vanishes for variations which are zero on the endpoints. Eq.(21) can be applied to the construction of actions for scalar multiplets if one applies an odd number of super-derivatives so as to obtain a composite fermionic multiplet, sometimes called the (fermionic) prepotential. A simple example of this construction is the free kinetic action for a scalar multiplet, constructed from the composite fermionic multiplet
!P bn =
~V21-'
x Vl-',
(26)
Inserting the components of this multiplet into the action formula (21) gives
, J (1. 2e
Skin =
dT
-
X
2
i ·+i- x1/Jx.) . + -1/J1/J 2 e
(27)
If one extends this formula to d free multipJets E" = (x", 1/J"), J1 = 1, ... , d, then using the appropriate minkowski metric it becomes the action for a free spinning particle in d-dimensional space-time. This is a special case of the most gcneral action involving only scalar multiplets and quadratic in proper-time derivatives of the bosonic co-ordinates x": the D = 1 (non- )linear u-model in a d-dimensional target space, constructed from a fermionic multiplet
(28) Here g"v( E) is a symmetric tensor in the space of the scalar multiplets, which can be interpreted as a metric on the target manifold. Using !P[g] in the action formula (21) gives the component expression
(29)
J.W. van Holten
180 when~
D denotes a target-space covariant derivative (30)
with [~/'(x) the Riemann-Christoffel connection. The action Ski,,[g] is manifestly covariant in the target space. The symmetries of this action have been investigated in detail in [30,31], with applications to special target manifolds like Schwarzschild space-time and Taub-:WT in [12,31,32]. The simplest action for scalar multiplets involves only one super-derivative. It starts from the general fermionic multiplet (super I-form)
P [A] = AI'(E) x
P~'I',
(31)
with AI'(E) an abelian vector field on the target space. Inserting the components of
P[A] in the action formula (24) gives the result Svec[A] =
JdT (AI'(x)xl'-IFl'v(x)1/JI'1/Jv),
(32)
with Fl'v the field strength tensor of the abelian vector field AI'(x), A similar construction can be carried out using arbitrary odd super p-forms [24]. For example p = 3 gives
P [H]
~ HI'VA(E) x P~'" x PE v X PEA.
(33)
This gives an action involving the anti-symmetric 3-tensor Ill'v>-' (x):
Soda [Il] =
~Hl'v>-,(X)1/JI'1/JV (x>-. + ~i x1/J>-') + ~()"Hl'v>-.(x)1/J"1/JI'1/Jv1/J>-'.
(34)
The inclusion of even p-forms is also possible, but requires one or more fermionic multiplets; details are given below. In the sallie spirit one can find odd p-form extensions of the kinetic term
P [G] = Gl'vl ...vp(E) x p 2 EI'
X
P~Wl ..•
X
EVP.
(35)
For a discussion of these unconventional actions I refer to [24]. Finally, actions with higher powers of p 2 and/or with P" (n 2': 3) lead to higher-derivative component lagrangians. I do not consider them here. Combining the results for scalar fields, within the restrictions we have imposed the general action for scalar multiplets is of the form
me2 .)' [L;] = mS'kin [g] + qSvcc [A] + aSadd [Il] + meSA - -2-5,,1,
(36)
This action describes a spinning particle in background electro-magnetic and gravitational fields, with the possible inclusion of torsion for ai-D. The first-class constraints obtained from the equation of motion for the einbein and gravitino are
D = 1 Supergravity and Spinning Particles
lng llv x·,"V 1jJ
W H 'lVA JjJIlJjJVJjJA -:3
--
mee
(11 -
(J
181
)•
Hcre f'"v(11) and F"" VA (H) arc the field strenghts of the vector All and 3-form H IlVA ' rcspectively. Eqs.(37) are the pseudo-classical cquivalcnts of the Klein-Gordon and Dirac equations. :\Tote that local supersymmetry can be used to chose a gauge (J = 11 in which the expressions on both sides of the second equation vanish. If e = 0 (absence of Snl and SA) the particle is massless. With the inclusion of Snl (e f- 0) the particle acquires a non-zero mass. If the kinetic terms arc normalized in the standard way, thc relativc co-cfficicnt q between the first two terms represents the electric charge of the spinning particle, as defined by the generalized Lorentz-force [II]. Then the anti-symmetric tensor D"V = QV,1l 1V represents the electric and magnetic dipole momcnts. The terms involving the ;l-form Hllv.\(x) combine to form an anti-symmetric contribution to thc Riemann-Christoffel connection, representing torsion indeed. S. Finally we turn to the construction of actions involving elementary fermionic multiplets. To begin with, there is the simple action formula (24) lincar in the components of a single fermionic multiplet. It involves no proper-time derivatives, and therefore it can constribute only to potential terms. A natural and straightforward generalization of this action involving r fermionic llluitiplets
with the Ui(E) a set of scalar-multiplet valued potentials. The component action then is
(39) As the equation of motion for fi requires all Ui(x) to vanish, this action by itself is useful only to impose constraints on the target manifold. This conclusion is modificd when additional (kinetic) terms are added to the action. More complicated actions obtained using the multiplet calculus with both ferlllionic and scalar multiplets must havc an odd total number of fermionic multiplets and super-derivatives. Therefore the next complicated type of action involves thc product of two fermionic multiplets including a super-derivative. The general form of the fermionic prepotential is
J.W. van Holten
182
(40) with J(;j(E) a scalar-multiplet valued symmetric matrix. The component action for this prepotential is
It contains kinetic terms for the fermionic variables r/, but the variables P only appear without derivatives and are auxiliary degrees of freedom. In combination with the potential term SpodVj its elimination turns the constraints Vi into a true potential, allowing the bosonic variables to fluctuate around the solutions of the constraints
Vi(X) =0. Other actions can be constructed by replacing some of the super-derivatives 'DEJ1. in the odd p-form prepotentials like rJ>[Il] (33) by fermionic multiplets. I give the details for the case p = 3. First consider a prepotentiallinear in fermionic multiplets: (42) The potentials SiJ1.v(X) define l' anti-symmetric tensors (2-forms) on the target space of the scalars. Thus this construction and its higher-rank generalizations allows the inclusion of even p-forms in the action. Substitution in the linear-multiplet action 5Ii", eq.(24), gives
Seven
[il]
iCJiB (),I J1.,I.V iJ1.V X 'I' 'f/ J dT ( -'2
-
. is' (),I.J1. • v TTJ ;J1.V X 'f/ X
1 isip.v (),I.J1.,I.V + '2XTJ X 'f/ 'f/
(43) Next consider the case of a quadratic expression in fermionic multiplets. The prepotential is (44) The vector field V;j,,(x), anti-symmetric in [ij], takes values in a gauge group G <;:; 50 (1') for l' even (in the quantum theory this is always the case [19]). Thus this action describcs the coupling to Yang-Mills fields. After quantization the fermionic variables TJi generatc a Clifford-algebra represcntation of the group G cmbedded in 50(1') on the particle wavc-functions. The explicit cxpression for the pseudo-classical action is
[) = 1 Supergravity and Spinning Particles
183
S'YM[Vj = Jdr (~2 TI i 1] j\l;'J" ( x )'''+=x 4 1] i 1] jp(Oj ij"v ( X ),I,",I,V+' 'f' 'I-' ze fi\l;'J" ( x )1] j,I,") 'I-'
'
(45)
Here FJ~) represents the abelian (linear) part of the field-strength for the vector field \1", The non-abelian part can be obtained by a proper choice of J
t
[1'] = ~ 7i J k(E)
X
X
3. The action constructed from this prepotential becomes
(46)
Comparison with the action Spot[U] shows, that this action represents the coupling to non-abelian scalar fields, where the generators of the group are again expressed in terms of the rank-r Grassmann algebra. Extensions of these results to higher-order forms are straightforward. 5.
Applications
The actions constructed above can be used to describe spinning particles in a ddimensional target space-time in various kinds of background fields: scalar fields, abelian and non-abelian vector fields, anti-symmetric tensor fields, rank-3 antisymmetric torsion, etc. (l\'ote that in four-dimensional space-time the rank-3 antisymmetric tensor is dual to an axial vector field.) In this section I discuss some special examples which are particularly useful in physics applications. 1. Yukawa coupling. One of the simpler cases is that of a spinning particle in Minkowski space-time interacting with a scalar field. This situation is described by the kinetic action with 9"v = 1]I'V' the Minkowski metric, extended with the action SfcT'" for the internal fermion variables in a flat background (J
(48) The full component action is
S'Yuk =
Jdr (-2 x" + ? m e
2
im ' _ 1/J,,1/J"
im
.
i
i
e
2
+ ---:c x1/J"x" + ? _ 1]ir, + -2 fi
(49)
184
J.W. van Holten
l'
The auxiliary variables equation
can be eliminated using their algebraic Euler-Lagrange
(50)
.\Ui(X). This gives the result
.'3Yuk =
J (-x dT
nt
2
2c"
im . irn ,i + -V 1jJ" + - v1jJ x' + - TI·TI· 2" c "" 2
I
1
e 2 2 - - AU 2
1
(51 )
+ i.\ev"oI'Ui(x)1]i).
+i,\Ui(x)XT/
The constraints from varying the action with respect to the gauge variables are
(52) TIIX"V"
+ c.\Ui1]i
o.
=
The model describes a spinning particle in a relativistic scalar potential .\2lJl(x), which may be dynamical. If this field has a vacuum expectation value .\2(lJ?) = mc2 , it generates a mass for the particle, showing it can act as a Higgs field. This mechanism of generating mass dynamically is an alternative to adding the non-linear multiplet action. However, in some sense the two mechanisms are the same, because the action for the linear fermionic multiplet (1], J) becomes identical with the nonlinear multiplet action Snl( 0"; c) if one imposes the constraint that f = c, a constant.
2. Yang-Mills coupling. A very interesting application from the point of view of particle physics is the case of a spinning particle (e.g., a quark or lepton) coupled to a vector gauge field V,,( x) [7,34,35], a supersymmetric generalization of Wong's model [:n]. Again, I consider ordinary Minkowski space-time and a flat internal space-time. Then adding the vector action:
Sgauge = rnSkin [TI"vJ and eliminating the auxiliary variable
+ Sjerm [OiJ]
l'
- gSYM [V],
(53)
by its Euler-Lagrange equation
(54) the component actioIl reads
Sgauge =
irn . ige ) J (2c X" + T v"V" + --;- Xv"xl' + 21]i1]' + gV"x" - ·2 F(V)l'vvl'v Tn
dT
2
im
.
i
..
.
V
(55) For convenience I have introduced here the Grassmann-algebra valued gauge field
•
1 Supergravity and Spinning Particles
J) =
185
VI'
(56)
and similarly for the field-strength:
Pry),," = -~1)i1)j}ij,,"(V)
f)"v"-aY"-g[v,,,v"].
(57)
The equation of motion for a particle in a non-abelian background gauge field then becomes d2 X I' = gF(V)" dx" -iJ£DI'F(V), .I,A.I," df2 " df 2 A"'f/ 'f/ , (58) where df = edT and DI' is the gauge-covariant derivative. The first term represents the non-abelian Lorentz force, the second one the Stern-Gerlach term responsible for non-abelian spin-orbit interactions [11]. m
3. Gravity. The actions above can be easily generalized to include gravity, by using a general curved-space metric 91'''( x) in the kinetic multiplet rather than the Minkowski metric 1]1'''' The internal-space metric [(ij(X) however remains flat. The only new feature is then to change the kinetic terms to the general form Skin[g], eq.(29).
4- Anti-symmetric tensor coupling. As a final example of the coupling of spinning particles to external fields we consider anti-symmetric rank-2 tensor fields Bilw(x) in curved space-time as well as internal space. The action to use is Stcnsor = rnSkin[g]
+ SjeTm[[(] -
yScven[B],
where y is a coupling constant. The auxiliary variables j I.' ij fj
=
2"Z
(1p , 1'8I' j' I. ij 1] j -
P now
Y B i,w 1/; I' 1p! " )
(59) satisfy the equation (
.
60 )
To solve it, we assume that [(ij(X) is invertible. Elimination of the auxiliary variables from the action then gives the component result .. im,,, if' + -im 9 1/;1' D1/;" + - 9 X1/Jl'x + Jd (2cm 9,," xl'x" 2 ,," e "" 2 T
e i J + -1)1) 8
i'J'
I." 1] 1]
-
j'-1 . V j' 1.'1.
("
"
a I.,) I\
"ij
'1',./." ye i 1p'f/--1)
1
(a AI.' j'
j'-I) j I. i
IJ
B ''f/1p1{' .1," ,",A )1'"
(61 )
186
.J.W. van Holten
Here Fi'WA(B) = Ij3(fJ AB;"v + avEiA" + O"H;VA) is the field-strength of the antisymmetric tensor field. When the internal metric is flat: J
6.
Space-time supersymmetry
In all previous examples the fermionic multiplets were used to represent internal degrees of freedom, connected with rigid or local internal symmetries. I conclude this paper with an application where the extra fermionic variables represent spacetime degrees of freedom. This example is the spinning superparticle [22,23], which possesses hoth (local) world-line and (rigid) target-space supersymmetry [36]. For simplicity. I consider only space-times which allow Majorana spinors (d = 2,3,;\ mod 8). In such a spac(~-time one can define, in addition to the usual coordinate multiplets E", a spinor of real fermionic supermultiplets
(62) with a = 1, ... , 2[~]. More generally, in an arbitrary spinor basis we do not require reality, but the Majorana condition If! = elf;,
(63)
where If! = If!t ro is the Pauli conjugate spinor, and C: is the charge conjugation matrix. Then the components (Oa' h a ) define an anti-commuting and a commuting Majorana spinor in then target space-time, respectively. The super-derivative of this spinor of multiplets is defined as in eq.( 15). Introducing the Dirac matrices "'I" in the d-dimensional target space-time, I next construct a d-vector of composite spinor multiplets [2" =
VE" -
vch"lf!.
(64)
The components of these spinor multiplet are
(65) From the spinor supermultiplets [2" it is straightforward to construct a fermionic prepotential which is a Lorentz scalar in target space-time:
(66) The component action derived from this prepotential is
D = 1 Supergravity and Spinning Particles
":;SUPCT
J [ dT
1 (
2e
.
--
X" - iB,~B - eh", h
)2 + 2i ( Jj,~ -
-
187
) d (
-)
h",B dT 1/;" - h,~O
(67) A superficld derivation of this action has been presented in [37]. We observe that h is an auxiliary commuting Y1ajorana spinoL Contrary to our previous actions, in "super these auxiliary variables in general have' a cubic and a quartic term, of the form ,~hJ,,~ It and (h,~h)2. However, owing to the Fierz identities these terms vanish in four-dimensional space-time, where the auxiliary variables only appear quadratically. The action Ssupcr has a huge number of symmetries. Except for local worldline supersymmetry, I mention rigid target-space supersymmetry, under which the gauge multiplet (e, X) is inert, whilst the linear multiplets 2-'~ and Pa transform with an anti-commuting Majorana spinor parameter e:
(68) bO =
bh = O.
f,
These transformations imply that the components of the multiplet n~ = (w", IJ~) in eq.(65) are invariant: bw" = bil" = O. Then there is the Siegel invariance with anti-commuting spinor parameter K on the worldline, which takes the form
(69)
bO=,·/h.,
be = 4iBK,
bX =
o.
Under these variations the components (w", il") transform as
'"il" -- 2i 0'" il K -4i O· 11" , -,,' -K
(J
e
e
(70)
In addition there is a bosonic counterpart of the Siegel invariance [22] with commuting spin or parameter 0':
188
J.W. van Holten
()x,· = 0,
00 = 0,
ok
=,' no -
2hho,
(71)
oe = -4e1w, ox = 0, resulting in
oIlI' = 20.,1',·
nk,
owl'
=0.
(72)
Still other symmetries can be found for the massless spinning superparticle, which I do not discuss here. If one assigns the space-time supersymmetry transformation 00- = to the non-linear fermion multiplet, addition of the mass term SgTav respects local world-line supersymmetry and space-time supersymmetry. However, in this case the Siegel transformations and their bosonic extension are no longer invariances of the model.
°
Acknowledgments The research described in this paper is supported in part by the Buman Capital and Mobility program of the European Union through the network on Constrained Dynamical Systems.
REFERENCES [1] Y.A. Gol'fand and E.I'. Likhtman,Jf;TP Lett. 13 (1971) 323. [2] V.V. Volkov and V.I'. Akulov, Phys. Lett. B46 (1973) 109. [3J .I. Wess and B. Zumino, Nuc!. Phys. B70 (1974) 39. [4] F.A. Herezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 336. [5] L. Brink, P. Di Vecchia and P. !lowe, Nuc!. Phys. B118 (1977) 76. [6] R. Casalbuoni, Phys. Lett. B62 (1976) 49. [7] A. Barducci, R. Casalbuoni and L. Lusanna, Nuov. Cim. 35A (1976) 377; Nuc!. Phys. B124 (1977) 93; id. 521. [8] A. Balachandran et aI., Phys. Rev. D15 (1977) 2308. [9] M. Henneaux and C. Teitelboirn, Ann. Phys. (NY) 143 (1982) 127.
[10] .l.W. van Holten, Proc. Sem. Math. Structures in Field Theories 1986/87; CWI Syllabu8 vol. 26 (1990) 109. [11] .l.W. van Holten, Nucl. Phys. B356 (1991) 3; Physica A182 (1992) 279. (12] .l.W. van Holten, Phys. Lett. B342 (1995) 47.
D = I Supergravity and Spinning Particles
189
[1:1] S.M. Shvartsmall, Mod. Pilys. Lett. A5 (1990) 943. [14] M. Strasslpr, Nucl. Phys. B385 (1992) 14.5. [15] D.G.C. McKeon, Ann. Phys. (NY) 224 (1993) 154. [16] C. Schubert, Phys. Lett. B331 (1994) 69. [17] M. Mondragon et al. Pltys. Lett. B351 (1995) 200. [18] E. d'Hoker and D. Gagne, hep-th/9508131. [19] J.W. vall Holten, prpprint ~IKHEF/95-050; hep-th/9508136; Nucl. Phys. B (in press). [20] J. de Boer et al. SU:--JY preprillt ITP-SB-95-32, hep-th/9.509158. [21] M. Green, J. Schwarz and E. Witten, Supcrstring Theory,2 vols. (Cambridge Univ. Press, 1987). [22] S. Aoyama, J. Kowalski-Glikman, J. Lllkierski and J.W. van Holten, Pltys. Lett. B201 (1988) 487; Phys. Lett. B216 (1989) 133. [23] J. Lllkierski, Pmc. Int. ConI. on Selected topics in QFT and mathematical physics, Liblice (Cz) (1989) 297. [24J lUI. Rietdijk, Class. Quant. G'mv. 9 (1992) 1395; Mod. Pilys. Lett. A7 (1992) 881. [25J J.M.A. de Vrpp, diploma thesis, Univ. of Amsterdam (1993) [26] E.A. Bergshoeff and J.W. van Holten, Phys. Lett. B226 (1989) 93. [27J A.I. Pashnev and D.P. Sorokin, Kharkov preprint KFTI 90-31. [28J J.A. de Azcarraga and J. Lukierski, Phys. Lett. B113 (1983) 1337.
(1982) 170; Phys. Rev. D28
[29] A. Frydryszak, Pltys. Rcv. D35 (1986) 2432. [:10] n.H. Rietdijk and .J.W. van Holten, Clas.~. Quant. (imv. 7 (1990) 247; J. Geom. Phys. 11 (1993) 5.59. [31] G. Gibbons, R.H. Rietdijk and J.W. van Holten, Nucl. Phys. B404 (1993) 42. [32J R.H. Rietdijk and J.W. van Holtpn, Class. Quant. Gmv. 10 (1993) 575. [33] S.K. Wong, N. Cim. 65A (1970) 689. [34] P.D. Jarvis and M.J. White, Phys. Rev. D43 (1991) 4121. [35J N. Linden, A..J. Macfarlane and J.W. van Holten, preprint NIKHI::F/95-049, DAMTP-95/37. [36] M. Grepn alld J. Schwarz, Pltys. Lett. B136 (1984) :167. [37] A.n. Kavalov and R.I,. Mkrtchyan, preprint YERPHI-1068(31)-88.
From Field Theory to Quantum Groups
SUPERALGEBRA STRUCTURES ON SO(3) TENSOR OPERATOR SETS PIERRE Mll\'NALlrr t A:'
MARL\( MOZRZYMASt
t Laboratoire de Physique Theorique, Universitc
Bordeaux 1
19, rue du SolariuIll, 33175 GRADIGNAN Cedex, France + + Institute of Theoretical Physics, University of WrocJaw pl. Maxa Borna 9,50204 WROCLAW, Poland
Abstract: In this paper it is shown that in addition to the well known Lie algebra structure exhibited by Racah, the 80(3) tensor operator sets can also been endowed with a Lie sllperalgebra structure, and that by suitable choices of subsets, several subsllperalgebras can be built explicitly.
1.
Introduction
In his fundamental papers on the classification of atomic spectra, Racah [1] showed that the sets of so(3) tensor operators carry a natural Lie algebra structure, and that it is possible to built explicit bases for the fundamental representations of the main Lie algebras such as ,~l( 11), 80( 11), 8p(2m) and even of the exceptional algebra G(2). Many developments in this direction has been made since [2,3], in particular in relation with the existence of non trivial zeros of the Racah coefficients [4]. In this paper, we show that besides the Lie algebra structure, it is also natural to define a Lie superalgebra structure on the 50(3) tensor operator sets, and that bases for several Lie superalgebras can be built explicitly.
2. 2.1.
Lie algebra structure on 80(3) tensor operator sets Commutation relations of so(3) tensor operators
s
The spin j tensor operators of rank L, TI:t(O S L 2j, -L S M S L), are linear combinations with Clebsch-Gordan coefIicients of the dyadic operators Ijm) Unl built
192
P. Minnaert and M. Mozrzymas
with the standard basis of angular momentum eigenvectors: j
7:{j
(I)
2: (jmljnLM)ljm)(jnl·
=
m==-j
Clearly, the operator T!>t maps the (2j + I )-dimensional space with basis lim) onto itself. Therefore, the set of tensor operators is closed for the commutator and for the anticommutator operations. Using the properties of coupling and recoupling coefficients one has the relation [T,~,Tt',]± = (_1)2 j +I-+L'(2j
+ 1)1/22:
[(_I)L+L'+J ± I]
J
X(2J+I)I/2 2.2.
{
I ]~
L' ]
J } (LM L' M'IJ ]
N)T~.
(2)
Lie algebra structures
Let us first review the algebras corresponding to the commutator operation [2]. The number of spin j tensor operators is (2j + 1)2. Altogether they form the gl(2j + I) algebra. It is evident that the operator T~ = I decouples from the other operators and forms a gl( I) subalgebra. The (2j + 1)2 - I remaining operators with L > 0 form the algebra .~1(2j + I). One can identify other Lie algebras in the $0(:1) tensor operator sets by looking at some subsets characterized by some specific values of L or M. For instance, because of the factor [( -1 )L+L'+J ± I] in Eq. (2), it is clear that the odd L operat.ors form a subalgebra. If j is integer it is t.he orthogonal algebra 80(2j + I) and if j is half-oddinteger it is the symplect.ic algebra 8p(2j + I). In the same way, because of the Clebsch-Gordan coefficient in the right-hand side of Eq. (2), one has N = M + M'. Therefore, the even M operators form a subalgebra which has been identified [5,6] as follows: if j is integer it is the algebra gl(j + I )8g1(j) and if j is half-odd-integer it is the algebra gl(j + ttJ gl(j + ~). Finally if one combines both preceding conditions: odd L and even M, t.he corresponding subalgebras are gl(j) for integer j and gl(j + ~) for j half-int.eger.
i)
3. 3.1.
Lie superalgebra structure on 80(:1) tensor operator sets The superalgebras gl(j
+ Ilj)
and gl(j
+ 111 + i)
The main point for the introduction of a superalgebra structure on 80(3) tensor operator sets is the observation that since N = 1''11 + M' both for the commutator and for the anticommutator operations, one can consider the even M operators as the even part Lr; of a superalgebra and the odd M operators as its odd part Lo , such that (3)
Superalgebra Structures on 80(3) Tensor Operator Sets
193
As mentioned before, the Lie algebra formed by the even part is gl(j + I) GJ gl(j) or gl(j + 8 gl(j + Therefore, the superalgebra generated by the spin j tensor operators is gl(j IJ + I) if j is integer or gl(j + + if j is half-odd-integer.
t)
3.2.
t).
W t)
The superalgebras sl(j
+ 11j)
and sl(j
+ W+ ~)/gl(IIO)
The operators Tl, L > 0 are diagonal and traceless, so if we decouple the 1~ = 1 operator the remaining even M operators form the algebra .~(gl(j + 1) EEl gl(j)), which is isomorphic to .~l(j + I) Gl sl(j) ill gl( 1). Therefore, the even M operators with L > 0 form the even part of sl(j + Ilj) and the complete set of 7'!:t, L > 0 form the superalgebra sl(j + I Ii). The case of half-add-integer j is more complicated because sl(mlm) is not semisimple. The superalgebra sl(rnlm)/gl( 110) obtained by removing the tensor 7~ is semisimple, hut in that case the remaining tensors only form a pseudorepresentation
[7]. 3.3.
The superalgebras osp(j
+ Ilj)
and P(m)
As in the case of the algebraic structure, one can look for subsets of tensor operators characterized by some specific values of L or M in order to define subsuperalgebras. It appears that the most interesting superalgebras that can be built in this way are characterized by a specific choice of Land M for the bosonic and the fermionic operators. More precisely, the selection of the bosonic and fermionic operators is made in the following way: bosons
even L : M = 4n +2 { odd L: M =4n
. {even L: fermlOns odd L:
M = 4n - I M = 4n + I
(4 ) (5)
where in each relation n is a relative integer that takes all possible values compatible with the constraints IMI ::; L. Because of the factor [( -I )LH'+J ± I] in Eg. (2), one has the following commutation relations: [even L , even L']_ = odd J, [even L , odd L']_ = even J,
(6)
[odd L , odd L']_ = odd .J , and anti-commutation relations: [even L , even L']+ = even .J , [even L • odd L'l+ = odd .J , [odd L , odd L'l- = even .J ,
(7)
P. Minnaert and M. Mozrzyrnas
194
between tensor operators. Then it is easy to check that the set of bosonic and fermionic operators defilwd by Eq. (4,5) is dosed under the superalgebra operations. Note that there is another st't with tlw same property where the bosonic operators are defint'd by Eq. (4) and the fermionic operators are chosen as
M = -1n + I 1.. 1. = 4n - 1
{even L: . fermlOns odd L:
(8)
One can show [8] that, according to the value of j, the superalgebras built in this way are either orthosymplectic superalgebras or the P( m) superalgebras, as shown in Table 1.
Table l.Superalgebras built by the sets defined ill Eq. (4,.5)
superalgebra
J
integer integer
4.
2n
+1
2n
half integer
2n -
half integer
2n
t
+~
osp(j Ij
+ 1) = osp('2n -
II'2n)
osp(j
+ IIJ) = osp(2n + 112n)
osp(j
+ W+ t) = osp(2nI2n) P(j - ~)
= P(2n)
Conclusion
II] this article we have shown that the so(3) tensor operator sets carry a natural supersymmetric structure and that by choosing specific subsets one can exhibit various sub-superalgebras. As a final remark we would like to mention that since the condition M + M' = IV is valid for the commutation and for the anticommutation operations, the sets Lr; of even A1 operators and L o of odd M operators defined in section ;3, satisfy the following relations:
(9) These relations define an algebraic structure that can be considered dual of the superalgebra structure in that sense that the commutation and anticommutation operations are exchanged.
Superalgebra Structures on 80(3) Tensor Operator Sets
195
Acknowledgments This work has been supported by the Cultural and Scientific Office of the French Embassy in Poland (Actioll lntcgrce f\,' 5270). We would like to thank Pr. J. Lukierski and Dr. A. Frydryszak for discussions on the algebraic aspects of this research.
REFERENCES [1] G. Racah, Phys. Rev., 62, 438 (1942), ibid., 63,367 (1943), ibid., 76,1352 (1949): G. Raeah, G1'OUp Theory and Spectroscopy, Princeton Mimeographed Notes, Princeton (1951 ). [2] H. R. Judd, Opemtor Teelmiques in Atomic Spect1'Oscopy, Me Craw·JIill Book Com· pany Inc. New York (196:1). [3] L. C. Fliedenharn and J. D. Louck, The Racah· Wigner Algebra in Quantum Theory; Encyclopedia of Mathematics and its Applications. Addison· Wesley, London, (1981). [4] J. Van der Jeugt, G. Vanden Berghe and H. De Meyer, J. Phys. A: Math. Gen., 16, 1377 (1983). [5] P. Minnaert,
1~'u1"Ophys.
Lett., 12, 97 (1990).
[6] P. Minnaert and M. Mozrzymas, J. Math. Phys., 32, 588 (1990). [7] B. Dc Witt, Supennanifolds, Cambridge H.P., Cambridge, (1984). [8] P. Minnaert and M. Mozrzymas, to be published.
From Field Theory to Quantum Groups
THE SUPERSYMMETRIC SL(2,C) KAC-MOODY ALGEBRA AND THE SUPERSYMMETRIC KORTEWEG - DE VRIES EQUATION ZIEMOWIT POPOWICZ
Institute of Theoretical Physics, University of Wroclaw Pl. M. Borna 9 50 - 205 Wroclaw Poland
Abstract: The deep influence of the nonexistence of the supersymmetric N = 2 S L(2, C) Kac-Moody algebra on the construction of the SUSY soliton equation is pointed out. The general method of the construction of the SUSY constrained KP hierarchy is described.
1.
Introduction.
The Kac-Moddy algebra plays a crucial meaning in the modern theoretical physics, especially in the quantum as well as in the classical field theory. This algebra is connected with the hidden symmetry of the hudge class of the nonlinear partial differential equations solved by the inverse scattering transformation in the so called AKl\'S framework. Moreover this algebra could be used to the construction of the second hamiltonian structure for these equations [I]. Several years ago Chaichian and Lukierski [2] have proved "no go" theorem on the non existence of the extended (IV = 2) supersymmetric version of the 5 L(2, C) Kac-Moodyalgebra. In this paper we show a deep influence this "no go" theorem on the procedure of the extended supersymmetrizations of the equations which belongs to the AKI\'S framework. The idea to usc the extended supersymmetry (SUSY) for the generalization of the solitou equations appeared almost in parallel to the usage of the SUSY in the quantum field theory [3,1]. The main idea of SUSY is to treat bosons and fermions operators equally. The first results, concerned the construction of classical field theories with fermionic and bosonic fields depending on time and one space variable, can be found in [5 8]. In many cases, the addition of fermion fields does not guarantee that the final theory becomes the SUSY invariant. Therefore this method was named the fermionic extension in order to distinguish it from the fully SUSY way.
198
Ziemowit Popowicz
We have at the moment many different procedures [9-21] of the supersymmetrization of the soliton equations. From the soliton point of view we can distinguish two different recipes. In the first we add to the theory the new anticommuting (;rassmann valued functions only while in the second case we also add the new commuting functions. Interestingly enough it appeared that during the supersymJlletrizations, some typical SCSY effects (compared to the classical theory) occurred. We mention few of them; the nonuniqueness of the roots for the SCSY Lax operator [17], the lack of the bosonic reduction to the classical equations [16] existence of non local conservation Laws [35] and the nonexistence of the extended SUSY extension of the S £(2, C) Kac-Moody algebra. These effect strongly relics on the descriptions of the generalized classical systems of equations which we would like to supersymmetrize. In the classical case the AKNS hierarchy is connected with the one component KP hierarchy and in the SCSY case it is tempting to use similar arguments and try to construct the SUSY version of AK!\S hierarchy. In this paper we focus our attention on this problem. From this classification of the supersymmetrizat.ion methods one can infer that the second approach is more important then the first because we extend our knowledge on the new commuting functions. However, it is not completely true. As we show, it is possible to carry out the supersymmctrization of the one component KP hierarchy in two different ways. We show that despite of using the superfcrmions in the first approach for the supersymmetrization of the one-component KP hierarchy, the bosonic sector coincides with the usual classical two-component K P hierarchy. Interestingly the bosonic part of the SUSY Lax pair of the one component KP hierarchy is matrix valued operator in contrast to the scalar Lax operator in the classical case. The paper is organized as follows. In the first section we describe the multicomponent KP hierarchy and explain its connection with the AKNS approach. The second contains the introduction to the supersymmetrization of this hierarchy which is developed in the next chapters. More precisely in the third section we describe the superfermionic approach while in the fourth the superbosonic. We use superfermions as well as superbosons in the fifth chapter in the supersymmetrization of our multicomponent KP hierarchy in order to demonstrate the third (mixed) possibilities. The last section contains concluding remarks.
2.
Multicomponent KP hierarchy.
The multicomponent KP hierarchy have been introduced by Sidorenko and Strampp [22] which is a straightforward generalization of the scalar case. This is a hierarchy associated with the following Lax operator
£ " = u,:)"
m
+ lln-2(.)"-2 +... + Un + '~" u.=1
j-I I'j,
qj(
(1)
The Supersymmetric SL(2,C) Kac-Moody algebra
199
The corresponding flows could be constructed by means of fractional power method [2:J]. For n= I, one has multicomponent AK:'JS hierarchy, which includes coupled Nonlinear Schriidinger [2·1] equation as an example. For the case n=2 and n=:J one has the multicomponent Yajima - Oikawa [25] and Melnikov hierarchy [25] respectively. We first consider multicomponent AK~S hierarchy which is given by n
L = 0 +L
(2)
qjO-lT'j ,
i=l
and the flows are
Ltk = [( Lk) + ' L] The bi-hamiltonian structure of these equations have been widely discussed in the literature recently [22,26,27] and it has the following representation
(4)
SO = (()
-1
1)
()
(5)
,
where 1 is m x m identity matrix. [JI is in the form [26]
(6) where S~,k(n. k = 1,2) are
Tn
x
Tn
matrices with the entries
(I3Lr = -B~l' and
* denotes
B~2 = {T'iO-1T'J
+ T'jO-lT'i}
the hermitean conjugation. In the special case
Tn
(8)
= 1 we obtain
2qa-I q,
H1 =
( a-2T'a- 1 q,
(9)
Interestingly, this Hamiltonian operator could be considered as the outcome of the Dirac reduction of the hamiltonian operator connected with the 5 L(2, C) Kac-Moody algebra. Indeed let us briefly explain the standard Dirac reduction [28] formula. Let U. V be two linear spaces with coordinates u and v. Let
Ziemowit Popowicz
200
Puu, 1'(11, v) =
(
(10) Puu,
be a Hamiltonian operator (Poisson tensor) on U EB V. Assume that Pvv is invertible, then (II) is a Poisson tensor on U. B J direct inspection, it is easy to confirm that the application of this procedure to the Hamiltonian operator connected with the 5£(2, C) algebra Ox - 2Jo 0
q]
p =
ax + 2Jo
0
-1'
-q
l'
-~ox
( 12)
( to the subspace where J o = 0 gives us the formula (9). :\1oreover if we apply second times the same strategy to the Hamiltonian operator given by (9) to the subspace were l' = I we obtain the Hamiltonian operator p = -
~8xxx + 8 . q + q . 8
(13)
2
which is connected with the Virasoro algebra. The Hamiltonians Ih may be computed from
in which Res denotes the coefficient standing in 8- 1 term. For the subsequent discussion let us explicitly presents the equations (6) for the two-component KP hierarchy in the two particular cases. For k = 2 these equations are in the form m
(15)
qi,=qixx+2q'Lqsl's, i=l m
1'i,
= -I'ixx -
21'i
LQs1's,
(16)
i=1
This is a vector generalization of the l'\onlinear Schrodinger equation considered first time in [24], For k =:3 m Qi, = QiXIX
+ 3Qi L s=1
Qsx 1's
+ 3Qix L
5=1
Qsl's ,
(17)
The Supersymmetric SL(2,C) Kac-Moody algebra m
1';,
= r;xxx
+ 31'; L 8=1
201
m
qsr sx
+ :.l1',x L
(18)
qsr s .
s=1
These equations could be further restricted to the known soliton equation. Indeed, assuming that m=1 we obtain that equations (15)-(16) reduce to the usual Nonlinear Schrodinger equation while the eqs. (13)-(14) for q = r to the modified Korteweg - de Vries equation or for r = I to the Korteweg - de Vries equation. The second Hamiltonian structure of the Nonlinear Schrodinger equation is given by the formula (9) while for the Korteweg - de Vries equation by the formula (13).
3.
The extended supersymmetrization of the multicomponent KP hierarchy.
The basic objects in the supersymmetric analysis are the superfield and the supersymmetric derivative. We will deal with the so called extended N = 2 supersymmctry for which superfields are the superferrnions or the superbosons depending, in addition to x and t, upon two anticommuting variables, 81 and O2 , (0 2 01 = -8 1 82 , O~ = 0i = 0). Their Taylor expansion with respect to the 0 is
(19) where the fields W.ll, are tu be interpreted as the boson (fermion) fields for the superboson (superfermion) field (1,(2, as the fermions (bosons) for the for the superbosoTl (superfermion) respectively. The superderivatives are defined as (20) with the properties
vi =
V~ =
o.
(21)
Below we shall use the following not.ation: (V; F) denotes the outcome of the action of the superderivat.ive on the superfield F, while V;F denotes the action itself of the superderivative on the superficld F. The principal problem in the supersymmetrization of the soliton equations could be formulated as follows: if we know the evolution equation for the classical function u and its (bi) hamiltonian structure or its Lax pair, how is its possible to obtain the evolution equation on the supermultiplet P which contains the classical function u? This problem has its own history and at the moment we have no an unique solution. We can distinguish three different methods of the supersymmetrization, as for example the algebraic, geometric and direct method. In the first two cases we are looking for the symmetry group of the given equation and then we replace this group by the corresponding SlJSY group. As a final product. we are able t.o obtain the SCSY generalization of the given equation. The classification
202
Ziemowit Popowicz
as the algebraical or geometrical approach is connE'cted with the kind of symmetry which appears on the classical level. For example, if our classical equation could be described in terms of the geometrical object then the simple exchange of the classical symmetry group of this object onto SGSY partner justify the name geometric. In the algebraic case, we are looking for thE' symmetry group of this equation without any reference to its geometrical origin. These methods have both advantages and disadvantages. For example, sometimes we obtain the fermionic extensions of the given equations only. In the case of the extended supersymmetric Korteweg-de Vries equation we have three different fully SUSY ext.ensions, however only one of them fits to these two c1assiflcations [14]. It seems that the most diflicult problem in these approaches is the explanation why a'priori SCSY extension of the classical system of equation should be connected with the SUSY extension of the classical symmetry of these equations. By these reasons we prefer to use the direct approach in which we simply replace all objects which appear in the evolution equation by all possible supermultiplets and superderivatives in such a way that to conserve the gradations of the equation. This is highly non unique procedure and we obtain a lot of different possibilities. However this arbitrariness can be restricted if we additionally investigate its super-bi-hamiltonian structure or try to find its supersymmctric Lax pair. In many cases this manner bring the success [15,16,21]. In the next we utilize this way. Let us now start our considerations of trying to find the Lax operator for the multicomponent SUSY KP hierarchy. The direct method suggests to assume that L depends on the vectors supermultiplets P, G its supersymmetric derivatives and on the derivative and superderivatives in such a way that finally it has the gradation 1. Therefore we postulate that the Lax pair is an operator in the form (22) In order to specify this form we have to assume the gradations of the supermultiplets
F and G. However we quickly recognize that we encounter three different possibilities of the gradations of P, G: 1.) All F, G are superfennions with the gradation 1/2, 2.) All F. G are superbosons with the following gradation: F has 0 while G has 1 (or symmetrically). 3.) The mixture of both prE'vious possibilities in other words son1(' of the F and G are superbosons and the rest are superfermions. In the next sections we investigate in more details these possibilities.
The Supersymmetric SL(2,C) Kac-Moody algebra
4.
203
The superfermionic approach.
We now assume that the components of the vectors supermultiplets F and G are superfermions which could be written down as
(23) (24) where Ij, gf are usual classical functions while (jk, 17f are Grassmannian valued functions. The Lax operator we choose in such a way that contains all possible combinations of "variables" in the (24) in such manner that each term has gradation 1. Then using the symbolic language REDUCE we verified that the following operator k
L=
a+ L Fi · a-I. DI . /)2 . G
(25)
i,
i== 1
generate extended supersymrnetric multicornponent KP hierarchy. Indeed, its second flow is
I-i, = Fixx
+ 2 Lk
r~ (V I V 2 GJil - Fi
L FsGs
(
k
8=1
G it = Gixx
) 2
,
(26)
8=1
k
+ 2 ~ Gs (VI V2Gir~) + G i
(
k
~ r~Gs
) 2
,
(27)
while the third is
(28)
1:, [(VIGjFi) (VIGlri) Fj + (V 2Gj ri) (V G Fi ) I'~]} 2
I
'jrixZ,
1=1
Git =
G;x
+ :l1:,
{(VjV2GiXFj)
+ Gj
(V jV 2G';FJ ) Gjx+
J=I
(29)
where
k
Z =
L
FiG;fojGj.
(30)
i,j=l
Let us now discuss several particular cases of the equations (26-29). For k=l, equations (26 27) reduces to
(31)
204
Ziemowit Popowicz (32)
In the components, using (23-24), we obtained that eqs. (31-32) are equivalent with l
(t = (;x
+ 2C
(1)1(2
fi = f;x - 2(1 (l(1 - f2 T/2L ft2 = f;x
+ 2(1
(/2
=
+ t l - 19l),
+ 2P
(1)1(2
+ T/ 2(1 + pg 2 - F q l)
+ 2f2 (1)\2 + T/ 2C + pg 2 2(1 (1) I C )xx + 2fl (giC - t1)l)x +
(giC - tT/IL
(;x -
2P (g2C - p1)l)x
+ 2(2 (1)2(1 + flg2
1): = -T/;x - 21)1 (1)
+ 2T/ I (l(1
2
(33) ,
(34)
f2 g1) ,
(a5)
(36)
- Pi),
C + P l - f i) , 2
(:J7)
- f21)t - 2g 1 (1)1(2
+ 1/ 2(1 + P l -
f2 gl) ,
(38)
g; = -g;x - 21)1 (gl(1 - PT/lt - 2l (1)1(2
+ 1/\1 + t l -
f2 gl) ,
(ag)
g: = -g;x
1); =
-T/;x
+ 21)1 (1) IC)xx -
2g 2 (g2(1 _ PT/lt
2yl (gl(1 - fl1)l)x-
+ 21)2 (1)1(2 + flg2
(40)
_ pg 1 ).
As we sec, this system of equations can be interpreted as the extended supersymmetric Nonlinear Schrodinger equation which have been widely discussed recently [17,29 :J2]. The bosonic part (in which all fermions fields vanishes) give us the equations (7) for m=2 with the following identifications (41 ) Interestingly, our Lax operator in the bosonic limit for k= 1 does not reduce to the scalar Lax pair (4). In our case, it has a matrix form
(42)
In this way, we have shown that our one component extended supersymmetric KP hierarchy in the bosonic sector is equivalent with the usual two-component KP hierarchy. Yloreover, in this bosonic sector, our equations constitute the bi-Hamiltonian structure given by (6-11), but we are not able to find its supersymmetric bihamiltonian counterparts. This fact is connected with the nonexistence of the extended (:'oJ =2) supersymll1etric SL(2,C) Kac-?vloody algebra what have been proved first time by Chaichian and Lukicrski [2]. Ylorcovcr, the applications of the direct method to the supersymmetrization of the formula (9) also does not give us the correct solution,
The Supersymmetric SL(2,C) Kac-Moody algebra
205
what we have checked using the symbolic computation program REDUCE. However it is rather surprising that the extended (N=2) supersymmetric version of thc Virasoro algebra exists hut it could not hc considered as the outcome of the Dirac reduction technique 011 the supersymmetric level. On the other hand our equations are Hamiltonian equations which can be written as
(43)
where F = (FI , F2, ...fkr, (; = (C I , (;2, ... Cd and I is a k x k identity matrix. The Hamiltonians H k can be computed by using the formula (14) in which now theRes denotes the coefficient standing in a-I D I D z term. Let us now discuss the equations (28-29) for k=1. In this case they reduce to
(44) (45) with the following bosonic sector (46)
ht = hxxx
+ :lgz UJ!z)x -
:19]
Vi) x'
(47)
9lt = 91xxx
+ 311 (9192)x
312
(gi) x'
(48)
+ 311 (9~t .
(49)
92t = 9zxxx - 3Iz (9192)x
-
This systcm of equation can be considered as the vector generalizations of the Modified Korteweg - de Vries equation. Now we can investigate different reductions of the eqs. (46-49) to much simpler equations. For example, by assuming that 91 =
II
=
h,
92 = 0,
(50)
we obtain the usual Modified Korteweg - de Vries equation. To finish this section let us notice that this superfermionic manner discussed in this section allows us to obtain some extension of the usual system of equations by incorporating anticommuting functions but we do not change the usual multicomponent K- P hierarchy. We show in the next sections that superbosonic or mixed ways of supersymmetrizations generalize our usual multicomponent K-P hierarchy in the class of the usual commuting functions.
206 5.
Ziemowit Popowicz
The superbosonic approach.
We now assume that the components of the vector supermultiplet F and G are superbosons and could be expressed as
(51) (52)
g;
where (f and Tit are Grassmann valued functions while If, are usual commuting functions. In order to find the proper Lax operator in this case we assume the following gradation on the functions
un = 1,
deg Ui)
= 0,
deg
(a) = 0.5,
deg
deg(gl)
=
deg
(Tit) = 1.5,
deg (g1)
(53) 1,
= 2, .
l\"otice that it is possible also to assume the symmetrical gradation in which we replace -> g, ( -> TJ but we will not consider such possibility because we obtain the same information as in the considered case. We postulate the Lax operator exactly as in the formula (25) and interestingly in this case we obtain the same flows, where now in contrast to eq. (2:l 24) F and G are superbosons. Therefore, they have different expansions in the components. Let us consider more carefully two particular cases (k = 1) of these flows. The second flow is d 2 3 (54) diP = Fxx - G F' + 2F ('D]'D 2 GF) ,
I
~G =
G xx
-
3
G F
2
-
(55)
2G('D]'D2 GF).
It is the extended supersymmetric Nonlinear Schrodinger Equation considered in [21]. The third flow is
~G =
G xxx
+ 3Gx ('D]'D2 GF) + 3G('D]'D2 G x F)
2
2
- 3F G G x .
(57)
From the last equation it follows that for F = -1, our equations reduces to the supersymmetric Korteweg - de Vries. As it is known there are three different generalization of the extended supersymmetric KdV equation which have the Lax representation and this can be compactly written down as
The Supersymmetric SL(2,C) Kac-Moody algebra
207
Here, 0 is just a free parameter which enumerates these three different cases. Our case corresponds to 0 = I, after rescaling the time and transforming G into -G. In the paper [20] the present author considered the nonstandard Lax representations for this equation. Here, as the byproduct of our analysis we obtained the usual Lax representation for this equation which could be connected with the extended supersymmetric AKi'\S approach. Indeed, our Lax operator in this case takes the form
(59) with the following flow
(60) Unfortunately, similarly to the superfermionic case considered in the previous chapter we have not found the bihamiltonian structure of this equation.
6.
The superfermionic and superbosonic approach.
We are now able to consider the mixed approaches to the construction of the SUSY IlIulticomponent KI' hierarchy. Therefore we consider now the following SCSY lax operator k
L= 8
+L i=!
m
Fia- I D1 D2 Gi
+L
B;ij-I
D 1 D 2 Ci ,
(61)
j=1
where now F and G are vector superfermions with the expansions (2:l - 24) while B and Care superbosons with the expansions (51 52). Using the same technique as in the previous sections we computed the second and third flows but the final formulas are complicated. lIence we present the second flow only which can be written down as
when>
(66)
208
Ziemowit Popowicz
As we sec, the last system of equations describes the huge class of interacting fields. In some sense, it describes the interaction of the superfermions with the superbosons.
7.
Concluding remarks.
We have constructed the extended supersymmctric version of the multicomponent K-P hierarchy in three different ways. We obtained a new class of integrable equation for which we were able to constructed the Lax operator and showed that they are Hamiltonian equations. Moreover, due to the existence of the Lax operator, we obtained til<' infinite number of conserved currents for our generalizations. Unfortunately we could not prove that these currents are in involution. In the soliton theory, in order to prove the involution of the conserved currents, we utilize the recursion operator. Magri [33] has shown that such recursion operator could be constructed if we know the bihamiltonian structure. However in our case we could not find such bihamiltonian structure. It does not mean that our system docs not possess the recursion operator. The excellent example of the situation, where we do not known the bihamiltonian structure, but we known the recursion operator is the Burgers equation [34]. Therefore, it seems reasonable that if we wish to find the recursion operator for our supersymmetric generalizations we should try, to follow the Burgers approach.
REFERENCES [1] L. Faddeev and L. Takhtajan: Hamiltonian Methods in the Thcory of Solitons, Springer 1987; A. Das: Integrable Models, World Sci. 1989; M. Ablowitz and H. Segur: Solitons and thc Inverse Scattering Transform, SIAM Philadelphia 1981. [2] M. Chaichian and J. Lukierski, Phys. Lett., BI83 (1987) 169. [3] .J. Wess and J. Bagger: Supcl'symmctry and Supcl'gravity, Princeton, :"lJ 1982; [4] S. Ferrara and .J.e;. Taylor: Intmduction to Supcl'gravity (in Russian) Moscow 1985. [5] II. Kupershmidt: Elements of Supel'integrablc Systems, Kluwer 1987. [6J M. Chaichian, P. Kulish, Phys. Lett. BI8 (1978) 413. [7] R. l)'Auria and S. Sciuto, Nucl. I'hys. BI71 (1980) 189. [8] M. Gurses and O. Oguz, I'hys. Lett. AI08 (1985) 4:37. [9] Y. Manin and R. Radul, Commun. Math. Phys. 98 (1985) 65. [10] C. ylorosi and L. I'izzochero, Commun. Math. Phys.158 (1993) 267. [11] 1'. :,lathieu, J. ;'v[ath. Phys., 29 (1988) 2499. [12] C.A. Laberge and P. Mathieu, Phys. Lett. B2I5 (1988) 718.
The Supersymmetric SL(2,C) Kac-Moody algebra
209
[13] P. Labelle and P. Mathieu, J. Math. Phys., 32 (1991) 923. [14] T. Inami and H. Kanno, Comrnun. Math. Phys. 136 (1991) 519. [15] C.M. Yung, Phys. Lett. B309 (1993) 75. [16] E. Ivanov and S. Krivonos, Phys. Lett. B291 (1992) 63. [17] P. Kulish, Lett. Math. Phys. 10,87 (1985). [18] W. Oevel and Z. Popowicz, Commun. Math. Phys.139 (1991) 441. [19] Z. Popowicz, J. Phys. A: Math. Gen. 23 (1990) 1127. [20] Z. Popowicz, Phys. Lett. A174 (1993) 411. [21] Z. Popowicz, Phys. Lett. A194 (1994) 375. [22] J. Sidorenko and W. Strampp, J. Math. Phys. 34 (1993) 1429. [23] L.A. Dickey: Soliton Equations and Hamiltonian Systems, World Scientific, Singapore 1991. [24] S. Manakov, Sov. Phys. JETP 38 (1974) 248. [25]
~.
Yajima and M. Oikawa, Progr. Theor. Phys. 56 (1976) 1719.
[26] V.K. Melnikov, Commun. Math. Phys. 112 (1987) 639. [27] Q.I'. Liu: "Hamiltonian Structure of Multi-component Constrained KP Hierarchy", hep-th/9502076. [28] II. Aratyn, J.F. Gomes and A.H. Zimerman: "Affine Lie Algebraic Origin of Constrained KP Hierarchies", hep-th/9408104 to appear in J. Math. Phys. [29] G.II.M. Roelofs and P.II.M. Kersten, J. Math. Phys. 33 (1992) 2185. [30]
J.e.
Brunelli and A. Das, J. Math. Phys. 36 (1995) 268.
[31] F. Toppan, Int. J. Mod. Phys. A10 (1995) 895. [:l2] S. Krivonos and A. Sorin: "The minimal N=2 superextension of the hep-th/9504084 to appear in Phys. Lett. B. [33] F. Magri, J. Math. Phys. 19 (1978) 1156. [34] W. Oevel and W. Strampp, Commun. Math. Phys. 157 (1983) 51. [35] 1'.11. Kersten, Phys. Lett. A134 (1988) 25.
~LS
equation",
Cllapter IV.
MISCELLANEA
From Field Theory to QWUltlllll Groups
THE GROUP OF DIFFEOMORPHISMS AND ITS UNITARY REALIZATION IN QFT ZBIGNIEW HABA
Institute of Theoretical Physics University of Wroclaw, Wroclaw, Poland
Abstract: It is shown that the Gaussian measure describing the time-zero two-dimensional quantum scalar field on the circle is quasiinvariant with respect to a change of parametrization of the circle.
1.
Introduction
The group of diffeomorphisms appeared in quantum physics in the string model [1]. The choice of a parametrization of the string coordinate should have no physical meaning. In such a case a unitary implementation of various parametrizations may be required. It has been recognized quickly that in physical models we can only expect projective representations of the group of diffeomorphisms because of the positive energy requirement. Instead of the group of diffeomorphisms, which is mathematically quite a complex object [7]' its algebra consisting of vector fields on the circle is usual discussed. Positive ellergy Hermitian projective representations of this algebra ( called Virasoro algebra ill physical literature) have been completely classified [2]. However, it is still all open problem which of the representations of the algebra are integrable to a group. Moreover, from the physical point of view a realization of the unitary representation in the Hilbert space of quantum fields is required. Physicists obtained a realization of the Virasoro algebra in terms of oscillator's creation and annihilation operators. Lukierski and myself [4] described a realization of the Virasoro algebra as well as string dynamics in a functional framework i.e. in the Hilbert space of fUllctionals of the string coordinate. The string coordinate consists of an infinite number of modes which oscillate according to the free field dynamics. In such a case each spatial component of the string coordinate can be identified with a scalar free field Oil the circle. The ullitary implementation of reparametrization means a unitary realization of this transformation in the Fock space of the free field.
Z. Haba
214
Recently, some important n~sults on this problem have been obtained [.'i] [3] [6]. In this paper we obtain a unitary implementation of the reparametrization transformation in the space of functionals of the time-zero free field defined on the circle i.e. in £2(dfl), where fl is a Gaussian measure. In such a case a unitary implementation means a guasiinvariance of the Gaussian measure.
2.
Quantum scalar free field on 8 1
We realize the Fock space F = ff/H n of the scalar guantum free field
(I) It follows from eq. (I) that
(2) We define the positive energy part of the time-zero field on the circle by the expansion
q,( 0) =
~L
(3)
an exp( inO)
y27r n>O
From egs. (2) and (3) we can compute the two-point function I
I
[.
2
I
I
[((0,0) = 27r1n sm ("2(0 - 0))
]
(4 )
The inverse operator to K has the kernel I I [ . 2 I( I ] 1 L(0.0)=-47rsm("20 -0))-
(5)
i.e. f{ [, = I. Let us denote q,(J) = J dO¢(O)f(O) Then, we can characterize the measure 11 by its characteristic function
J
dfl exp(
+
= expCl f{ 1)
(6)
However, if the r.h.s. of eq. (6) is to be finite, then f has to be restricted to a subspace L6(dO) of £2(dO) consisting of functions whose zeroth Fourier component is equal to zero I.e. (7) dOf(O) = 0
J
The field itself has til(' property (7) i.e. J dO and f{ into an orthonormal basis of £2(dO) we can check that eg. (6) follows from eq. (I). Let us note that we could extend the field
The group of diffeomorphisms
215
disk D = {izi ::; I} and (0) into thf:' exterior r'D of the unit disk ( where r denotes a reflection in the circle). In such a case
Jdfl1>(rz')1>(z)
=
1
-In(z' - z)
(8)
27r
The extension of the field to the disk can be related to its evolution in an imaginary time as it is well-known that eq. (8) describes the two-point Schwinger function of the holomorphic part of the Euclidean scalar free massless field in two dimensions.
3.
The action of a diffeomorphism
Let It : 8 1 -+ 51 be a continuous invertible map. We define a transformation T( h) of the field
(9) where It-I denotes the inverse map. The second term on the r.h.s. of eq. (9) ensures that T 1> has no constant term in the Fourier expansion. The Fomier coefficients are determined by (T(h))(lI) = ;121l'
12>- 1>(It-
I
(O)exp(-inO) -
0
Strictly speaking the transformation
1 -OnO 21l'
1
2 "
d01>(It- J(O))
(10)
0
1'1> is defined by its dual T* acting in q(5 J )
I.e.
(T1»(f) = 1>(1'*n
(11 )
where
(T*(h)J)(O) = dh f(h(B)) _ dh dO dB
fh f(h(B')) dh dB'
Jo
dB'
( 12)
We assumed in eq. (12) that h is differentiable. Let us denote by fm = v2r: h exp(imO) the orthonormal basis in U(dO). Then we can expand Tfm again in the basis. Hence
(T(h)fm)(B) = I.e.
A(h)mr = - 1
fa
L
Amr(h)fr(B)
2 ".
dOexp(-irB
+ imlt-J(B))
( 13)
(14)
21l' 0 If It-I is C= then by a stationary phase method and integration by parts we obtain the following estimate (sec rd. [5] for details)
(15) for any positive integers n, m and k. Let us note that A nm
=
A(-n)(-m)
(16)
z.
216
Haba
Hence, the matrix A has the block structure
( 17) where both indices of (\ are positive, the first index of (J is negative whereas the second is positive. The matrix A fulfills all identity which follows from a covariance of wit.h respect to diffeomorphisms. So, let us denote {) = h- 1 (O). Then, inserting 0 = h(iJ) into eq. (13) and differentiating over 17 we obtain
fa
.. . dh({)) . lrnexp(lrn{)) = ~ Amrlr~exp(lr1?)
( 18)
On the other hand by complex conjugation of eg. (13) we have also exp(-imJ) =
LA ns exp(-i8h(iJ))
( 19)
We multiply eqs. (lH) and (19) and integrate over iJ on bot.h sides. Assuming that h is a diffeomorphism we change again the coordinates 19 ~ () in the integral on the r.h.s .. Then, performing the integrals we obtain the identity (here m, n > 0)
rnb mn =
LA
mr
(20)
A nr l'
r
In terms of the matrices (\ and
fJ
the identity (20) can be expressed as follows (21)
Eq. (21) means that A is a symplectic matrix.
4.
Quasiinvariance of the Gaussian measure under diffeomorphisms
The transformation T* of the test functions induces a transformation Jl. the measure according to the formula
~ Jl..
of
(22) where 1\1' = TI\T or in matrix notation
I\~n =
Lrs Amrl\rsAsn
(24)
The group of diffeomorphisms
217
For the unitary implementation of the transformation (10) the following theorem [8] plays a crucial role Theorem l(Kakutani) The measure fl. is continuous with respect to fl if and only if the operator R = f{-t(/{T - J{)/C t
(25)
is a Hilbert-Schmidt operator in L6(dO) i.e. TT(R+ Ii) < CXJ. Applying Theorem 1 and some formulae of sec. 3 we can prove our main result Theorem 2 The Gaussian measure (6) is guasiinvariant under any Coo diffeomorphism of the circle 8 1 Proof:Applying eg. (21) we can express R solely in terms of the matrix (3.
(26)
R=J{-t(3/{(3+/{-t
The matrix II has the first index negative and the second positive. In such a case the estimate (15) holds true. As a conseguence for any positive :'J there exists C(:'J) such that (/?+ R)mn C(N)(lml + Inl)-N (27)
:s
It follows that R+ Ii has a finite trace. The Kakutani theorem gives also a formula for the Radon-l\ikodym derivative
dfl. ( ¢i) = [del(l{-t dfl 5.
/{T IC
tJt exp[ ~¢;( (/<)-1 2
-
(/{Tt 1)¢i]
(28)
Discussion
The quasiiuvariallce of the measure fl allows a unitary implementation of the reparametrization transformation. ;'\amely, the transformation
(29) is unitary in U(dfl). Its infinitesimal form leads to the Virasoro algebra. The formula (28) shows a relation between infinitesimal diffeomorphisms and the Schwarz derivative. This comt's out because the infinitesimal variation of (/{T)-l = L T in eg. (28) (where L T is the transformation of L defined in eg. (5)) is expressed by the Schwarz derivative [9]. TIJ(' functional realization of the group of diffeomorphisrns, its relation to til(' hololllorphic maps of the plane (cf. eq. (8)) and to the Schwarz derivative could be interesting also outside its original domain of applicability (quantum field theory) e.g. in the ergodic theory of the group of diffeomorphisms. Acknowledgments
This paper is dedicated to my teacher Jurek Lukierski on the occasion of his 60th birthday. Its contell-t is closely related to the subject of my master thesis when under Jurek supervision I started my adventure in theoretical physics.
218
z.
lIaba
REFERENCES [1] M.A. Virasoro, Phys. Rev. Dl (1970) 2933. [2] V. Kar, in Led. Notes in Physics, 94, p. 441, Springer 1979. [3] R. Goodman and :;'.R. Wallach, J. Reine. Angew. Math. 347 (1984) 67; J. Funct. Anal. 63 (1985) 299. [4] Z. Haba and J. Lukierski, Nuovo Cimento 41A (1977) 470. Z. Haba and J. Lukierski, "Stochastic method in the string model", in Theoretical Physics. Memorial Book on Occasion of Professor Rzewuski's 60th Bir·thday, University Publishers, Wroclaw 1976. [5] G. Segal,Cornrnun. Math. Phys. 80 (1980) 301. [6] Yu.A. Neretin, Funct. Anal. Appl. 17 (1983) 85; Comrnun. Math. Phys. 164 (1994) 599; A.A. Kirilov, Funct. Anal. Appl. 21 (1987) 35. [7] J. Milnor, in Les Houches, Session XL, B. de Witt and R. Stora, cds., North Holland, 1984. [8] S. Kakutani, Ann. Math. 4 (1948) 214. [9] D.A. lIejhal, Memoirs AMS, No. 129, 1972. E. Witten, Cornrnun. Math. Phys. 114 (1988) 1.
From FielcI Theory to Quantum Groups
CHIRAL SYSTEMS ON GROUP MANIFOLDS!
z.
HASIEWICZ
Institute of Theoretical Physics University of Wroclaw pl.Maxa Borna 9 PL-50 205 Wroclaw
Abstract: The class of chiral systems is introduced. Their general properties: canonical structure and geometry of chiral splitting are discussed. For the case of loop group, corresponding to WZW model the non linear Poisson algebra of its chiral sector is described explicitly. PACS 0220,0240,0:350 1.
Introduction
It is common impression that many of geometrical models in Field Theory share some common and universal properties, despite the technical complexity of descriptions of the models. For each of these models a specific 'machinery' has been developed, making the comlllon features less visible. Maybe the most prominent example of this class is the Wess-Zumino- Witten Field Theory [15]. Both classical Hamiltonian formulation [II] and corresponding quantum theory [.5] are being viewed on the ground of the results which are known or are expected to be obtained. This indicates that in spite of the great development done recently [a] [6][4], sOIJIe fundamental background is missing. It is our hope to shed some new light on those questions by formulation WZW model as chiral canonical system. The starting point for definition of the class of chiral models is the group structure on the corresponding configuration space. The next ingredient is the non-canonical lifting of the left and right actions of the group on itself to the cotangent bundle (the phase space). The phase space is equipped with a symplectic form which is invariant under the above (lifted) actions and allows 1 Neither
this work as a whole nor its part is supported by any grant
z.
220
lIasiewicz
for their Hamiltonian realization (by momentum mappings). The Hamiltonian defining the dynamics is just a quadratic function of these momentum mappings, and this gnarantees that the equations of motion are 'chiral'. Hetter understanding of the geometrical nature of those models may be very helpful in quantization, as for instance it allows one to usc the data of the representation theory in a more conscious (and efficient) way. The paper is organized as follows: the first section contains a canonical description of the general chiraJ system living on a group manifold. For the case of the loop group the Hamiltonian description of WZW model is obtained. The second section is devoted to brief description of the geometry of chiral splitting for til(' ge'H>ral chiral system. In the case of loop group the symplectic descript.ion of the chiral sector of WZW theory is given. The third section is devot.ed to presentation of Poisson algebra of functions on the loop group.
2.
The Chiral System
The configuration space of the general chiral system is an arbitrary group manifold G (in particular infinite-dimensional). It is tacitly assumed that the objects under consideration do exist (even in the infinite-dimensional case). This is automatically satisfied in the cases of groups of maps from compact manifolds (with the loop groups as a special case corresponding to the WZW theory) if proper meaning of the cotangent space is given [12]. In order to describe the cotangent bundle it is convenient to trivialize the bundle by means of the left action of the group. The points of the phase space (T"G) are thus described by pairs (9, p) where 9 E G and pEg". The left and right action of the group on itself can be lifted to the action on the phase space in the following way:
+ Ad;:glO(90)) CP;J9,P):= (gg;', Ad;op + (1(90)) cp~Jg,p) := (g09,P
(1 ) (2)
where 0 is arbitrary go-valued group cocyele, i.e. it does satisfy: (3)
and Ad" is the coadjoint action. The formulas (1:3) and (14) do not look symmetrically because the left trivialization is used. In tll(' case of right trivialization the nonlocality appears in the expression for cpr. The canonical symplectic structure of the cotangent bundle is given by the differential of the Liouville form:
Chiral systems on group manifolds
n := det;
where et:= (p,g-ldg)
221
(4)
and obviously the pairing is evaluated in 9 - the space of values of the canonical left-invariant form g-1 dg. It is not difficult to check that neither the Liouville form nor its differential are invariant under (13),(11). This means that the actions (13,14) cannot be realized in a Hamiltonian way. In order to obtain an invariant symplectic form one has to add to [l an additional term:
(5) where E denotes the derivative of the cocycle 0 at the group unity:
2.,'(X)
:=
lim dd O(e lx ) VX E 1-0
t
9.
(6)
By definition it is linear operator on 9. At this point it is assumed that E is (,) -antisymmetric. In the literature [13] the cocycles with antisymmetric derivatives are called symplectic. l\'otice that one is free to add to (17) an arbitrary closed and both left- and rightinvariant 2-form on G. On a semi-simple and finite-dimensional group however the only one such form is 0 [2]. The actions (13),(14) are Hamiltonian with respect to (17) and admit the following (weak) momentum mappings:
f(g,p) := -Ad;p - O(g)
(7)
r(g,p) := p - Ad;_,O(g).
(8)
The Poisson algebra of (18) and (19) with respect to [linv has the following form:
+ 2C(X, Y)
(9)
{F(X), r(Y)} = F([X, Y]) - 2C(X, Y)
(10)
{J/(X), F(Y)} = 0
(11 )
C(X, Y) := (E(X), Y)
(12)
{f(x),j/(Y)} = JI([X, Y])
where and 2.,' is a derivative the cocycle as in (17). The dynamics is introduced by defining the following quadratic Hamiltonian:
'H :=
~(J((f, f) + I«r, F))
(13)
Z. Hasiewicz where f{ is some quadratic form on go. If f{ is Ad'·invariant then (27) together with (17) give the following equations of motion:
:i J dt
I
= 1-'(.Ji)
~~ J" =
(14)
(1.5)
-E(jr).
where.i is the 1{ - dual of J. These equations are called chiral, as in the case of G being a loop group they are precisely the equations of motion for the chiral currents in WZW theory (see below). The equations of motion for a group point take the simplest form when expressed in terms of affine momenta:
d 1 'r '/ -dt 9 -- -2 (gJ - J .g) . .
(16)
One can observt' that the Hamiltonian is in fact a collective one in the sense of [7], i.e. it is a pullback of a function on g' x g' by the chiral momenta (18,19). This fact has profound geometric consequences and some light will be shed upon it in one of the following sections. The example given by the configuration space being the loop group t:G corresponds to WZW theory. In this case, since the group is infinite-dimensional, one should from the very beginning extract from t:G' the smooth part [12] by identifying it with t:G via the non-degenerate form on the Lie algebra t:G, defined as follows: 1 fa2~ (X(O'), Y(O'))dO' 21l' 0
J{(X, Y) = ; where ( , ) is an Ad-invariant form on
( 17)
g.
In this case one takes takes non-trivial cocycle on t:G: k d O(g)=-(-d g)g-l;kER 2 cr and then
k d
(19)
\'---
.~
(18)
- 2 dO'
The chiral currents do satisfy the following Poisson commutation relations:
{J/(X),;t(}/)} = JI([X, Y]) {J"(X), F(Y)} = J"([X, Y])
2~ [~(X(cr), d~Y(cr))dcr k (2-rr
+ 21l' Jo
d (X(O'), dO' Y(O'))dcr
which are recognized as the affine algebras with underlying Lie algebra g.
(20) (21)
Chiral systems on group manifolds
223
By introducing the chiral derivatives: .
(h
iJ
:= ;:;ut
a
± k-:-) crr
(22)
we can write the chiral equations of motion as:
(23) The chiral derivatives (37) take the standard form after appropriate redefinition of time variable or equivalent rescaling of the Hamiltonian.
3.
Chiral splitting
In order to make the description of chiral splitting most transparent it is most convenient. to discuss the underlying geometry of general case. As noticed already the Hamiltonian is a pullback of a function on g* x g* by the chiral momenta (18,19). This means that one can try to describe the motion in terms of g* - valued momenta as their equations of motion arc much simpler. Thus one should consider the image of T*G in g* x g* under the map (24) From the definit.ions (18,19) it follows that the chiral momenta are not independent, or to be more precise: (25)
I.e. they are on the same orbit of the affine action defined by the cocycle 20:
A(g) = Ad;
+ 20(g)
(26)
Therefore the image of T*G under (24) is precisely the fibered product defined by projection 1f on the space of affine orbits:
J(T'G) = g*
x~
g*
(27)
where 1f assigns to each element ~ E g* its conjugacy class with respect to the action A. It is clear that the point in a fibred product contains less information about the system than a point in T*G. In order to see what is missing one should look at the fiber of J over the point (~,() E g* x~ go. The fiber is clearly isomorphic to the affine stabilizer subgroup of ~ (or any other point conjugated with ~ by A). Let W be the set of conjugacy classes (affine orbits). Defining the projection 1f J := 1f 0 J of T*G on W one can see that for any point w E W
(28)
224
Z. Hasiewicz
where If is the afline stabilizer of some 1/ in class w. Obviously If as a subgroup depends on the choice of 1/ (w fixes it up to an isomorphism only). One can thus immediately see the possible obstructions for the fibers to fit together into the structure of a smooth bundle over W. However, the situation is not hopeless because in many int('J'esting cases the mapping 7l" defines a trivial fibration over the open subset W o of W. The inverse image 7l" J I(Wo) =: (T*G)o is called the set of regular points. For the compact Lie groups the space W o can be identified with the interior of the Weyl Chamber in to, where t is the Lie algebra of some maximal torus of G. H is the maximal torus generated by t. Then the fibers of (28) fit together to give the principal bundle:
If
'-t
(T*GL .:!.., (Y* x rr 9* ~ Gj If x G j If) ~ W o
(29)
For the loop group £G with simply connected G the situation is similar. The set of classes can be parameterized by alcove [2] A in t and as in the previous case lJ IS identified with the maximal torus of t. The corresponding principal bundle has analogous structure:
11
'-t
(1'"£G)o':!'" ((UG x GjH) x (UG x Gjlf)) ~ W o,
(30)
where UG denotes the space of loops based at the group unity. Any element 9 E £G is a product 9 = U90 of 'U E UG and 90 describing the basepoint. This decomposition depends on the elwin> of the "beginning" of the loop and breaks the invariance with respect to rigid rotations. This is not the end of the story. because above structure factorizes in the general case provided the bundle obstructions are absent. Introducing two copies PI,r of the manifold W o x G. endowed with the following symplectic forms: D/ = dh,9/-1dgl) + (E(gi1dgd,g/-ldg/) (31 )
Dr = d(l'r,dg r g;l) + (E(dg r 9;I),dgr 9;1) For P = Pt x Pr being a symplectic manifold with the form f2 = fl l one can prove that
(32) -
Dr. (33)
where j denotes the symplectic reduction by the set of (first class) constraints. The above statemcnt is called the chiral splitting. In the case of £G the gcneral formula for the symplectic form of the left sector (the superscript I is suppressed for the sake of simplicity) (32) can be written according to the decomposition of loop onto basepoint and based loop:
(34)
Chiral systems on group manifolds
22<'i
The variable I' above parameterizes the set of conjugacy classes and satisfies f E A. It describes the "spin" content of the quantum theory as it labels the representations of the current algebra in the Hilbert space of chiral sector.
4.
Poisson Algebra
The structure of the (complexified) Lie algebra 9 can be most conveniently described in terms of Chevalley system [2]:
9
i'H EB(CTo
::=
+ CL o )
0>0
O'(h)TO • N o13 To +p
[To. T_ ex ] 0'
::=
-H o
+ f3-1'Oot,
(36)
The invariant form on 9 will be denoted by ( , ). The corresponding loop-algebra is spanned by the following elements:
(37)
mEZ
and {h)} is some basis of the Cartan subalgebra i'H. The left invariant I-form w on [G can be written as follows: W
D),mlLj,m == '" L...J \./"
+ '" L...J
mEZ,j
no,m Ta,m
Jt
+
(')-a,-m Jt T-o:,-m,
mEZ,o>O
where the forms 8 j ,m and [l0.m are left invariant duals of (37). Csing the above expression one can write the symplectic form (34) of the chiral sector:
n
:}-
(I}lr" hj)fjJ + '27rk L. imff m II e)J·-m(h/, It
7l'
+
j
L
j )
mEZ,/j
((1', I/o)
+ 47l'imk( To, Lo))nO,m II n-o,-m)
(39)
mEZ,o>O
The symplectic form introduces the Poisson bracket structure on the ring of functions on [G. On compact groups the rings of functions arc generated by the matrix elements of all representations. It is then natural to consider an analogous set in the case of loop group. One may take the functions defined by the Fourier modes of the matrix elements of the representations pulled-back onto circle by the loop maps.
226
Z. lIasiewicz
For any matrix representation l' : G ....... End( V) one can consider the corresponding functions on the circle and their Fourier expansions:
1'(g((1))
=L
T(ll)(g)c ina
(40)
llEZ
Using the homomorphism property of T and denoting by [ the corresponding representations of the Lie Algebra 9 one can easily find the Poisson brackets of Fourier modes of any pair of representations 1'1,1'2:
}& --
T(m) r(ll) I , 2
(41)
"\" 1'(m+s) W 1'(n-s) ( _ _1_' [(h) (j<; ~ I 2 2ks I I -
[(h )Mlj) 2
J
s;tO,]1
+ L
T?n+s) Q9 1'J"-s)
((r'2~") + 2iks(lf", H_,,))-I [I(L,,) 'Xi [2(To )
O>O,S
L 7~(m+s) ~\i 1~n-s) ((1", H,,)
_ 2iks(lf", H_,,)) -I [I(T,,)
Q:) [2(L,,)
271'
,,>o,s
The matrix M above is the inverse to the one formed by the scalar products (hi, hj ). One is now able to calculate the Poisson brackets of the matrix functions on the group of basepoints given by evaluations of the loops at O. These fnnctions are the sums of all Fourier modes of (40): T(go) = T(m)(g) (42)
L
tnEZ
For this reason in order to evaluate their Poisson brackets one needs some regularization procedure. The rule of symmetric (with respect to 0) summation eliminates the contribution from the first term in (41) and defines the following bracket: ']'}:9 {T ' I, 2
_
-
'(1'
1
c'
I 'co)
']') "\"
2
~
0>02
k(
71' T(Xl T_"
) ct
h (1', flo) ( ) L1A(~), 4k T", T_" U
(43)
where (44) The bracket of (43) neither defines the Poisson structure on the group G nor it has Poisson properties on the extended space G x A as in both cases it breaks the Jacobi identity. This anomaly is introduced by the regularization procedure employed in the definition of (43). Taking the limit k ....... 00 which corresponds to the loop length going to 0 one gets the Poisson bracket: (45)
Chiral systems on group manifolds
227
of the chiral sector of the point particle model OIl the group manifold [9]. The above gelleral classical analysis of chiral systems can be thought of as an introductory step towards their quantization. It is rather hopeless to expect that the quantization problem can be solved on the same level of generality. The construction of the correspondiIlg quantum system in any particular case seems to be of great importance. The reason is twofolJ First of all it can give physically illteresting qualltum model (as WZW or loop formulation of Yang-\1ills theory). Secondly it may shed some new light Ollto the difficult questions related to the quantization of slightly more complicated classical phase spaces. The construction of the quantum counterparts for the chiral models on the group manifolds seems to be a reasonable step beyond the class of "linear" quantum systems. Acknowledgments
The author would like to thank Jerzy Lukierski for his warm hospitality at the Institute of Theoretical Physics (University of Wroclaw). The author would also like to thank the Polish Government for regular payments of unemploymellt allowallces for his work at the University.
REFERENCES [1] A. Alekseev, L. Faddeev, Commun. Math. Phys. 141 (1991) 413-422. [2] N. Bourbaki, Gmupes et Algebres de Lie, Hermann, Paris 1968. [3] M. Chu, P. Goddard, I. Halliday, D. Olive, A. Shwirnrner, Phys. Lett. B266 (1991) 71-81.
[4] F. Falceto, K. Gawedzki, Talk given at 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, June 3-7,1991. [5] L. Faddeev, COmln1m. Math. Phys. 132 (1990) 131-138. [6] K. Gawedzki K, Commun. Math. Phys. 139 (1991) 201-213.
[7] V. Guillernin, S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge 1984. [8] R. Hamilton, Bull. Am. Math. Soc., 7 (1982) 65-222. [9] Z. Hasiewicz, P. SicmioIl, W. Troost, Int. J. Mod. Phys. A9 (1994) 4149-4168.
[lOJ P. LiebermaIlIl, Ch-M. Marie. Symplectic Geometr'y and Analytical Mechanics, D.Reidel, Dordrecht 19k7. [11] G. Papadopulos, B. Spence, Phys. Lett. B295 (1992) 44-50. [12] A. Pressley, G. Segal, Loop Groups, Oxford University Press, Oxford 1986.
228 [I;~] J.-~1.
Z. Hasiewicz Souriau, Structul'e des systemes dymamiqu€s, Dunod, Paris 1969.
[14] S. St('rnber~, Lectures
Oil
Differential Geometry, Prentice Hall, Englewood Cliffs 1964
[l.,)] E. Witten, Commun. klath. Phys. 92 (1984) 455-472. [16] R. Butt, Ellseign. Math. 23 (1977) 209-220. [17] LN!. Gelfand, D.H. Fuks, FUllct. Anal. Appl., 2 (19GB).
From Field Theory to Quantum Groups
QUANTUM POTENTIAL AND QUANTUM GRAVITY
.T.
KOWALSKI-GLIKMA:-I
Institute for Theoretical Physics, University of Wroclaw Pl. Maxa Born 9, 50204 Wroclaw, Poland e-mail; jurekk at ift.uni.wroc.pl and jurekk at fuw.edu.p
Abstract: The quantulIl potential approach makes it possible to construct a complementary picture of quantum mechanical evolution which reminds classical equation of motion. The only difference as compared to equations of motion for the underlying classical system is the presence of an additional potential term being a functional of the real part of the wavefunction. In the present paper this approach is applied to the quantum theory of gravity based on Wheeler De Witt equation. We describe the derivation of the 'quantum Einstein equation' and discuss the new features of their solutions.
1.
Preface
It gives me great pleasure to contribute this article to the volume in honor of Professor Jerzy Lukierski. I clearly remember when, still being a student, I have met him for the first time during one of the winter Karpacz schools in late 1970s. Since then he was always close to mc, first as an advisor and refcrcc of my PhD thcsis, and then as a closc collaborator in many research projects and a very good friend. I cannot possibly fully pay my debt to him; let this article be at least an expression of acknowledgement of what lowe him.
2.
Introduction
The enigma of quantum gravity is probably thc most challcnging problem of modern theoretical physics (for thc rcccnt reviews see [I], [2], [3], and [4]). It is for the difficulty of the problem that in spite of its importance, a very little progress has been made so far. Among many approaches, the canonical quantization seems to be the most natural one for at least two following reasons. First of all, we understand
230
J. Kowalski Glikman
quite well the point we start at: the Einstein theory of general relativity and the Dirac theory of quantization of constrained systems. Secondly, other more exotic approaches (most notably the string theory) introduce, as a rule, a number of their own difficulties. It is the lesson resulting from the modern developments that the quantum theory of gravity must be non-perturbative. This fact comes from the careful analysis of the condition of diffeomorphism invariance. From this point of view the nonrenormalizibility of perturbative quantum gravity is not surprising; indeed the modern interpretation of this fact is that it just shows that the perturbative expansion of quantum gravity docs not make sense. Most of the recent work on the canonical approach to quantum gravity is related to the loop formulation (see [4] and [o'l].) This formulation has the virtue that some part of constraints (Gauss law) is in terms of loop variables solved automatically. However, it turned out that this formulation has its own problems, the major of whose are the fullowing: 1. In quantum theory all the composite operators require regularizations. In particular it turns out that in the loop representation the metric is a composite operator and there seems to be no regularization which is consistent with diffeomorphism invariance of the theory. Therefore, even if the theory can be eventually quantized in this representation, it is presumably not equivalent to the quantization in metric representation. (One may argue however that the tetrad representation, which is the starting point for Ashtekar approach, is the more fundamental because only in this representation the coupling to fermions exists) . 2. It is well known that contrary to the quantum mechanical case, there exist many representations of quantum field theory which are not equivalent. Thus one must ask the question as to if some prediction of the theory like quantization of volume and area operators 6 ), which are claimed to indicate the discrete structure of space-time at sub- Planckian regime, arc solid facts and not artifacts of the chosen representation. As a matter of fact, it is unclear if in this formalism one can construct any operator with continuous spectrum at all.
:l. One of the major arguments in favor of using Ashtekar variables was the simplicity of the hamiltonian constraints. But it turned out that because of the absence of background metric it is very hard to regularize this operator, and solving the regularized version of it may well be as complicated as it is in the case of the Wheeler- De Witt operator. I made the comments above just to justify the statement that it is not outrageous to be a little bit old fashioned and base our discussion on the De Witt formulation of quantum gravi ty 7). For, if both metric and Ashtekar formulations arc equivalent,
Quantum Potential and Quantum Gravity
231
using them we look at physical reality from two equivalent and yet distinct points of view. If, however, these formulations arc not equivalent, they will compete as to which is the correct one. In both cases, therefore, investigating the prediction of metric formulation is quite important. I would like to stress however that the results described below may well be applied to some other formulations of quantum gravity. Quantum gravity theory faces yet another set of problems. Assume that sooner or later we will have in our possession a class of solutions of this theory. These solutions will be presumably of the form of some wavcfunctioll being a solution of a set of complicated equations defining the theory. Now the question is how should we interpret such a wavcfunction? It seems that if we accept the orthodox interpretation of quantum mechanics, we will immediately face a number of problems like: what is the meaning of the wavcfunction, how are we to interpret superpositions of states (universes) etc'? There are many attempts to address these kind of problems (sec e.g.,
[2]). In my personal opinion it would be fruitful to try to translate the information carried by the wavefunction to the language which would be easier to comprehand. It happens that such a language exists and is provided by the 'pilot wave' or 'quantum potential' approach to quantum mechanics (see [8],[9], [10]). Before discussing the merit of this approach as applied to quantum gravity, to set the stage, let us consider some simple examples.
3. 3.1.
Quantum potential in action: quantum mechanical examples The standard quantum mechanical system
Let us consider the simple one dimensional quantum mechanics. The time evolution of the wave function of particle of mass m is governed by the Schriidinger equation ')
ih~l/J 8t
h2
')2
= ---(-l/J + V(x)l/J. 2
2m 8x Let us now consider the polar decomposition of the wave function l/t(;r, t) = R(x,
(I)
tJ exp GS(x, tJ) ,
where both Ii and .'i are real functions. Substituting this into the Schrodinger equation, we obtain two equations for real and imaginary parts, to wit
8 I (85)2 +V(x)----=o h 18 R R ax
(2)
iJ I iJRiJS I 82 5 -R+---+-R-= O 8t Tn 8x Dx 2m iJx 2
(3)
2
-5+- at 2m Ox and
2m
2
2
J. Kowalski-Glikman The interpretation of Eq. (3) is clear: this is noting but the continuity equation. Eq. (2), on the other hand, can be interpreted as the Hamilton-Jacobi equation, with the additional term proportional to h 2 and resulting from the wave function of the system. This potential is called the 'quantum potential' and it is customary to denote it by Q( x, t). Observe that this potential is, in general, time dependent. It is a well known fact that the Hamilton-Jacobi equation contains the whole dynamics of the system. Thus, interpreting Eq. (2) in such a way, we can immediately write Hamilton equations of motion: p = {p, H}, (4 )
x=
{x, H},
where p2
H = 2m
+ V(x)
(5)
h 2 1 iF Ii - 2m Ii Dx 2
(6)
is the effective hamiltonian containing t.he quant.um pot.ential t.ernl. Let us discuss interpretation of t.hese equations. To set t.he stage, let us consider Schrodinger equation first. Here we have to provide one initial condition for ljI, ljIo = ljI(x, t = 0) which corresponds t.o two initial conditions for Ii and S. It should be observed that Eq.% (:3) expresses conservation of probability. ]\"ow, in the quantum potential approach t.he situation is quite different. First of all, Hamilton's equations (4), (5) govern the evolution of x(t), p(t). We need therefore two initial conditions for p and x. Second, we have Eq. (:3) which shapes time evolution of R and therefore t.he t.ime dependence of quantum potential. The initial condition for t.his equation plays double role. First., R 2 (x, 0) is to be interpreted as t.he probability distribution for various initial conditions for x, and second R(.r,U) is itself an initial condition for R time evolution. This shows that in the case of this equation we have to do with some kind of personality split. To cure this disease, we should, in principle, proceed as follows. Recall that ~; = p. Thus, Eq. (3) can be rewritten as [)
1 [) Ii
1
[)2.')
at
m Dx
2m
ax
-R+--p+-R-,2 =0. If we could express ~:~ as a function of p, q, t, we would be able to turn the equations governing the quantum potential dynamics to closed form. Moreover, we will be able manifestly take into account the back reaction of p and x on the quantum pot.ential itself. This procedure is currently under investigation, and in this paper I will ignore this equation whatsoever assullling that it is ident.ically satisfied. To finish this subsection let me make some remarks. The dynamical equations above describe trajectory of a particle. One may treat this trajectory as some unphysical (i.e., not related to reality) auxiliary 'picture' of
233
Quantum Potential and Quantum Gravity
the particle lwhaviol'. This point of view is especially fruitful in the cases when interpretation of the wave function is not clear, for example in quantum cosmology II) . However, as stressed by Bohm and Hiley in [8], one may take the 'ontological' stand and assume that the trajectories are real, that is that there exists a 'real' particle moving along the trajectories. It should be mentioned that even if such an interpretation of quantum mechanics contains 'hidden variables' (the trajectory itself) it docs not contradict the Bell theorem 9l , because the theory is clearly non-local. This fact is not by itself very much fearsome since, as careful analysis of Einstein- Podolsky·· Rosen paradox shows, in the standard Bohr - Von Neumann interpretation of quantum mechanics, the theory is non-local as well. Let me now turn to morc complex example, which I will call
3.2.
Cosmo quantum mecha,nics
In this subsection I will consider an example of the so-called one dimensional parametric systems. Systems like that possess a symmetry which makes them independent. of a paramet.rization of the world line; this symmetry is the one dimensional diffeomorphism invariance which is analogous t.o the diffeomorphism invariance in four dimensions. On the classical level the whole dynamics of the parametric particle is given by one constraint being the hamiltonian; in quantum case, according to the Dirac met.hod of quantization of constrained systems, the hamiltonian operator must annihilate physical state, and thus t.he wave funet.ion is, by virtue of the Schrodinger equation, time independent. Now consider a parametric model wit.h simple hamiltonian (which nevert.heless is of importance in quantum cosmology, see [11])1
where gij may 1)(' x-depeudenl. Let us assume again that decom posi tion Ijt
Ijt
has the following polar
= R(xi)exp (kS(xi))
(8)
with both Rand S real. Inserting (8) into (7), we obtain two equations corresponding to real and imaginary part, respeet.ively. These equat.ions read
h[S(x)]
1 85 as . h2 1 = -2.%~-;--) . + V(x') = -2 -R OR , uX' (Xl , i j as aR - 0 R O., +2g ~~- . l
ux' ux
(9) (10)
1 In what follows all quantum mechanical operators will be written in sans serif type face: a, b, ... , A. B etc.
J. Kowalski Glikman Equation (10) will not concern us anymore. As in the previous example it corresponds to probability conservation. We assume that the wave function IjI is a solution of equation (7), and thus this equation is identically satisfied (even though we may not know what the explicit form of the wavefunction is.) On the other hand, equation (9) is of crucial importance. This equation can be used to derive the time dependence and then serves as the evolutionary equation in the formalism. For, let us introduce time I through the following equation
dx i _ ij 5'H[S'(x)] dl - g 5(fJS'j8x j )'
(II)
This equation defines the trajectory xi(t) in terms of the phase of the wavefunction S. Now we can substitute back equation (II) to (9). Assuming that the matrix gi j dX') Ilas t he ·Inverse gij, we fim(I (x. i = dt I . . 2%X'X)
+ V(x
i)
=
n?
I
2R DH .
(12)
We see therefore that the quantum evolution differs from the classical one only by the presence of the quantum potential term
on the right hand side of equation of Illotion. Since we assume that the wave function is known, the quantum potential term is known as well. Equation (12) is not in the form which is convenient for our further investigations. To obtain the desired form, we define classical momenta
and cast equation (12) to the form I ..
'H =. _g'J p 'P 2' 'J
.
+ V(x')
tt 2 I - --DR = O. 2 R
( 13)
We regard 'H as the generator of dynamics acting through the Hamilton equations Pi
x'
a'H ox i 8'H
(14)
°Pi
The time evolution is therefore governed by equations (14) subject to the constraint for initial conditions (13). This completes the technical part. However a number of remarks is in order.
Quantum Potential and Quantum Gravity
235
1. The quantum potential interpretation may be used to obtain a well defined semi-classical approximation to quantum theory. Indeed, it can be said that the system enters the (semi-) classical regime if the quantum potential is much smaller than the regular potential tern!. 2. One of the major advantages of the quantum potential approach is that it provides one with an affective and simple way of introducing time even if the system under consideration has a hamiltonian as one of the constraints. In particular this approach serves as a possible route to final understanding of the problem of time in quantum gravity. 3. Related to this is the problem of interpretation of wave functions which are real. This problem has been a subject of numerous investigations, but from the point of view of quantum potential the resolution of it is quite simple. We just say that real wavefunctions (of the universe) represent a model without time evolution (and therefore time) at al1. 2 It is clear from the formalism: :i; is just equal to zero, so nothing evolves and therefore there is no clock to measure time. On the other hand, equation (12) means that for the real wave function the system settles down to the configuration for which the total potential (i.e., classical plus quantum) is equal to zero. But there is one important modification of the theory for real wave functions. Such wave functions, which form a degenerate subclass of all wave functions being solutions of the theory, should be regarded as ones which impose additional constraints on the quantum potential theory, namely that the momentum is zero. This constraint, together with H = 0 form a second class system. This means, first, that one can take them as strong equalities (up to the standard manipulations with Dirac bracket), and this leads to the abovementioned equality of classical and quantum potential. What is more important, however, since the hamiltonian is now a second class constraint, it does not generate gauge transformations (time reparametrization) any more. Roughly speaking, the quantum potential theory is "anomalous" for real wave functions. It should be mentioned that similar effect tums out to be present in the full quantum gravity theory, and without the interpretation given above it may lead to apparent paradoxes, one of whose was presented in [12]. I will discuss the quantum gravity case below. 4. The definition of time by equation (11) is, of course, not unique. In fact, we can use a more general expression ·i
x
= N( ) 8H[S(x)] t 8(8SjfJx i ) '
(15)
2Recall that the standard (time-dependent) Schrodinger equation does not have any real solutions.
J. Kowalski-Glikman
236
where, in the case of gravity. the function N is to be identified with the lapse function of ADM formalism. 4.
Wheeler - De Witt equation and its formal solution
Now we can turn to the real thing .. the quantum theory of gravity. This theory is defined by two sets of constraints, one of whose are generators of three dimensional diffeomorphisms operators: Ha = -2\7bP~, (16) and the second consists of a single but very complicated operator, called the Wheeler -- De Witt operator
(17) Here ft = (16rrG)-I, G is the Newton's constant, (18)
pab are momentum operators related to the three metric hab, (3)'i? is the three dimensional curvature scalar, and A the cosmological constant. As it stands, this operator is meaningless. It contains the product of two functional derivatives acting at the same point; as a result, while acting on a wave function, the product of two delta functions at the same point appears. This is, of course standard property of second order operators in Schriidinger representation. The way out of this problem is to regularize the kinetic term of the operator, that is, to replace Il WDW above by
=..!....[:, 2ft reg +flVh (2A -
(3)'i?)
,
(19)
where in the limit when the parameter t goes to zero,
limKabcd(x,x';t) = Yabcd(:r)b(x - x'). 1-0
(20)
The function K can be found, for example, by using the heat kernel equation
8 ' ) =\7(x·)K-(x,x;t 2 , ) 8tK-(x,x;t
(21)
with the initial condition (20). This approach was proposed in [13] in the context of Yang - Mills theory, and then applied to the theory of gravity in [14]. In this latter paper the action of the regularized WDW operator on some simple expressions built from
Quantum Potential and Quantum Gravity
237
the metric and curvature was also found. The virtue of this approach is that mathematically meaningless expressions (like products of delta functions at the same point) are now replaced by well controllable singularities of K(x; t) = lim x _ x ' K(x, x'; t) for small t. Each singular ternl can be then replaced by a renormalization constant. The problem is however that at the first glance it seems that the number of such independent constants is infinite. However, the action of the operator has been tested on some simple expressions (sec [14]) and one may hope that the physical wave functions, i.e., the wavefunctions annihilated by all constraints (or, to be more precise, the matrix clements constructed with the help of them) will eventually contain only finite number of them (hepefully two - one corresponding to the renormalization of the :"Jewton's constant, and second to the renormalization of the cosmological constant.)3 Be as it may, let us assume that we have constructed the regularized Wheeler - De Witt operator. Then one can easily write down the formal expression for the physical wave function being annihilated by this operator. This can be done by observing that a physical state is of the form 4
(22) where ljJ[h] is any wave function(al) of the metric, that is
(23) Even though the expression above is very formal, it can be used to obtain a perturbative expansion (in J-l) of a solution. Second, and this will be quite important in our analysis below, using this formula we can find out some properties of the physical wave function. The next important property of the formula (2:3) is that it may be used as the starting for construction of the inner product. It should be recalled that one of the major problem of the Dirac quantization procedure is that it does not offer any constructive way of Ending what the inner product of physical states is. I will not dwell on this problem any longer as it is the subject of the forthcoming papcr [I.')]. Let me only mention that somc partial results and discussion are contained in [11]. Now we are ready to extend the discussion of one dimensional parametric systems presented in subsection :l.l to the case of full theory of gravity. This will by the subject of the next section. would like to thank .Jeff Greensite for long discussion concerning this point. 4The commutator of two wnw operators is proportional to the diffeomorphism operators (see [14] for discussion of the algebra of regularized constraints in quantum gravity.) Therefore it is not needed to apply diffeomorphism constraints separately. 3(
J. Kowalski-Glikman
238
5.
Quantum potential for quantum gravity
As in subsection :1.1 we start with decoIIlposition of the wave function to the polar form (from now on h == G == c == I)
I/I[h] == R[h]eiS[hJ • Substituting this into the WDW equation, one easily finds (sec also [12])
_ ~K 8S 8S :2 /l abed 8h ab 81!ed
6 + /lVrrII (,2A _ (3)) n +~ 2 /l
re9
].>
R _
O.
-
(24)
t
If one identifies momenta with the (functional) gradient of S, to wit p ab( x) =
oS ( x ) , cy;U
(25)
ab
this equation turns to the lIamiltonJacobi equation for general relativity _
~I'
2/l '>Iabed
pabped
+ IIVh (211 r'
_
(3)n) + ~ 6 r <9 R = 0 2/l
Ii
'
(26)
with the additional last term corresponding to quantum potential. This equation (without the potential term, of course) was first analyzed by Gerlach 16). Equation (26) is only one of two equations which result from action of the WDW operator on the wave function. As in the case of the parametric particle mechanics, we ignore the second equation, which could be interpreted as an equation shaping the quantum potential and/or guaranteeing conservation of probability. The wave function is subject to the second set of equations, namely the ones enforcing the three dimensional diffeomorphism invariance. These equations read (after decomposing into real and imaginary part) a 8S a V' == V' Pab = 0 oh ab
(27) (28)
Each of these equations has different interpretation. The second expresses invariance of R with respect to spatial diffeomorphisms - this invariance is actually guaranteed by the fact that Ii is the modulus of a wave function being a solution 0 WDW equation; the first is to be interpreted as a constraint equation. Thus our theory is defined by two equations (26) and (27). ~ow we can follow without any alternations the derivation of Gerlach 16) to obtain the full set of ten equations governing the quantum gravity theory in quantum potential approach
o=
Ha
== V' a pab ,
(29)
Quantum Potential and Quantum Gravity
~r.
pabpcd
239
+ VIh (2A _ (3)R) + ~ 6,e9 Ii
0= Hl-
_
;tab(X,t)
{hab(x, t), H[N,
(31)
pab(X, t)
{pab(X, t),
(32)
2Jl '::tabcd
I/.
r
2Jl
n
Nl}, H[N, N]}.
Ii
'
(30)
In equations above, {*, *} is the usual Poisson bracket, and
H[N, N] =
J
d3 x (N(x)Hl-(x)
+ Na(x)Ha(x))
(33)
is the total hamiltonian (which is a combination of constraints). A number of comments is in order 1. Let us observe that we somehow "solved" the problem of time in quantum gen-
eral relativity. Indeed out of sudden time appears in Eqs. (1), (2). It is a matter of taste if this is to be considered as a solution of this problem, nevertheless, in this formulation the problem of time in quantum general relativity is as simple (or as difficult) as the analogous problem in classical theory of gravitation (see
[2].) 2. The very important problem related to Eqs. (29 - 2) is under which conditions these equations are equivalent to ten four dimensional covariant equations (" quantum Einstein equations".) To see what this problem is about, let us observe that Eqs. (29), (30) are constraint equations. This means that 1ta and Hlmust have (weakly) vanishing Poisson bracket with H[N, N]. But this means that Poisson brackets of H a and Hl- must form closed algebra. Yet it is well known that it is quite difficult to close the bracket
{Hl-(x), Hl-(Y)} for arbitrary quantum potential. Indeed one can easily check that this Poisson bracket does not close if Hl. contains, for example, terms quadratic in curvature. s I\'ow, a simple inspection of Eq. (23) shows that r]Jphys will contain higher order terms. It should be observed however that quantum potential term 6. rJi R is essentially non local and the standard argument may be not applicable. This problem will be the subject of the separate paper.
-h
Assuming however that for some quantum potentials the bracket above does not close. Then we have to do with secondary constraints which will be second class (this follows from simple counting of degrees of freedom.) This just means that the 3 + I symmetry is not promoted to the full four dimensional diffeomorphism 5This fact can be understood by observing that the effecti ve theory in four dimensions is, in this case, a higher derivative theory.
240
.J. Kowalski Glikman invariance after quantization. This fact is without doubts very important, for example it may be related to the appearance of minimal length in quantum gravity (see [17]). [t may also mean that either 3 + I splitting being the first step of quantization procedure is incorrect and/or that the four dimensional diffeomorphism invariance is just a low-energy phenomenon.
;3. Talking anomalies, let us return to the apparent puzzle discussed in [12]. [n this paper the author argues that for real solution of wnw equation there exists the transformation generated by the hamiltonian H.l which transforms the solution of Eqs. (I), (2) into configuration which is not a solution. But if the wave function we start with is real, the momentum of gravitational field vanishes. Then the quantum potential must compensate the classical "potential term"- the three-curvature. As in the case of parametric particle, this means that the constraint H.l together with pab becomes a second class system and the time shift invariance of the theory is lost. 4. Let me stress once again that one of the crucial points of the construction is an appropriate regularization of the kinetic (derivative) term in H.l' This problem is not discussed in most papers concerning (semiclassical) quantum gravity in metric representation. However without proper understanding of regularization and renormalization of the theory it is impossible to find any solution and to even start the program described above. 5. Finally, it is feasible to extend the above discussion to the case of Ashtekar variable (in either connection or tetrad representation.) This is especially important because one clearly should include matter fields and it is well known that we need tetrad formalism to take care of fermions (for bosonic matter fields there are only obvious technical modifications of the formalism.) To conclude, the quantum potential program may provide us with an important insight into the meaning of the physical wave function of quantum gravity. However, we need to find a wave function of the universe first. As for now, therefore, this set of ideas can be only applied in the context of the minisuperspace models. Let us hope that this situation will soon change. Acknowledgments
I would like to thank Prof. .Jeff Greensite and Mr. Arkadiusz Alaut for discussions concerning the problems discussed in this paper.
REFERENCES [1] C.J. Isham in Canonical Quantum Gml1ity: from Classical to Quantum, Springer Verlag, Berlin, 1994.
Quantum Potential and Quantum Gravity
241
[2] C.J. Isham in Integrable Systems, Quantum Groups, and Quantum Field Theories, L.A. Ibort and M.A. Rodriguez eds., Kluwer Academic Publishers, Dordrecht,1993. [3] C.J. Isham Structural Issues in Quantum Gravity, GR14 plenary session lecture, gr-qc 9.510063. [4J A. Ashtekar, Non-Pertw'bative Canonical Quantum Gravity, World Scientific, Singapore, 1991. [5] C. Rovelli, Class. Quant. Grav. 8,1613 (1991). [6] C. Rovelli, L. Smolin Nuel. Phys. B 442,593, (1995). [7] Bryce S. De Witt, Phys. Rev. 160, 1113, (1967). Rohm, Phys. Rev. 85, 166, (1952), Phys. Rev. 85, 180, (19.52); Bohm and B.J. Hiley Phys. Rep. 144,323, (1987), Bolllll, R.J. Hiley, and P.]\;. Koloyerou Phys. Rev. 144,349, (1987); Bohm, R.J. lIiley, The Undivided Universe: An Ontological Interpretation of Quantum TheoT'y, Routledge & Kegan Paul, London, 1993.
[8] D. D. D. D.
[9J J.S. Bell Speakabh and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987. [10] K. Berndl, M. Daumer, D. Diirr, S. Goldstein, :'I. Zanghi, A Survey of Bohmian Mechanics, to appear in II Nuovo Cimento and references therein. [11] A. Blaut and J. Kowalski-Glikman, gr-qc 9509040, to appear in Class. Quant. Grav. [12J YII. Shtanov Pilot Wave Quantum Gravity, gr-qc 9503005 [13] Paul Mansfield, Nuc/. Phys. B 418, 11:1 (1994). [14] T. lIorigllchi, 1\. Maeda, M. Sakamoto, Phys. Lett. 344 B, 105, (1995). [IS] .J. Kowalski-Glikman On the Inner Product for Dirac Quantization Scheme, in preparation. [16] Ulrich II. Gerlach, Phys. Rev. 177, 1929 (1969). [17] L.J. Garay, Int. J. Mod. Phys. A 10, 145 (1995).
From Field Theory to Quantum Groups
THE GENERAL FORM OF THE LAGRANGE FUNCTION FOR CLASSICAL TWO PARTICLE EQUATIONS OF MOTION COVARIANT UNDER GALILEI TRANSFORMATIONS J.
LOPUSZANSKIt AND
P.C.
STICH ELl
t Institute
of Theoretical Physics University of Wroclaw PL-50204 Wroclaw, Poland
t Fakultiit
fur Physik Universitiit Bielefeld D-33615 Bielefeld, Germany
Abstract: We are concerned with a system of two nonrelativistic classical massive point particles in 3-dimensional Euclidean space. We assume that the Lagrange function describing the motion of these particles, exists, docs not depend explicitly on time (autonomous system) and is regular. Further the equations of motion transform covariantly under the Galilei group transformations and the classical Poisson brackets of the position as well as velocity coordinates become under the special Galilei transformation a multiple of the original ones. Taking these hypotheses into account we present the most general expression for the Lagrange function. We discuss the results obtained by us, pointing to the symmetry properties of the Lagrange function and properties of the classical Poisson brackets. The present paper generalizes the results of D.R. Grigore [1] in an elementary manner.
1.
Introduction
The main goal pursued by us in this paper is to discuss in an elementary manner the case of two nonrclativistic classical massive point particles moving in a 3dimensional Euclidean space. We assume that this motion is controlled by a Lagrange function which does not depend explicitly on time and is regular. We assume further that the equations of motion transform covariantly under the Galilei group transformations and the classical Poisson brackets of the position and velocity coordi-
J. Lopuszariski and P.C. Stichel
244
nates become under the special Galilei transformation a multiple of the original ones. All these assumptions are pretty restrictive. For instance, the equations of motion do not need to follow from a Lagrange function as its Euler-Lagrange equations nor the Lagrange function has to stay in the same invariance class (cf. the Theorem of Henneaux [2]) under the special Galilei transformations. These restrictive premises guarantee, however, that we still get some formulae in a closed form. We shall try to relax the imposed restriction in our future work so that still the computations do not get intricate. It turns out that the most general form of the Lagrange function, obtained by us, is in general not invariant under the Galilei group transformations. In contrast to the results of D.R. Grigore [1], the Lagrange function need even not be invariant with respect to space rotations. Such a peculiar behaviour is well known for the one-particle case, as shown by Houard [3] and subsequently in an elementary manner by the present authors [4]. We do not claim that all the results obtained by us are new. Similar problems were already tackled by Helmholtz. What is new is the presentation used here by us.
2.
Assumptions and comments
The subject of our investigation is a system of two classical massive point particles moving in a 3-dimensional Euclidean space. We denote the coordinates of the A - th particle, A = 1,2 by
their velocities by
where t stands for the time variable and We make the following assumptions:
J:{A)
is understood to be a function of time.
i) The Lagrange function, describing the dynamics of the system of these two particles, docs not depend explicitly on time; in other words L = L(j:,
il, J:, Ii)
(1 )
,
where M=Lm(Al,
A = 1,2,
(2)
(A)
(3)
and m(A) are some constants> O. They can be, in principle, arbitrary constants for which M i- O.
245
The General Form of the Lagrange Function ii) The canonical momenta
aL
A_ = O.(A)
.
Pi
(4)
x-
do not vanish a.e., as well as
which ensures the invertibility between the variables (.:f(A), j;.lA)) and (.:f(A), [/A»). iii) The equations of motion are covariant under the Galilei transformations. To avoid confusion let us state explicitly what we mean by this covariance. For the Euler-Lagrange equations
L
Aj:R)(.:f(I),.:f(2)i(l),i(2»)xiR) + BjA)(.:f(I),.:f(2),i(l),i(2») = 0
(R)
we require ' " A(AB)( ~
jk
.:f
(I)'
,.:f
(2)" (I)' . (2)') .. /(B) .:f ,.:f xk
+ B(A)( (I)' (2)" (I)' . (2)') _ j.:f,.:f,.:f,.:f-
(8)
= R
Jk
{A(AB)(x(l) x(2)x(l) x(2»)x(B) kl
-
,-
-
,-
I
+ B(A)(x(l) x(2) x(1) x(2»)} k ,- ,- ,-
where xj
= RjkXk
+ Uj + Vjt
and the real matrix R = R = (R-I)T det R = 1 represents rotation in 3dimensional Euclidean space, IJ. and 11. are constant vectors, corresponding to translations and special Galilei transformation resp .. iv) Under the special Galilei transformation the classical Poisson brackets of .:f(A) and i(B) become a k-multiple of the original ones. In other words we have >l
[
X(A) 1
x(B)] 'J
(A).(B)]
L
(A)
~
0
~ a (G) o( k,G x k
'"
(B) Xj _ )(G) PL k
L
k(Q-J[xlA),x;Btv
.(A) . (B)] ,Xj L [ xi
k( Q ) [XI.(A) , X.(B)] L1) J
[Xi
,Xj
0 (B) 0 (A) ~ xi 0 xk(C) a( PL )(G) k i,j=1,2,3
=k
v
[X(A),x(B)]
U,
J
Lv
A,B=1,2
(5)
where k(Q-J is a constant and Lv
According to (4)
==
L(i,E + 11.,.:f,E. + 11.t) .
(6)
246
J. Lopuszariski and P.C. Stichel
v) We assume that the functions under consideration are smooth a.e., unless otherwise stated. Some of these assumptions require a comment. In classical mechanics one relates usual1y the term-equations of motion - to the Kewton equations, viz. ..(A) _ j(A)( (1) (2) • (1) • (2») j J;. ,J;. ,J;. ,J;. . Xj
If the third law of Newton is not satisfied, i.e. there do not exist such which
(7) m(A)
> 0 for (8)
then the relation between the masses of the particles remains indefinable. As in our considerations the case when (8) is not satisfied will occur, the constants m(A), appearing in (2), do not need necessarily to be considered as masses of the particles. The next observation concerns also the equations of motion. These equations, written in the Newtonian form (7), do not need to fol1ow from a Lagrange function as its Euler-Lagrange equations. It can, however, wel1 happen that the equations, equivalent to the original Newtonian equations, i.e. equations belonging to the same set of solutions as the Newton equations, can be deduced from a certain Lagrange function as their variations. As a matter of fact there can be several Lagrange functions yielding the same equations of motion. To make things clearer let us consider a simple one-dimensional mechanical model obeying the equation x - j3x = 0 . (9) There does not exist a Lagrange function yielding (9) (friction!). Nevertheless, the Lagrange functions £1
= xlnlxl + j3x
or
has as its Euler-Lagrange equation
T(x, x)(x - j3x) = 0 where for L 1
and for 1. 2
' j3x 7 2=1---;-. x
(10)
247
The General Form of the Lagrange Function It is obvious that (10) is equivalent to (9) a.e .. It can be easily shown that
L
2
aLl
_
=I
dcP(x, t) dt
where a is a constant as well as that [x, IlL, :::: I
=I a[x, IlL,
::::
a~(J x - x
.
The last two observations bring us to the further comment concerning the use of Poisson brackets in the assumption iv). The motivation is the Theorem of M. Henneaux which reads as follows: if Land L' are two Lagrange functions yielding equivalent equations of motion or equivalent Euler-Lagrange equations and the same Poisson brackets in the sense of relation (5), i.e. (A) • (B)] _ k[ (A) . (B)] , etc. [ x, ,xJ L X, ,xJ L then
,_ dcP _ (I) (2) L -kL+dt,cP-cP(;£ ,;£ ,t).
(11)
The rcverse statcment is also true. From this thcorem follows that two different Lagrange functions, yielding the same equations of motion, but whose corresponding Poisson brackets do not conform to (5), do not satisfy (II). Thus thc Poisson brackets can be uscd to label incquivalent classes of Lagrange functions in the sense of relation (11). To illustrate this theorem let us consider the following four Lagrange functions of one particle 1'2 1'2
..
1'2
..
1'2
"
2";£ , 2"x l + X2 X3, 2"x 2 + XIX3, 2"x 3 + XlX2 yielding all precisely the same equations of motion, viz.
These Lagrange functions are not equivalcnt in the sense of the quoted theorem, as e.g.
[XI,I2]
0,0,0 and 1 resp.
[X2' I3]
0, 1,0 and
°
resp ..
In other words, the Euler-Lagrange equations read
(Ii/ij) whcre T::::
U° n,u 0 I
° 1),(1 0
1
n,o ° n ° 0
1
I 0
resp ..
248
J. Lopuszanski and P.C. Stichel
Therefore these Lagrange functions do not satisfy the relation (11). A good illustration to this theorem gives also our one-dimensional example, discussed before, where the Lagrange functions L 1 and L 2 belong to two inequivalent classes and the Poisson brackets, corresponding to them, differ nontrivially from each other. The physical sense of our assumption iv) is that the Poisson brackets remain the same for a resting and uniformly moving frames of reference (up to the multiplicative constant k). The last comment concerns assumption iii). This hypothesis, in the form chosen here by us, is quite restrictive. The weaker assumption requiring that the equations of motion in the i'iewtonian form (but not necessarily the Euler-Lagrange equations itselves) are Galilei covariant, would created an opportunity to enlarge the class of Lagrange functions leading to Euler-Lagrange equations having the same set of solutions. A good example is again our one-dimensional model discussed before. It is easy to check that equation (9) remains unchanged under the transformation X'
= x + ~el3t ~
where (} is a constant. However, neither equation (10) with T = T1 or T = T 2 nor the corresponding Poisson brackets stay invariant with respect to this change of variable. The reason not to weaken our requirement iii) is that the general problem, tackled here by us, would in this case become too complicated and intricate to get clear-cut results, if any.
3.
Question to be answered
What is the most general form of the Lagrange function, which yields Galilei covariant equations in the sense of iii) as its Euler-Lagrange equations and satisfies the requirement i), ii), iv) and v).
4. 4.1.
Attempt to give an answer to the posed question The Lagrange functions
i) Because of assumptions iii) and iv) we conclude, taking into account the Theorem of Henneaux, that
( 12)
Here Lv is given by (6). For obvious reasons we have to have
The General Form of the Lagrange Function
249 (13)
It follows from (12) for small I.!!.I, taking into account (13), that
Let us differentiate both sides of (14) twice with respect to t. We obtain
(15 ) Here we made use of assumption i) and introduced the following notation
of
ok
j ~Iv=o uv· - - == k
j ~Iv=o uv· - - == F
Expression (15) is linear in ;i;, and consequently we have
.
(16)
J
}
k and holds true for every choice of these variables, (17) (18)
and
F) - 2 ~Fot2 }
+ Vlt + F 2 }
}'
(19)
It follows from (17), (18) and (19) that Fjo
= const.
(20)
and (21 ) Let us insert (19) into (14) keeping in mind (20) and (21). We get
J. Lopuszariski and P.C. Stichel
250
or, handling t as an independent parameter
in . oFj2 . - . - kjL = --Xk oRj OXk oL oFjl . oR] = OXk Xk
+
oF
.
,
j2 + --R k + rj1
(22)
oRk
oFjl · • ORk Rk + /'jO
(23)
.
The integrability condition for (23) reads
0 2Fjl . OXkORe Xk
[j21'~1'
+
8 2 Fl1 . k oRkoReR = OXkoRJ Xk
8 2 Fn
.
+ 8Rk 8Rj Rk
or
Hence
oFjl 8Fn 8R - oR = 2aje = -2aej = con,~t . j e
(24)
The most general solution of (24) reads (25) Let us insert (25) into (23). We get
in oR J
021jJ.
o21jJ.
.
,
= ORJOXk Xk + oRjoRk R k + ajkRk + f jo
This expression integrated with respect to L =
.
E. yields
d1/J.
. .
dt + ajkRj Rk + FjoRj + >"(;r,;r, R)
.
If we gauge away the first term (this amounts to 1/J = 0) we get the new Lagrange function (26) as well as froIll (25) Fj1 = ajkRk .
(27)
We assume>.. to be a smooth function a.eoo From (26), (27) and (22) follows, by differentiating the latter with respect to 11,
. 0 2Fl2 aje - keajkRk - kerjo = ---Xk OXkoRj
+
8 2 Fn oRkoRj
. Rk + alj
The General Form of the Lagrange Function
251
or
(28)
2ajt = +ktFjo
fPF/. 2
(29)
oXkoR j = 0 8 2 Fn oRkEJR j
= -ktajk
(30)
.
Because of (24) and (30) Then or ~otice
(3)k j
kj == 0
0')
either
of.
0
and
f: o =
(31) (32)
0.
that in both cases
ajk = 0 .
(33)
The latter implies, according to (26) and (27),
L=
FjoRj
+ ..\(.f.,,t, R)
(34)
and
Fjl
= o.
In case 0) (kJ == 0) we have to distinguish between two possibilities
0)0)
PjO
0)(3)
Fjo
0
of. o.
In case 0')0') we have a translationally invariant Lagrange function o. We shall return to this case below. In case 0')(3) the Euler-Lagrange equations with respect to 11, read
L as
well as
kj ==
(36) According to assumption iii) these equations should be rotationally covariant. It is shown in the Appcndix A, that this entails Fjo = O. Thus wc may discard thc case
0' )(3). In case (3) (kj of. 0, Fjo = 0) the Lagrangc function L is again translationally invariant. It is shown in the Appendix II that in this case we run into contradiction with assumption ii). Thus we arc left with the case 0')0), namely
(37)
J. Lopuszariski and P.C. Stichel
252
From (29) and (30), taking into account (33), follows (38) where
Cjk
is a constant matrix. From (22), (35) and (38) we read off that (L = L) (39)
The integrability condition for (39) is
(40) Then from (39) follows (41 ) where we assume £ to be smooth function a.e.. For the Lagrange function (41) we have
k = 1. I1itherto b,J;f) as well as £(;;:.,;;:.) were arbitrary functions. So far we did not yet exploit fully the assumption iii), which allows to impose some restrictions upon these functions. The equations of motion with respect to 11 read
,
..
2c;jRj
ah; ..
a 2 h;
..
+ --;:,-Xj + ~XjXk =0 . UX UXJUXk
(42)
J
Because of rot.ational covariance we should have
(43) We normalize the Lagrange functions by requiring
(44) The rotational covariance entails also r
== 1;;:.1
(45)
The General Form of the Lagrange Function where
Q
253
is constant. Notice that
.. MR= _cPh _J
(46)
dt 2
J
violates the third law of Newton as soon as
d2 h dt 2J =f- 0 . Equation (46) was obtained from (42), (43) and (44). Equations of motion with respect to ;r yield (47)
Here we made use of (46).
4.2.
Equations of motion and the two-particle force
We are free to express f in (41) as f
1m = - 2
(1)
m
(2)
.2
M!f.
_
U(
.) ;r,;r .
(48)
U being again a smooth function a.e.. This we do for obvious reasons to adjust our scheme to the standard one. By virtue of (43), (44), (45) as well as (48) the Lagrange function (41) reads
The equations of motion with respect to the variables
;r(A),
A = 1,2, read
(50)
where the index A is understood modulo 2 and
254
J. Lopuszanski and P.C. Stichel
Here we madt' ust' of
ah i aXj
ah j
= ah i
.
To make the equations (50) Galilei covariant, in accordance with the assumption iii), we have to demand that 1 (51 )
as well as
iJU d (iJU) dt
OXi
(52)
OXi
transform under the rotations as a covariant tensor and vector resp. Notice that U itself does not need to be a scalar, as will be made plain in an example below. Equations (50) can be solved with respect to i(A) almost everywhere in the space of if(A) and ;i:(A). Thus the equations (50) can be brought into the form of Newton's equations; as mentioned earlier there does not need, however, to exist a Lagrange function which has the latter as its Euler-Lagrange equations, e.g. in case, when the third law of Newton is violated. To avoid tedious computations, resulting in rather intricate final formulae, we are going to inspect two special simpler cases, namely when m(1)
1 = m(2) = -M == m
2
.
(53)
Notice that this can be always achieved in case when the third law of Newton is violated. In case both particles under consideration are identical, the equation of motion should not change when if(1) is replaced by if(2) and vice versa. It follows then immediately from (2), (3), (45) and (49) that d2 h· dl 2J (-x)
=
d2 h· dt 2J (x)
d2
= - dl 2 (x j h(r))
d2 = dt 2 (x j h(r))
=0
or
(54)
her) = 0 . By virtue of (53) and (54) equations (50) reduce to
(
8 IJ m
_~) .. (A) ;). [J'
UXi
Xj
XJ
~ .. (A+l)
+ a'Xi a'Xj x
J
-
-
(
(A)[JU [J
Xi
a2(j
(A)
+(
n' f)
UXi
Xj
'. XJ •
If we add two equations for fixed i and A = 1,2 we get from (55) m
LiJA) =
0 .
(A) 1 Notice
that
i.(A),
A = 1,2, appears only (linearly) in the two terms on the l.h.s. of (50).
(55)
The General Form of the Lagrange Function
255
Hence the third law of l'\ewton is here satisfied. Let us choose U in (55) as
U = AI(;rJXI + B(;r,) . Then from (55) follows (56) It is clear, that as soon as as we have
A transforms under the rotations as an vector, i.e. as soon Al(;r,) = xIA(r),
the first term on the r.h.s. of (56) docs not contribute. To get a nonvanishing contribution of this term we have to give up A being a rotationally covariant vector and consequently U being a scalar. If (53) remains valid but the particles are no longer identical the terms involving h do not, in general, vanish. To make things easier let us assume in this case, III addition to (53), that U = U(r) . Then equations (50) reduce to r
(
Vi'Tn J
ahi) x· oo(A) + {(A) -aXj J
= -(
(A)
Xi dU
2
1 a hi . .
- - - ----XkXj r dr 2 aXkaXj
(57)
Let us introduce the following notation
This matrix is Galilei covariant. The inverse matrix to etA) reads 2 (58) and is also Galilei covariant. Hence we get from (57) and (58)
(59) 2These matrices are not invertible for values of ;r(A) for which thus we have to exclude e.g. f(A)m + h = Cr- 1 , C a constant.
f(Al m
+ h(r) = 0 or = -r¥;
256
J. Lopuszariski and P.C. Stichel
where Xi
- m + ((Alh
{c(AldU lxsxsdh lxsXsXtXtd2h ----;- dr + 2"-r-dl' + 2 r2 dr 2
((A)Xk4!!.
- r(m
+ ((A)h :r((A)r~)
1 XsXsXk dh +2"--r- dr
[((Al Xk dU
XkXtXt dh
-d;:- + --I'-dr
-1-'
2
1 XsXsXtXtXk d h]} r2 dr2
+ 2"
Xi X/Xt dh m+c(Alh r dr'
(60)
It is easy to see that there do not exist, in general, such m(A)
> 0 that
I.e. the third law of Newton is not satisfied. Although there does not exist, in general, a Lagrange function corresponding to (59) and (60), there exist nevertheless a Lagrange function (49), giving rise to equation of motion (57), the set of solutions of which coincides with the set of solutions of (59) and (60). Let us go back to the case of Lagrange function (41), taking into account (43), (44) and (45). The equations of motion with respect to fl read (see (42))
idt (Mk + dhj(;r)) dt
=
O.
J
This implies
L(m(A)8ij (Al Tn
+ c(A)mij)x
J
= Ci = cont.
XiXj dh(r) = 8··h(r) +- -dr'J I'
'J -
a generalized notion of conservation of the motion of the mass center in case the third law of !\'ewton does not need to be observed.
4.3.
Invariance properties of the Lagrange function and the Poisson brackets
The Lagrange function (49) consists of four terms, the first term of which,
is invariant under rotations and translations but is not invariant under special Galilei transformations, The second term
U(;r, ;1;.)
257
The General Form of the Lagrange Function
is invariant under translations as well as under special Galilei transformations, it does not need, however, to be invariant under rotations. The third and fourth terms, namely dhk(;r.) it
----;[t
k
are invariant under rotations as well as translations, but are not invariant under special Galilei transformations. To construct the Poisson or Lagrange brackets, the latter reciprocal to the former ones, we need the notion of canonical momenta, viz. (A) = oL Pi - fJx .. (A)
_ -
TIl
(A)' A _ (A) ( Xf
l
aU
m(A)
ahl . .
(A)R'
a'Xl + M aXj X J + (
ohk OXl
k "
•
Then the Lagrange brackets are as follows
(61 )
( Xi A) ,x. j(8))
(x(A) x(B))
=
(x(B) x(A))
m(A)t5
_
(A)(B)~ +
)
,
t
AB
t5.
1
,
J
=
aXjax j
l)
1
+_(m(A)(B)
M
(62)
fJh + m(B)(A))_' aXj
.(A) .(B)) ( Xl ,.T
= 0
J
•
(63)
To get (61) and (62) we made use of
Dh j
_
OXj -
oh j OXj •
The relations (61), (62) and (63) are obviously invariant under special Galilei transformations, as requested by assumption iv), as well as under translations. They remain also unchanged under rotations. To show that notice that
LR
_
L(X(I)',;r.(2)',i:(I)',iY)') = L
+ U(;r., i) -
U(;r.',i')
J. Lopuszariski and P.C. Stichel
258 and (A)' _
Pi
iJL H
_
(A)
= ~ - Pi OX
(A)
- (
i
oU(;r', i') ') .,
( Xi
+(
(A)
au -a: . Xf
Here
X:
= RijXj
etc..
We conclude that the only thing to be proved is that (64) and
0 2 U (;r/, ;£')
a U(;r', ;£')
[PU(;r,i)
ox/3x J
aXjaXi
ax/h j
2
o2U(;r,i) OXjOXi
(65)
The relation (64) coincides with the requirement (51). Formulae (65) follow from (51) and (.52). This can be shown as follows. We demand that
OU
d OU
---aXi dt aXi transforms covariantly under rotations. Because of (64)
au
a2 u.
-----x· aXi OXiOXj J
(66)
should then also transform covariantly. From (66) follows that the same is true for (67) Using again (64) we conclude from (67) that (65) holds true. Similar results one gets for the correspol)ding Poisson brackets mutatis mutandis. The considerations presented above prove also that in case of rotations we have
L R. .
-
L = d1/J dt (x(1) - , -X(2) , -x(1) , -X(2) , t·, R) .
To summarize: we assumed that a set of Galilei covariant equations of motion derived from a Lagrange function is given. This Lagrange function L has the property not to depend explicitly on time t. Moreover, all Lagrange functions obtained from this given L by a special Galilei transformation stay in the same equivalence class as L. Then the most general Lagrange function satisfying these requirements reads (see (49))
259
The General Form of the Lagrange Function
where m(A) > 0 an' COllstants and (! as well as h(l') (see relation (45)) are arbitrary (smooth) functions. Then all Lagrange functions obtained from L by a Galilei transformation remain in the same equivalence class as L. Let us wind up with the following comment linked to the problem of the equivalence classes of Lagrange functions. The Galilei covariant equations of motion (50) were obtained from the Lagrange function (49). These equations can be put in the following form O(A,B) .. (B) + n(A) = 0 XJ
1)
Oii'B)
=
I
Oi:,B}(:[(1),:[(2),;i;(1),;i:,(2)),ni A )
,
=
n j(A),(:[(1},;r.(2),;i:,(1),;i:,(2)) ,
As we assumed Oi:,B) to be invertible a.e., so we may rewrite (68) Newton's equations, viz. itA) t
= _(0- 1 )(A,B) niB)
the form of
(69)
J'
I)
J[l
(68)
We may multiply both sides of (69) by some matrix (I) (2)' (I) . (2)) 1,(C,A)( kj :[ ,:[ ,:[ ,:[
with det T i- 0 a.e. and demand that the equations of motion obtained in this way follow from a Lagrange function L' as its Euler-Lagrange equations. This condition implies the following restrictive relations, viz. (PL'
=T(A,B)
8±(A)a±(B) •
(70)
'J
J
and ;PL' ±(B) iJ±(A} 8x(B) J •
J
_
iJL'
= _T(A,B)(O-I)(B,C)n(C)
8x(A)
'J
Jk
(71)
k'
•
Notice that neither (70) nor (71) need to be Galilei invariant anymore. The considerations of the solution of this problem will be left by us for future investigations. Acknowledgments Wer are grateful to Dr. J. Cislo for many valuable discussions and remarks. It is a pleasure to acknowledge the financial support of the Alexander von Humboldt Stiftung. One of us (J.L.) expresses his thanks for the warm hospitality extended to him during his stay at the University of Bielefeld and at the Max Planck-Institut fur Physik in Munich.
REFERENCES [1] D.R. Grigore, Int. J. Mod. Phys. A7 (1992) 7153. [2] M. IIcnneaux. Arm. Phys. (N. Y.) 140 (1982) 45.
[3] J.C. Houard, J. Math. Phys. 18 (1977) 502.
[4] ,I. Cislo, J. Lopuszanski, P.C. Stichel, to appear in: Fortschritte der J'hysik43 (1995).
260
J. Lopuszanski and P.C. Stichel
Appendix A: The covariance of equations of motion (36) written in extenso requires that
U<>2,\'
U
+ ( - 8Rjax~ + 8RjaR~
Tn (2))
R
M
:.(2) keXe
+
(A.I) 8 2\ A
_ R jk { (
-
82\ A
oR k 8x e + 8R k 8Rl
m
(1)) ..
M
(1)
xf
+
Here
\' ==A;£,;£,_ - \( , ., R")
A
where det R = I . Sincc,\ does not dcpend on ;¥:(A),A = 1,2, we get from (A.I)
(A.2)
8 2 ,\'
t "·, oR·Juxk
8 2 ,\
= RjsRkt
oR' s o'Xt
in other words, the usual tensor transformation. \[
8 2 ,\ -1<00
1 -.
8 RiJx k
thcn from
for
£=0
(A.3)
The General Form of the Lagrange Function
261
follows
Fjo = 0 .
(AA)
If, however. say, 02 ,\ 1 . - -f·(x x, -R) ' '" -. J -, -
oRjUXk
where
fj
#0
Xk
f
,
is a continuous function for which
(A.5)
for
then (AA) does not follow; even we can have
i.e. usual vector transformation with ~ # o. To show that (AA) always holds true we differentiate (A.5) with respect to and use the symmetry in the indices j and 8, namely
J!,(.!.- fj) aR Xk
=
s
R.
-!(.!.- fs) oR Xk j
These relations entail
0!J _ 0 fs
oRs - oR j or
f - ot1>(;£,i,R) J-
(A.6)
ak J
ep should be a continuous function of its arguments. Let us insert (A.6) in (A.5). This yields 0'\ 1 . (A.7) ~ = --;-t1>(;£, i, BJ + 1Pk(;£, i) .
UXk
Xk
Let us differentiate (A.2) with respect to;£' 03,\'
03,\
aRjaR~ox; = RjkRs/R " oRkoR/ox" t
and insert (A.7) in here. We get
or, after simple manipulation, "" = x.,,",,.,,, 1 '" .... t L..J ''\-t" --;- .... " x"
"(3 + x"R" t kQk + xt
J. Lopuszariski and P.C. Stichel
262
valid for any T,;£, if and any t. Let us choose for XI X2X3
Then
. I
rJ>1(£',!£',fi) = 0 for any choicc of i
i'
i'
0 such a rotation RP that
x: = x~ = 0
for
j, i, j = 1,2,3. By continuity we gct also
Finally from (A.5) an (A.6) follows '-l \I
· '" I VA • I1m L.Jxk-'-- = !1m
i:-Q
for arbitrary
;£'
and
k
k.
oRj8xk
8J.I(
i:-O
'¥
I
•I
;£,;£,
R'I
.
8Rj
. I'j = 0 = 11m
i:-O
This concludes the proof.
Appendix B: We are going to inspect the case whcn
and
Feo=O. Notice that in this case the Lagrange function (34) is
(B.l) From (22), (35) and (38) we read off that
a£ . 8h j . - . - kL = -Xk DR] J f}xk
.
+ 2ckRk J
(B.2)
where
hj = hj (;£) and
Cjn
is a constant matrix. To solve (8.2) with respect to
L we
use
The General Form of the Lagrange Function
263
or
(B.3) The r.h.s. of (H.:n depends linearly on ;i: and .il, the l.h.s. is symmetric with respect to the interchange of j and f. Therefore we have
kj uh i = k oh j i
OXk kjCik Cji
=
(BA)
OXk
(B.5)
kicjk
= Cij
(B.6)
.
From (HA) we have Hence
+ bj ,
hj = kj:.p(;£)
Q- a constant vector.
(B.7)
a - a con8tant.
(B.8)
From (H.5) and (8.6) follows
Let us insert (8.7) and (B.8) in (8.2). We get
aL
-
- . - kL =
oR j
J
ocp(x).
k----Xk J aXk
.
+ 2akk i Ri J
.
The solution reads
-
.
ocp
L = exp{kR}f(;£,;i:) - -;:;-Xk - 2a(1 UXk
.
+ kR)
or, discarding the gauge terms,
L=
(B.9)
exp{ kR}f(;£,;i:) .
The equations of motion with respect to
.. £(;£,;i:)( kR)
E. obtained
from (B.9) are
d
+ d/(;£,;i:) = a .
These equations are not rotationally covariant, irrespectively of t.he choice of the function f. This violates our assumption iii). The Lagrange function (8.9) is also not admissible, since it violates also assumption ii) as the Hessian 2 L
a
I uR;aRj 1= a
which entails also
From Field Theory to Quantum Groups
GREENSITE-HALPERN STABILIZATION OF AK SINGULARITIES IN THE N -+ 00 LIMIT
J.
MAEDER AND
W. RUHL
Department of Physics, University of Kaiserslalltern, P.O.Box 3049 67653 Kaiserslalltern, Germany E-mail: [email protected]
A bstraet: The Greensite-Halpern method of stabilizing bottomless Euclidean actions is applied to zerodimensional O( 1'\) sigma models with unstable A k singularities in the N = 00 limit. 1. Classical actions which are unbounded from below do not define Euclidean quantum field theories because the partition functions diverge. A method to modify the classical actions in such a fashion that convergence is guaranteed on the one hand whereas the classical actions are only minimally changed on the other hand has been proposed by Greensite and Halpern!). We refer to this method as "Greensite-Halpern stabilization". Modifications of a theory are considered minimal if the stabilized and the original "bottomless" theory have the same
I. classical limit; 2. perturbative series;
:J. IV
-+ 00
limit.
In [I] it has been proved for typical models that these requirements are indeed fulfilled. The Greensite-lialpern stabilization applied to a stable theory leaves it unchanged. A famous example of a classical bottomless theory is Euclidean gravity. The same problem of instability arises in matrix models of pure gravity. Applications of Greensite-Halpern stabilization to these models can be found in [2,3]. The most popular method of stabilization is analytic continuation of the classical action in a coupling constant. Expectation values are then not necessarily analytic 4 ) but it seems that the perturbative series is always invariant under continuation. So the three axioms of minimality formulated by Greensite and Halpern may also be fulfilled. It is, however, known that both stabilization methods are inequivalent.
J. Maeder and W. Riihl
266
We want to apply the Greensite-lialpern stabilization method to zero dimensional sigma models that exhibit A k singularities with k > I, (k = I appears in [I J). In these cases we have to perform double scaling limits, where N goes to infinity and coupling constants {Ir} tend to their critical values {I;}. There arise scale invariant variables {(r }~-I and the singular factor in the partition function is a generalized Airy function depending on these variables (see [5] for the details). The cases A k with k = 2n are unstable. If k = 2n + I there are two signs A~'+l one of which (the "wrong sign" A 2n +d is also unstable. The generalized Airy functions are given by integral representations. In the stable cases the integral contours are the real axis. Mathematical textbooks 6 ) teach us that we have to choose complex contours in the unstable cases. Though this leads to well-defined Airy functions, it is not clear whether they are suited for a probabilistic interpretation in at least a subdomain of the variables {(r}. At the end of this article we will make a clarifying comment on this problem. On the other hand the Greensite- Halpern stabilized theories have an obvious probabilistic interpretation for all {(r} E IRk-I. 2. We consider zerodimensional sigma models
JII d¢;a N
Z =
e- s ,
(1)
,pa E IR
a=1
s=
I
+L k
-;-1. ¢ 2
f
~N-r+l(<,D.
r=2 21'
<,Dr
(2)
,p·¢=Nz -
S(z)
=
I
NS
(3)
I ~ Jr = -z + L.J -z
2
r=221'
r
(4 )
.
Angular integration gives
Z =
(rrN)lf r('2)
--y-
1
00
0
I
I
-
dz- exp N( -log z - S(z)). z 2
(5)
The exponent in (5) may exhibit a singularity An(n ::::: k) which in the limit N -> 00 allows us to expand Z and allY expectation value in a series of fractional negative powers of N. In the present context we will deal with only the leading term which for Z gives a generalized Airy function. The starting point of the Greensite-Halpern stabilization is the Schrodinger-equation (6)
Greensite-Halpern Stabilization
267
with normalized ground state wave function 'l/Jo( ¢) and eigenvalue Eo. The ill-defined probability density
I -s ZC is replaced by IV·o(1))I2. Change of the coordinates (:I) and action (4) gives
a + NV(Z) ] 'l/Jo(z) - =
2z fP [- N DzZ - oz
l\'ext we apply the N
-
Eo'I/Jo(z)
(7)
'l/Jo(¢) = ;f;o(z(¢))
(8)
V(z) = ~z(S')Z - ~S' _ z 5". 2 2 N
(9)
-> 00
limit to the equation (7)1,7): We factorize
(10) and rescale the equation
2z {P [ - N OZ2
+ N ( 8zI + V(z) ) + 0(1) ] 'Po(z) =
Eo'Po(z)
(II)
in the neighborhood of the singularity. If this singularity is AI, its location Zo is determined from - 2I
8z o
It has been shown in
p]
+ V '( zo)
( 12)
= O.
that the left hand side factorizes
-~ + V'(z) 8z
= F1 (z)F2(z)
( 13)
with -
1
(14)
F1 (z) = 5'(z) - 2z Fz(z) = zS"(z)
+ ~,~'(z) + ~. 2
4z
(15)
If
(16) we have an Al singularity in the action (5) as well. An additional branch of Al singularities in the potential of the Schrodinger equation (11) arises at (17)
J. Maeder and W. Riihl
268
We will not consider it here (sec, however, [I J). The ground state energy is in this approximation
Eo
N
(8~0 + V(zo))
"21 N Zo ( 8'( zo) ) 2 ::::: 0
( 18)
which remains valid in the A k , k > 1, case. To complete the discussion of the Al case we prove stability (i.e. Al is Ai). We expand the potential to next order
-1 + V(z) 8z
1 + V(zo) 8z o
= -
1 + -(z -
2
2
zo) w
2
+ O((z -
~
zo)')
(19)
and r~(zO)F2(ZO) + FI(zo)r~(zo) r~(zO)F2(ZO)
(20)
if (16) holds. Now from (14), (15) we obtain
r2(z)
= zF;(z) + ~Fl(Z)
(21)
so that once again from (16) (22)
and we have (local) stability. We will later see that any A n + 1 singularity in the action (5) implies a (stable) A n + 1 singularity in the potential of the Schrodinger equation. Other singularities in the potential (such as Al (17)) are not automatically stable. The ground state energy Eo is to next order
(_1_ + V(zo)) + 8z o
Eo = N
(23)
(I
where to leading order now 2 Zo fJ2 [ -!II 8z2
1
+ "2 Nw
2(z -
zo)
2] 'Po(z) =
fl'PO(Z).
(24)
This equation is rescaled by
(25) so that the oscillator equation 1 D 1 w 2] + ;--x 'Po(z(x)) = [ -;--.-.' '20.£2 24z 2
2
o
(1
-'Po(z(x)) ,1zo
(26)
Greensite-Halpern Stabilization results. It follows
269
I
£.1
= zJw
and
(27) I
~
'Po(z(x)) = A . e -2~x 3. An A n + 1 singularity in (5) shows up at
F1(m)(zo) = 0,
Zo
2
.
(28)
if
0:::;
Tn:::;
Ft+I)(zo)
n
i- O.
(29)
If in (2) and (4) we choose k = n + I (the "minimal set" of coupling constants) there is exactly onc such singularity and corresponding critical coupling constants {f;}~+1 (see [5]). Since from (21)
(0) (29) implies
FJml(zo) = 0,
0:::; m :::; n - 1
FJn)(zo)
i- O.
(31)
At such point Zo
(FI(z)F2(z))(2n+lllzo =
C::/)
Zo (F1(n+I)(zo)f > 0
(32)
whereas
It follows that at leading order in z - Zo the potential in the Schrodinger equation is
+ N. with
92n+2 =
92n+2 (z _ zo)2n+2 2n + 2 zo(Fi n+1)(ZO))2 n!(n + I)! > O.
(34)
(35)
So the Greensite-Halpern program produces a stable potential 1Il the Schrodinger equation for each A n + l . If the Schrodinger equation is rescaled at N -> 00 in analogy to (24), (25) we obtain
liP ( ---.-,' 2 iJ:r2
I I 2n+2) +--x 'Po(z(x)) = 2n + I
N £.1 = - . - . 'Po(z(x)) ,X2 4z o
(36)
J. Maeder and W. Riihl
270 where
x
=
(37)
.\(z - zo)
and
I
.\ = (N 292n+2) 2n+'
(38)
4z o
So 'Po(z(x)) is a universal function of x and
(:l9) where tl is a universal number (depending on n). The function 'Po(z(x)) = Xo(x) is symmetric in x and for x ~ 00 behaves as (n > 0)
Xo(x) = Aexp { -
x n +2
1
+-(n+l)logx (n+l)2(n+2) 2 I
(n
+ l)t (IX-n + O(x- n - 2 )}. n
(40)
Squaring this function and substituting (37), (38) wc obtain the Grccnsite-Halpern probability distribution over the real z-axis for large Izi
(and analogously for large N). Here the effect of the stabilization can be clcarly seen: all "wrong signs" are eliminated. l\ow we considcr a deformed A n+1 singularity: the coupling constants {Jr} are different from the critical ones
{In
(42) but with N
~ 00
these B r go to zero in such a fashion that G(x;{O)
=
Z}
lim N { S(z) - S(zo) - -21 log~
N_~
n (
'" ~xr
~ r!
xn+2
+ f.,--------:-;(n+2)!
(43)
(( = ±l for even Il)., Thus ill terms of .\ (37) (the normalization in (38) is marginally changed)
(44)
Greensite-llaipern Stabilization
271
and the point Zo is kept fixed by the requirement that the power of order n + 1 (4:l) vanishes. In this case the saddle point integration of (5) gives
, Zsing
=
1
(7rN)lf I r( If) AZo c dx exp{ -G(x; {(m
III
(45)
where C is a chain running from infinity to infinity along which the integral converges exponentially. The integral is a generalized Airy function. ?\low we apply the analogous procedure in the Greensite-Ilaipern stabilization program, which results in a measure dpGII ~ = exp{-G(x;{(})}
J
dpGH
In order to calculate
= 1.
(46) (47)
G we repeat the rescaling of the Schrodinger equation using
(21), (44) and (37) and get (48)
and
I .
xo(x) = exp{ -2"G(x; {(})}.
(49)
It follows for x -+ +00
G(x;{(})
cG(x;{(}) + log(c
axa G(x; {(}) (50)
+0(1).
4. Now we consider an example: the singularity A2 • The Airy function is the proper one 7r Ri( ()
lYJ dx{exp(-~x3+(x)+sin(+~x3+(x)}
1dxexp(-~x3+(x) c
(51)
3
where C is deflIled as follows. Let Cq , q E CQ, denote the contour along the ray {l'e 2rriq , 0 :S l' < oo} oricnted froIll zero to infinity. Then
C
= Co -
1
-2(Cl3
+ Ci). 3
(52)
J. Maeder and W. Riihl
272
How can this Airy function be used to calculate expectation values? Consider a polynomial M
PM(x) =
L
urx r .
r=O
It is natural to define then
(.54) In order that a probabilistic interpretation is possible, the matrix P M (()
(x) (x 2 )
2 (x ) (x 3 )
(xl\,j) (X
M
+1 )
1
(55)
(x 2M ) must be positive (for some (M and all M) at least for
(>
(56)
(,\1.
From the asymptotic expansion of the Airy function ([8], eqn. 10.4.63) follows
((PM(X))2)
(PM((!))2 (
+ lower order terms (57)
....... 00
so that PM(() has one positive eigenvalue for large (. It can be shown that the other eigenvalues arc positive for large (, too. With this knowledge it suffices to calculate (58)
For low M we find (for ( ....... (0) e.g.
(59)
(60) Assume that DM (() > 0 for ( ....... 00 has been shown. Then (,\1 is the largest zero of D M . For M = I we obtain (using the tables in [8])
(61)
(1 = 0.4003.
Finally we have to prove
Co
= sup (10,1 < M
00
(62)
273
Greensite-Halpern Stabilization which is so far only wishful thinking. In the Greensite- Halpern approach we have to solve
(63) By symmetry we have (64)
Moreover we find e.g.
( (+ lower order terms for ( ---7 2 (X )ml = O(I(lt) for (---7 -00
l
as compared with
(X 2) =
00,
(
(65)
(66)
from Airy's differential equation ([8], eqn. 10.'1.1). The differencp between the two approaches becomes more striking if we compare the dispersions
(67) (68)
REFERENCES [1] J. Greensite, M.ll. Halpern, Nucl. Phys. B242 (1984) 167. [2] J. Ambj0fn ..J. Greensite, S. Varsted. Phys. Lett. B249 (1990) 41 J. [3] .J. AIllbj0rn,.J. Greensite, Phys. Lett. B254 (1991) 66. [4] A. Turbiner, Phys. Lett. B276 (1992) 9.5. [5] W. Riihl, "Double scalin!!; limits and Airy functions for O(N) vector si!!;ma models with elementary catastrophes or the catastrophe X 9'" Ann. Phys. (N. Y.) to appear. [6] V.l. Arnold, S.M. Gusein-Zade, A.r-;. Varchenko, Singularities of Differentiable Maps, 2 Vols., Rirkhauser, Basel 198.5 and 1988. [7] E. Witten, in /97.9 Cal'gcse lectures: Recent developments in gauge theories, I'd. G. t'Hooft, Plenum, New York 1980. [8J M. Abramowitz, l.A. Ste!!;un, Handbook of mathematical functions, 10th pro Dover, New York 1972.
From Field TIICory to Quantum Groups
SOME ASPECTS OF SOLITON UNWINDINGS
B.
PIETTE AND
W.,J.
ZAKRZEWSKI
Department of Mathematical Sciences University of Durham, Durham DR] 3LE, England e-mail H.M.A.G.Piette«[email protected]
Abstract: We investigate the unwindings of solitons as seen in many numerical simulations. We study a simple model in (2+ 1) dimensions which involves two scalar fields 1>1, 1>2 with an interaction term which favours the fields to lie on a unit circle (i.e. to satisfy the constraint 1>~+1>~ = 1). We show that as the discretisation increases, or solitons become "thinner", the importance of the constraint decreases thus suggesting that in numerical simulations of 52 solitons the unwinding proceeds through the constraint not being satisfied in-between the numerical lattice sites.
1.
Introduction
Over the last few years we have performed many simulations of the scattering of soliton-like structures in various (2+ I) dimensional models [1 J. When the simulations involved the pure 52 (often called also the 0(3)) 17 model, the solitonic-like objects behaved very much as solitons except for their instability with respect to shrinking due to various perturbations, introduced either through their interactions with other objects, or even due to the lattice discretisation effects. This was due to the conformal invariance of the underling theory; the solitons can have any size and in any simulation the mode of changing their size was very easily excited. In practice this meant that solitons would start shrinking and then become so "thin" that the numerical simulation would become unreliable. If we ignored the numerical instability we would see that the solitons would "unwind", a process that is not possible in the continuum but which is allowed in a discrete version of the model. As most physically relevant soliton-like structures have fixed sizes we modified our original (pure 8 2 ) model and added to its Lagrangian density further terms
276
B. Piette and W.J. Zakrzewski
("Skyrnw" and potential terms) which fixed the size of our solitons and cured the numerical instability (in the simulations the solitons would not become so spiky as to take us into tlw unstable regime). The parameters of the extra terms (and so of the soliton sizes) were fixed by various "physical" arguments and so we could proceed to investigate "physical" properties of our solitons [2]. Of course, although most of these simulations have been performed for the simplest (J' models in (2+ I) dimensions one can easily extend them to more complicated systems, like those discussed in ref [3]. Later, one can hope to go to higher dimensions and study the scatterings of various extended structures that exist there, i.e. of the monopoles, vortices, skyrmions clc. Cnfortunately, at the moment, this is not possible due to still too big computer power requirements that the study of such processes necessi tates, However, many physical systems are indeed discrete [4] and so the unwinding we have observed in our simulations may be a "physical" phenomenon. But what was really the mechanism of the observed unwinding? As the phenomenon is not allowed in the continuum it must be related to the discreteness of the lattice. But how does it proceed'! To answer these questions we have decided to look at a simpler model which possesses all the features of our (2+ I) dimensional system - namely an appropriately defined model in (I + I) dimensions. We present our model in the next section; and in the following sections we discuss our results.
2.
A simple model in (1+1) dimensions The model we want to study is defined by the Lagrangian density
(I) where; = (¢" ¢2) and where a, b, c and d are arbitrary parameters. This model is a generalisation of the, so-called "pure" S' model in which the field ; is restricted to lie on the sphere ;2 = I and goes over to it in the limit a -+ 00. When (l # 00 the field can move away from lying on the unit sphere when such a motion is energetically favoured. The last term in Eq.( I) breaks the 0(2) symmetry and can be considered as a "potential" term. It favours the field to take the values ¢2 = I and, if 1;1 ~ I, ¢1 ~ O. As such Eq.( I) is a (I + I) analogue of the model that has been used in our studies [I] of interactions of soliton-like structures in (2+ I) dimensions. The equation of motion for Eq.( I) is given by
(2) It is not easy to find any solutions, even static ones, of Eq.(2).
277
Some Aspects of Soliton Unwind ings
static and which Note, however, that if we restrict ourselve s to fields that are Eq.(I) (in rewrite can we nt satisfy IJI = 1 then, using the Bogomo lnyi [5] argume the case when b = d = 2) as
Then as
1';1
= 1 wc can put ¢1 = 8in( (),
(4)
>2 = cos( ()
the minimu m of the and we see that the last term in Eq. (3) is a total divergen ce and solves action is attained by the field ((x) which (x
= y'C(1
(5)
- cos(()).
Of course, it is easy to solve Eq. Ui) Its solution , which goes to respecti vely, is given by ( = -2 arccot( vex) + 211".
°and 211", as x
--->
±oo
(6)
soliton. To see This field can be thought of as describi ng a (I + I) dimensi onal and that then, 1 = >2 0, = >1 by given this we note that the field at x = -00 is again given by is 00 = x at and space 2 = I. The cnergy "unwind " itself long as IJI does not move away too much from I, the field cannot analogu e of onal dimensi (1+1) a is field the property this In (in the continu um). 1';1 does not nced to the (2+ 1) dimensi onal skyrmio ns [1 J. Let us stress here that what matters is that be equal to 1 for this field to represen t a soliton- like structur e; to +00. As Eq.(4) is the angle bctwecn
J
°
effects ). Eq.(4) but for all In fact our studies have shown that this is not only true for at x = -00 are at which fields c. i. es properti similar "reason able" fields which have space so that 1, the in around rotate then which 1 and 0, >2 given by 2 0, = >1 by at x = 00 are again given bly stable reasona be to seem them of all and rations configu like describi ng solitonfrom "unwind ing'· 2 and for the field The (2+ I) skyrmio ns involve an 8 field and, in the continu um This is because ". "unwind lying on the sphere (0,0, -I) at limit'; the takes which the skyrmio n field correspo nds to a field'; Such a field is infinity. spatial at I) (0,0, ---> .; the position of the skyrmio n and
=
=
278
B. Piette and W.J. Zakrzewski
characterised by a topological number which, in appropriate units, corresponds to the number of solitons and which cannot change during its temporal evolution. However, all numerical simulations are performed on discrete lattices and so these general topological arguments do not apply to them. As a field is defined only at the lattice sites all configurations which differ from each other off lattice sites cannot be distinguished from each other - this allows the fields to vary rapidly "in-between" lattice sites thus leading to the "unwinding". In fact this is what has been seen in various numerical simulations of the 82 solitons. In the pure 52 (J model [6] any perturbation of the soliton (either through its interaction with other solitons or even due to the effect of putting the system on the lattice) tends to destabilise the solitons. In most simulations the solitons begin to shrink (they are more localised) soon reaching a configuration in which each soliton extends only over a few lattice sites. At this stage the fields vary a lot between lattice sites and the numerical procedure breaks down. Hence, to overcome these effects extra terms are often added to the Lagrangian [1]. They have the effect of making it energetically disfavoured for the solitons to change their sizes from their optimal values fixed by the interplay of all the terms in the Lagrangian.
I
I
energy denslty-
I
I
r~
>, u -.<
>, u -.<
~
•
•
•
r.
11
>-~
" c• •
energy density-
, -10
)
\
Fig.l a: Energy density for the field Eq.(4).
'"6:
•" •
r.
.
)
\
-10
Fig.l b: Energy density for the field Eq.(7).
However, one may wonder what happens in systems in which the soliton size is not fixed, i. t. in the original "pure" 52 model. How do the solitons "unwind" and what is the mechanism of this "unwinding"? Our (1 +1) dimensional model Eq.(l) has all the properties to allow us to find at least partial answers to these questions. Clearly, if we start with a field configuration corresponding to a perturbed soliton we can follow its evolution looking at its "unwinding". We can study the dependence on the lattice spacing and we can control
Some Aspects of Soliton Unwindings
279
the "unwinding" by varying a i.e. the strength of the term which encourages the field to lie on the sphere 1>~ + 1>~ = 1. Thus we decided to study the time evolution of various "soliton-like" field configurations varying the strength of the parameters and the lattice spacing (i.e. dx). In particular we looked at Eq.(4) and Eq.(6). We also looked at the field configuration given by 1>1 = sin(1l'[tanh(ax) + 1]), (7) >2 = cos(1l'[tanh(ax) + 1]), where a is a parameter. It is easy to check that Eq.(7) is not a static solution of Eq.(2) Hence if we take Eq.(7) as the initial configuration the field will evolve at time progresses. in fig. la and fig. 1b we plot the energy density for these two configurations (corresponding respectively to Eq.(4), Eq.(6) and Eq.(7)). We have performed several simulations on lattices with different lattice spacings and for different values of all parameters of the model, namely a, b, c and din Eq.( 1). We have found that the fields corresponding to these two initial conditions evolved in a very similar way. lienee we chose to use Eq.(7) for most of our simulations. We discuss them in the next section. 3.
Numerical Simulations
The evolution of the field Eq.(7) depends on the value of a. We have looked at various values of a and simulated the evolution for different lattice spacings. However, in concrete lattice simulations we replace ox1>i by ox1>i = <1>. (x+d:2-.(x) ; hence we see that there is a relation between a, dx and the parameters a, b, c and d of the model which is a symmetry of the problem; hence in practice we have fixed a = 1 and studied the dependence of the dynamics on the other parameters. All simulations started with 0' = 1 (and corresponding to "reasonable" values of dx) have shown that, initially, the field configuration tries to shrink and so to "unwind" itself; at the same time it emits waves which travel to ±oo. Not to have any reflections we either took very large lattices or absorbed the waves at the edges of the lattice. In most of the simulations we saw "unwindings" - i.e. the fields unwound themselves with the field 1>2 being almost always close to + 1 (modulo small oscillations and small waves) and the field 1>1 ,..., O. Looking in detail at the mechanism of this "unwinding" (i.e. at values of time at which the fields changed from their "initiallike" to "final-like" configurations we have found that these "unwindings" proceeded through the fields going through zero (i.e. ;;,..., 0 for x,..., 0 i.e. around the position of the "soliton".) We then studied the dependence of this phenomenon on all parameters of the model. First we kept dx fixed (in fact we used dx = 0.04 and took 2001 equally spaced points) an varied a. For small values of a we always see the "unwin dings" ; then as a increases suddenly the unwinding stops. This is true for most sensible values
280
R. Piette and W.J. Zakrzewski
400
b-2.d-4 -_. b-4.d-2 ...... b.2.d,2 - -
bd.d~4
llJ
3:)C
L~O
en
200
ISO
10C
\0
10
1\
20
C
Fig.2. Critical a as a function of c. of b, c and d. Of course, our claims are somewhat imprecise as, strictly speaking, we would have to wait infinitely long to be sure that the unwinding docs not take place. However, for small dx (i.e. for dx rv 0.05 - 0.5) there was always a sharp transition i.e. a well defined value of a below which we had unwinding and above which the unwinding did not take place. Next we looked at the dependence of this critical value of a on c and in fig.2 we present our results. Below the curves we have the "unwinding" above them the "unwinding" is heavily suppressed. We see very little dependence on band d. The dependence on c shows that with the increase of c to suppress the "unwinding" we have to take larger values of a (i.e. come closer to the nonlinear a model constraint 14>'1 = 1). Of course, this is to be expected as the increasing values of c steepen the soliton (i.e. the soliton becomes more localised) and hence it is easier for it to "unwind" . ='Jext we investigated the dependence on dx for fixed c and for d = b = 2. The preliminary results have shown that in all cases, for small values of dx (like those mentioned above), there was a monotonic increase of a with the increase of dx. For larger ~alues of dx there were some irregularities so we decided to study them in more detail. In fact this lead us to look at the "lattice field configurations" at various times. We looked at them for a few values of c; we chose them close to zero to decrease the
Some Aspects of Soliton Unwindings
281
effect of the shrinking mentioned before, and we studied the dependence on the initial configurations corresponding to a few values of a.
Fig.3 a: Potential a( 4>i + 4>~ - 1)2 + c(l - 4>2)2: a = 1.67, C = 0.1
Fig.3 b: Potential a(
Our results can be summarized as follows: for dx ::; 1 there is a rapid growth of the critical a with the incrcase of dx. In all thcse cases, if there is an "unwinding" it takes place very rapidly (i. c. we did not have to wait beyond t rv 20), for larger values of dx, say, dx rv 1.5 oftcn one has to wait longer for the "unwinding" to take place (i.e. up to t rv 500 or even 1000). Hence one can easily miss it and think that the curve of critical a versus dx, increases, turns over and then decreases again. However, a little thought shows that this is incorrect. Thc clue can come from noticing that the "late" unwinding seems to proceed through a different mechanism; during the unwinding I~I decreases but never vanishes! In fact, this phenomenon is associated with the discretcncss of the lattice. Our initial configuration Eq.(7) is very special and so leads to "metastability". This cffect is associatcd with the fact that for large dx the field configuration is effectively described by a few points which we can consider as material points, coupled to each other by elastic forces (the second term of Eq.(I)), and placcd in the potcntial givcn by last two terms of Eq.{l). This potcntial is shown in fig 3.a and 3.b for b = d = 2, a = 1.67 and c = 0.1 and c = 0.5 respectively. In fig 4. (a,b,c and d) we present the field configurations (4)1> versus 4>2) (for the points for which 4> < 0.99) as seen in the simulation (with dx = 1.0), performed for c = 0.1, a = 1.67). The fields are shown at a) t up to 150, b) t = 180, c) t = 200, and d) t = 250. We see that the "metastability" is due to the symmetry of our initial condition Eq.(7) for :r -+ -x giving us the
B. Piette and W.J. Zakrzewski
282
t-=: 50
tc180 •
•
"
.. -C.':>
-2
\----'-_'---'-_.1.--'-_.1.--'----' -2
-O.!>
-1.
0
-O.!>
0
0.':>
Fig.4 b: Field at t=180
Fig.4.a: Field for t :::; 150
t=250 •
t=200 •
..
I
~.
..
_____________._1 __. _
"'-"1--"'-"---"--
0_'
:,,:>
Fig.4. c: Field at t=200
I
-2 L--,-_,---,-_L--,-_,--~---' .J -C.':> ;)
Fig.4 d: Field at t=250
x - Xo for some small value of Xo, say Xo = 0.1. This modification does not alter the unwindings at small dx but speeds up the ullwindings at larger dx. Hence, due to its lack of symmetry the modified field configuration is a better guide of what is going on in a general case. For larger values of dx the effects are even more pronounced. In fig 5. we present the curve showing the dependence of a on dx (for c = 0.1, b = d = :2 and for this more generic case). This curve demonstrates very clearly the rapid increase of the critical value of a with dx.
4.
Scattering of 2 solitons
Next we decided to see what happens if we scatter two soliton-like configurations. Will they pass through each other or will they unwind? Will the unwinding happen when the solitons are "on top of each other" and how do the results depend on all parameters of the model?
Some Aspects of Soliton Unwindings
283
aj. ~
2.\
0.1
0.2
0.3
0.4
O. ~
0.6
0.7
0.8
0.9
1.1
dx
Fig.5. Critical a as a function of dx. To do this, first of all, we looked at Eq.(6) and studied it numerically and, as expected, it did not change much in our simulations. Of course I~I moved away from 1 but this change was not very significant and only restricted to small regions of x. This change can probably be associated with Eq.(6) evolving to become a solution of Eq.(2). We then decided to look at two such solitons and send them towards each other. To do this we had to find an appropriate expression for two moving solitons. Guided by our experience from (2+ 1) dimensional systems[l] we decided to take our field in the form Eq.(4) but this time we took
(8) where Xl = X - XO, X2 = X + Xo. This configuration corresponds to two static solitons located at ±xo. To study their interactions we boosted the solitons towards each other. This was achieved by replacing x ± Xo in Eq.(8) by XjXQ±~t and then calculating ~(x, 0) I-v and *(x,O) for this configuration. We performed several simulations for different values of v and for different values of various parameters of the model (a, b, c and d). The results reproduced our expectations based on the results of the simulations reported in the previous section. In some simulations the solitons scattered through each other, in others they unwound either right at the very beginning or at the time
284
13. Piette and W.J. Zakrzewski
of their passing through each other. To obtain more quantitative results we fixed b = 2, C = 1, d = 2, v = 0.4 and thcn varicd a. For very small values of a the solitons would "unwind". As a increased the solitons were stable until they reached each other. Then, as they passed through each other they would unwind resulting in pure waves. At a = 11.660107 there was a sharp transition to a regime where the solitons would not unwind. At that and higher values of a they passed through each other emitting some radiation; hence their velocity after the scattering was smaller than at the beginning. However, this difference of velocities was not very large. We have also repeated these simulations by breaking the symmetry of the problem, i.e. shifting Xl by Xl -> Xl + hx with hx ~ 0.01. The results were very similar except that the critical value of a decreased somewhat (down to about a = 11.66009. In a similar way reducing v decreases the critical v too as to be expected from what we have said in the last section. We also performed some simulations at higher values of velocities without putting in the relativistic corrections. Of course, such initial conditions do not have the correct relativistic deformations of the solitons and so increase the overall initial energy and the radiation effects. Indeed, this was seen in our simulations. Moreover, in one series of simulations we found that as we increased a we moved from unwinding solitons to the scattering ones through the intermediate range where effectively only one soliton would unwind (we had two incoming solitons at the beginning and a stationary soliton at the interaction place and some radiation after the scattering). Of course, for this to happen the conditions have to be somewhat unusual - we have not succeeded in reproducing this intermediate result for the relativistic initial conditions. 5.
Conclusions
We have seen that, in general, the discreteness of the lattice makes it possible for our (1 + I) and also (2+ I) solitons to unwind. In models in which the "topological" constraint 11>"1 = I is not enforced but energetically favoured, we have found that with the increase of the lattice spacing (i.e. dx) it is harder to suppress the unwinding. The coefficient controlling the energetically disfavoured term grows rapidly with dx. The unwinding proceeds through the field slipping over the barrier. At larger values of the lattice spacing the system can be treated as a set of points, coupled to each other by elastic forces and lying in the potential field with a barrier. However, given their discreteness the field can unwind by the particles moving away f~om each other (as much as possible) and then getting over the barrier by going round it (from both sides). Hence 11>"1 does not vanish at any point. What is the relevance of these observations to the unwinding of solitons as seen in the simulations of solitons in pure 8 2 u models'! In most simulations of these models the constraint 11>"1 = I is either solved explicitly or imposed at various stages of the numerical simulation. In this last case the initial fields satisfy the constraint but then all the fields
Some Aspects of Soliton lJ nwindings
285
each other thus leading to small deviation from the constraint. Then, after each time integration step, the fields are renormalised to satisfy the constraint and evolved further. This means that, effectively, we are in the situation studied in this paper; the procedure of renormalisation ctc. introduces an effective force which allows the fields to wander away from the constraint l
REFERENCES
[1] I.l.M.A.G. Piette and W.J. Zakrzewski, Durham University preprint DTP-93/41 (1993) - to appear in Chaos, Solitons and Fractals 1995 [2] H. PieHl', B.J. Schrol'rs and W.J. Zakrzewski, Zeit. Phys. C65 (1995) 165-174; Nucl. Phy". B439 (1995) 205-23R. [3] A.M. Din,.I. Lukierski and.J. Lukierski) Nucl. Phys. B194 (1982) 157 [4] see e.g. P.G. De Gennes and J. Prost, The Physics of Liquid Crystals Oxford Science Pub!., 1993 [5] F:.B. Bogomolny Sov. J. Nucl. Phys. 24 (1976) 449 [6] see e.g. W.J. Zakrzewski -Nonlinearity 4 (1991) 429 [7J Martin Zapotocky - private communication
From Field Theory to Quantum Groups
BV-ALGEBRAS OF W-STRINGS KRZYSZTOF PILCH
Department of Physics and Astronomy University of Southern California Los Angeles, CA 90089-0484, U.S.A. E-mail; [email protected]
Abstract: I review some recent results and conjectures obtained in collaboration with P. Rouwknegt and J. McCarthy on the BY-structure of the algebra of physical operators of noncritical W-strings.
Introduction
1.
W-gravities l provide interesting examples of gauge theories based 011 a lIonlinear algebra of constraints. It is expected that the underlyillg geometry, when properly understood, may provide a generalization of geometry, in which the diffeomorphism invariance of ordinary gravity is extended by higher spin tensor structures. However, despite a lIumber of prelimillary studies on the geometric interpretation of W-gravity [6] [11], a complete theory has 1I0t yet been developed. While it is difficult to study W-gravities in four dimensions, ill the past few years there has been some progress ill ullderstandillg those theories in two dimensions, where the algebra of cOllstraints is the W-algebra generalization of the Virasoro algebra. In particular, there is a quite satisfactory understanding of t.he geometry underlying algebras of physical operators in a special class of W-gravities, the socalled noncritical W-strings. This development started with the observation [12,13] that a subsector of physical (i.e., BRST invariant) operators .fJ[W2 ] of the W 2 (Virasoro) string is modeled by polynomial poly vectors on the complex plane. Subsequellt work [15]-[17] revealed that the full algebra of physical operators has the structure of a Batalin- Vilkovisky (BV-) algebra, with the BV-algebra of polyvectors arising as a quotient. I
For a general review of W-algebras, see, e.g., [1], while of W-gravity [2] [5].
288
K. Pilch
/\ similar. albf,it significantly more complicated, description of the operator algebra. .fJ[W3 ], of the W 3 string was given in [18]. This construction relied on an earlier observation [19,20] that the proper geometric framework for studyiug the W" string is the base affine space [21] of SL(n). The most convenient framework for studying W gravities is the BRST quantization, in which the space of physical states and the corresponding operator algebra, .fJ[W], are determined as the BRST cohomology, H(W,Q:). Because of the nonlinear structure of the W-algebra, the existence of a required BRST complex, such that the underlying chiral algebra, Q:, admits a stringy interpretation, is by no means obvious. For the first nontrivial case, which is the W 3 string, the BRST complex was constructed in [22]. Later, a systematic procedure for constructing BRST complexes for a large class of W-strings based on higher rank W-algebras, includiug the W n algebras, was given in [21]. Oue should note, however, that a direct computation of the BRST cohomology in the higher rank cases appears almost impossible with present techniques due to an enormous algebraic complexity of the problem. In this respect a recent result in [25] (sec, also [26]) which, at least conjecturally, describes the BY-algebra of an arbitrary noncritical W[g] string, may be quite useful. Here W[g] is the W-algebra associated with a simple, simply laced Lie algebra g by means of the quantized Drinfel'd-Sokolov reduction with respect to the principal S(2 embedding in g (see, e.g., [I]). It turns out that there exists a natural BY-algebra, BV[g], whose underlying graded commutative algebra is the Lie algebra cohomology, 1I (n+, £( G) ~ 1\( n+ \g)), with respect to the maximal nilpotent subalgebra n+ C g, where £(C) is the space of regular functions on the complex Lie group C of g. The main observation in [25,26] is the following conjecture, which has been verified thus far for the cases g = S(2 [25,26] and S(3 [25].
Conjecture 1. 1 There is a 13 V-algebra isomorphism .fJ[g] ~ BV[g] ,
(1.1)
whm .fJ[g] == .fJ[W[g]] == If(W[g],Q:) and BV[g] == If(n+,£(G) 01\(n+ \g)). One might find this conjecture rather surprising, as it relates two objects that arc quite different. On the one side, there is the semi-infinite (BRST) cohomology of an infinite-dimensional nonlinear W-algebra. Apart from the very existence of the BRST complex, rather little is known how to compute such spaces in general. and the only explicit results are available for 9 = S(2 [27,28] and S(3 [18]. On the other side, one recognizes a fairly familiar mathematical construction, namely the Lie algebra cohomology (see, e.g., [29]) of n+ with values in finitedimensional modules. in particular, finite-dimensional irreducible modules, L(A), A E 1\. of g. (The latter arise upon decomposition of £(C) with respect to g.) In that case one is led to a classical problem in Lie algebra cohomology - the computation of
BV-Algebras of W-Strings
289
H( n+, L( 11)) [30]. Although an explicit computation of H (n+, £( G) @ A( n+ \g)) is still quite iuvolved, it appears to be tractable using standard techniques of homological algebra. Also, as pointed out in [26], this BV-algebra is nearly isomorphic with the BV-algebra of the I\odaira-Spencer cohomology of the base affine space of G, and perhaps can be determined using methods of topological field theory developed in
[31]. In the following I will try to explain the above conjecture in more detail and by doing so summarize some of the main results in [18,25]. In section 2, I introduce the BV-algebra Si[g] and then, in section 3, the BV-algebra BV[g]. I conclude in section 4 with comments and open problems. In the appendices I present some explicit examples from the case 9 == 5(3, i.e., the W 3 string. ~any years ago Professor Lukierski asked me to give lectures on Kac's paper on Lie superalgebras. After a brief examination of the paper it became clear that I first needed to improve my understanding of ordinary Lie algebras, and, in the end, this is what my lectures were about. Since the effort to prepare them turned out to be such a good investment, I am very happy to dedicate this note on yet another facet of Lie algebras to Jurek on the occasion of his sixtieth birthday. 1.1.
Notation
The conventions and notation here will be the same as in [18,25]. In particular, is a complex, simple, simply-laced, finite dimensional Lie algebra and G the corresponding complex Lie group. The generators eA of 9 satisfy the commutation relation rCA, eB] == fABccc. It will be convenient to fix the Cartan decomposition 9
(1.2)
of 9 and choose CA as the Chevalley generators, (c a , Ci, e_ a ), i = I, ... , fl, 0 E .1+. The corresponding structure constants, fAB e , are then particularly simple. 2.
2.1.
The BV-aIgebra of the W[g] string The eohomology problem
The algebras W[g] have a realization in terms of fl = l'ank(g) scalar fields coupled to a background charge, 0u. Those realizations generalize the familiar Feigin-Fuchs realization of the Virasoro algebra [;32]. In particular the stress-energy tensor is given by T(z) == -W1¢(z), 8<1>(z)) - ioop· 8 2 ¢(z) , (2.1) where ¢(z) (
e-
K. Pilch
290
The noncritical W[g] string can be introduced, by analogy with ordinary twodimensional gravity coupled to eM = I conformal matter and quantized in the conformal gauge using the DDK ansatz [34,35]. The W[gJ gravity sector is then described by Liouville fields, which are scalar fields coupled to a background charge at = 2i. The matter sector consists of free scalar fields with no background charge, i.e., aiY = 0 and eM = Since (aiY)2 + (a~)2 = -4, there is a well-defined ERST quantization procedure of such system [22][24], by which the space of physical states is determined as the semi-infinite (BRST) cohomology, H(W[g], F(A M, aiY) ~ F(A L, at)), of W[gJ with coefficients in F(AM,a;Y) 1& F(AL,ak). In the following I will be interested in a particular subsector of the W[gJ string that is defined by restricting the momenta (AM, -iA 1,) to lie on the lattice
e e
e.
L =: {P,Il) E P x PI,\ - Il E Q},
(2.2)
where P and Q are, respectively, the integral weight and the root lattices of g. More precisely, let C be the complex
EB
C=:
F(AM,O)CY F(A L ,-2i) 09 F9 h ,
(2.3)
(AM .-iAL)EL
where F9h is the Fock space of the (be) systems of W[g] ghosts. The choice of the lattice L is partly dictated by the fact that by summing the matter momenta over the integral weight lattice P, the complex C becomes a module of the affine Lie algebra 9 at levell, while the chiral algebra2 1£ corresponding to C can be equipped with the structure of a Vertex Operator Algebra. The URST complex structure lifts to 1£, on which the ERST operator, dw , acts as the charge of a spin-I (ERST) current, J(z), (dwO)(z) =
dw
-.J(w) O(z), fcc,27l"z
(2.5)
where the contour C z surrounds the point w = z counterclockwise. An explicit form of the HRST current is known only for the lowest rank Walgebras and already in the simplest case of the W3 algebra [22,33J it is given by a complicated expression involving quartic couplings between the fields. The latter feature is a consequence of the nonlinear structure of W;h and is directly responsible for much of the difficulties in studying this cohomology explicitly. To determinE' the spectrum of physical states of the W[gJ string one must first compute the cohomology, H(W[g], 1£). (using the equivalence between the "states" 2Recall that the I-I relation between an operator 0(.) E 1£ and the state 10} E C is given by
10} = lim O(z) IO},
,-0
where IO} is the vacuum state.
(2.4)
BY-Algebras of W-Strings
291
and the "operators," one can switch freely between the "state" and the "operator" cohomology.) Since the underlying complex has the structure of a Yertex Operator Algebra, one also wants to know how much of this structure passes to the cohomology. While the computation of the cohomology is quite difficult (c.f., appendix A) it is rather straightforward to identify the algebraic structure of the cohomology, which I will now discuss.
2.2.
The BV-algebra structure of .\'.I[g]
A BY-algebra [36]-[38], (Q(, " ,1), is a tz-graded, graded commutative algebra Q( = EI1nEz Q(n with an additional structure of a BY-operator ,1 : Q(m --+ Q(m-l, which is a second order derivation on Q(, i.e.,
,1(a· b· c)
= ,1(a· b)· c + (-I)la la . ,1(b· c) + (_I)Ual-!lIb1b· ,1(a· c) -(,1a)· b· c - (_l)la la . (,1b)· c - (_I)lal+1b la . b· (,1c) ,
(2.6)
for all a, b, cEQ(, satisfying ,12 = O. BY -algebras are particular examples of Gerstenhaber (G-) algebras [39]. The latter are tz-graded, graded commutative algebras as above, but with the second order derivation replaced by a bracket operation, [-, -] : Q(m X Q(n --+ Q(m+n-I, with respect to which Q( becomes a graded Lie algebra, e.g., [a, b] = -( -I )llal-I)(lbl-!)[b, a] for a, b E Q(. Moreover, for each a E Q(, the operator [a, -] acts as a derivation of the algebra of degree lal- l. Recall that D : Q(m --+ Q(m+IDI is a (first order) derivation of degree IDI if it satisfies the Leibnitz rule D(a· b) = (Da) . b + (_l)laIIDl a . (Db). (2.7) The bracket operation that introduces on a BY-algebra the structure of a Galgebra, is given by
[a,bl=(-I)la l(,1(a.b)-(,1a).b-(_1)Ia 1a .(,1b)),
a,bEQ(,
(2.8)
and measures the failure of ,1 to be a derivation. The above structures arise naturally in the chiral algebra .\'.I[g] == II(W[g], It). The product and the BY-operator are defined as follows (see, (12,15]): The dot product of two operators in .\'.I is given by /'" = -I. ( v·O)(z) 27rl
i
dw /" , --v(w)O(z).
C, W -
(2.9)
Z
It is graded commutative according to the ghost number of the operators. It is also equivalent, in cohomology, to
(O·O')(z)
=-
lim O(w)O'(z).
w~z
(2.10)
K. Pilch
292
The action of the BV-operator. denoted by bo, is
(boO)(z) =
1
I cz
dw (w _ z) b[2](W)O(z) ,
(2.11 )
2Jrl
when> b[2](Z) is the antighost field associated with the Virasoro subalgebra of W[o]. Finally, one finds that the corresponding bracket is simply obtained as
[O,O'](z) = (_i)9 h(O)
1 2dw. (b~~O)(W)O'(Z). Ic z Jrl
(2.12)
It is clear from the definitions above that the subspace iJO[o] is an Abelian algebra with respect to the product. I will refer to it as the ground ring and denote by R[o]. Furthermore, iJI[O] is a Lie algebra with respect to the bracket and acts on iJ[o] by (infinitesimal) BV-algebra automorphisms. Among those automorphisms one finds 0, generated by the zero modes of the 9 currents acting on IL, and (Ul)l corresponding to the Liouville momenta, _ipL. With respect to 0 ED (uJlf the cohomology decomposes into a direct sum of finite-dimensional irreducible modules. By extrapolating from the known cases of W n , n = 2, :~, one expects that the following holds. i. W(W[o], IL) = 0 for n 11.
< O.
The decomposition of the ground ring R[o] == J[O(W[O], IL) as a 0 EB is given by
R[o]
~
EB
L(A):;~:)(CA,
(UI)f
module (2.13)
AEP+
i.e., R[o] is a model space of 0, meaning that each finite-dimensional irreducible module L(.I), A E J\, of 0 arises exactly once. Remarks: 1. This description of the ground ring of W n gravity in terms of the was anticipated in [40].
S(n
model space
2. Despite apparent simplicity of this result, there is no general proof of either (i) or (ii) available. In the known cases (i) and (ii) have been verified directly by an explicit computation of the cohomology. There is a natural BV -algebra, (<:p[o], . ,d s ), associated with the ground ring, where <:p[o] is the algebra of polyderivations of R[o]. Recall that a polyderivation R[o], of order n, is an alternating n-linear map on R[o] that is a derivation with respect to each of its arguments [42,43]. The BV-operator, d s ,
BY-Algebras of W-Strings
293
underlies the natural bracket operation on '+I[g], called the Schouten bracket [41,42]. An explicit construction of L1 s will be given in section 3. :'-low. one would like to know precisely how these two BY-algebras, fJ[g] and '+I[g], are related. The structure theorems proved in [15] for 512 and [18] for 513 suggest the following generalization. I.
There is a natural map Jr : fJ[g] ----+ '+I[gJ which is a HV-algebra homomorphism onto the BV-algebra of polyderivations ('+I[g], " L1 5 ).
II.
There exists an embedding 1 : '+I[g] satisfies Jr 0 1 = id.
----+
fJ[g] that preserves the dot product and
Remark: The projection, Jr, is defined by induction on the degree n [15]. For n = 0 it is just the identity map, as in degree zero '+I0[g] == R[g]. It is then extended to n > 0 using the condition (2.14) Jr(a)(x) = Jr([a, x]) ,
for any a E fJn[g] and x E R[g]. The polyderivations '+I[g], through the embedding 1, parametrize only a subsector of the cohomology, corresponding essentially to operators with the shifted Liouville momentum -iA L + 2p restricted to the fundamental Weyl chamber, i.e., -iA L + 2p E P+. An algebraic description of the remaining cohomology in terms of (generalized) polyderivations is rather cumbersome, see [18], and it is more advantegous at this point to seck a more geometric realization - first of the polydcrivations, '+I[g], and then of the entire cohomology, fJ[g].
3. 3.1.
The BV-algebra BV[g] The base affine space A( G) and its algebra of polyvectors
Consider the space, E( G), of regular functions on G. It carries the left and right regular representations of g. The operators representing the action of a generator eA will be denoted by [J~ and lJ f, respectively. The Peter- Weyl theorem asserts that as a 9 (i) 9 module (3.1 ) E(G) ~ L(A") ~ L(A),
EB
AEP+
where L(.1) and L(A") are the irreducible finite dimensional g-modules with highest (dominant integral) weights A and A", respectively, where A" = -woA is the conjugate of A. Given the Cartan decomposition (1.2), the base affine space of G [21] is defined as the quotient A = N+ \0, where N+ is the subgroup generated by n+. The space of regular functions E(A) consists of those functions in E(G) that are invariant under
294
K. Pilch ~
N+, and carries a representation of with respect to ~ GJ 9 yields E(A) ""
EEl g. Using (3.1), the decomposition of E(A)
EEl
!CA'
0 L(,1),
(3.2)
AEP+
i.e., E(A) is a model space of g. Roth algebras, E(A) and R[g] are finitely generated Abelian algebras. In fact, one expects that they should be isomorphic; the claim that is verified in the 5(2 [12,40] and 5[3 [18] cases by explicitly examining the set of generators of each algebra. The space of polyvectors, 'll( A), is defined as the space of regular sections of the homogenous vector bundle G XN+ 1\ b_, where 1\ b_ is an n+-module through the identification 1\ b_ ~ I\(n+ \g). Clearly, there is a grading 'll(A) = EBn'lln(A) induced from the decomposition
1\ b_
D
""
n
E9 1\ b_ ,
D
(3.3)
n=O
There is a natural action of 'lln(A) on E(A)0 n that identifies 'll(A) as polyderivations of the ground ring introduced in section 2. In fact, the two algebras, 'll(A) and 'll[g], are isomorphic. Later on 1 will need a convenient parametrization of 'll(A). It can be given by introducing a set of ghost oscillators {b a , e a }, a = -cr, i, with nontrivial anticommutators [b a, eb] = bab and associated ghost Fock space Fbc with vacuum Ibe) satisfying balbc) = O. One can identify 1\ b_ with Fbc, where the n+ action is given by 3 II~c = fob c cc bb. In particular, this identification induces a graded commutative product on pbc. Moreover, pbc is also an ~-module with the ~ generators flyc = fia b Cb b". Consider E( G) (9 1\ b-, the space of regular functions on G with values in 1\ b_. It has a natural structure of a graded, graded commutative algebra and carries commuting actions of 9 EEl ~ and n+, defined by the operators J1!} and fl{ + /ltc, and fl~ + fl~c, respectively. Both 9 EEl ~ and n+ act by derivations of the algebra product. In terms of E(C) ® 1\ b_ the poly vectors 'll[g] arc simply given by the n+-invariant elements, i.e., n
(3.4) While the polyvectors of order 0 are simply determined as the ground ring, the computation of higher order polyvectors is more involved due to the typically reducible, but indecomposable, action of n+ on 1\ L (see [18,21]). 3The summation convention used here is that the collective b_ and 9 indices are summed over their ranges.
BY-Algebras of W-Strings 3.2.
295
The cohomology problem
There is a nat ural generalization of the algebra '.P( A) ~ '.P[g] that is suggested by (3.4). Namely, one may consider instead of invariants of the n+ action, the Lie algebra cohomology
(3.5) of n+ with coefficients in E(G) \>9 /\ b_. Let me summarize this construction. First one introduces yet another set of ghost oscillators {a". w,,}, ex E .1+. corresponding to n+, with nontrivial anti-commutators [a",wll] = 8"1l and associated ghost Fock space Fa", with vacuum law) satisfying w"law) = O. The n+ action on Faw is given by n~w = -f"Il"allw", and the ~ action w by = - fi"" a" W". Again, there is a natural graded commutative product on
nr
Faw.
The cohomology H(n+,E(G) ® /\L) is now defined as the cohomology of the differential
d
=
a"
aw ) (n L + nbc + 111 2 0'
0:
Q'
,
(3.6)
acting on the complex C( G) == E( G) \>9 Fbc ® F"w. The complex C( G) is bi-graded by the be-ghost and the o-w-ghost numbers (gh(e a ) = -ghW) = (0, I), gh(o-") = -gh(w,,) =:: (1,0)), with d of degree (1,0). Clearly, this bi-degree passes to the cohomology. I will denote by /in (n+, E( G)@/\ b_) the cohomology in total ghost number n. l\'ote that the complex C( G) is a 9 G ~-module under and
Il!i
(3.7) With respect to this ~-action, the weights of b-" and w" are ex, whilst those of 0-" and c" are -no Since d commutes both with the action of 9 and ~, there is a direct sum decomposition
BV[g] =
EB EB
H(n+, L(A*)
8::!\ b_»,C\i £(A),
(3.8)
MP+ AEf'(C(A))
where erA) = £(A) 12: Fbc ® F"w, and P(V) denotes the set of ~-weights of V. The decomposition (3.8) reduces the problem of computing HV[g] to that of computing cohomology of finite-dimensional modules, £(A) (2) /\ L. The computation of the cohomology when the weight A lies sufficiently deep inside the positive Weyl chamber, P+, is relatively st.raightforward, and the result is given by the following theorem proved in [2ii]. Theorem 3. 1 Let A E P+. l.
The cohomology H"(n+, L(A) ® /\ b-)A' is nontrivial only if there exists awE W and'\ E p(/\k b_) such that A' = w(A + p) - p +,\ and n =:: I'(w) + k.
296
K. Pilch
ii. For 11 E P+ in the bulk, i.e. (A, a,) :::: N(g) jor some N(g) EN 8ufficiently large (in particular N(sln) = n - I),
p.9) Remarks:
I. The first part of the theorem is a necessary condition that holds for all weights A E f\. 2. Part (ii) shows that the cohomology in the bulk is computed as the standard Lie algebra cohomology of an irreducible g-module. The latter is given by the Kostant theorem [:lO],
/l(n+, L(I1))
3;'
E9 Cw(Mp)-p)'
(
(3.10)
wEW
In the cases where A lies close to the boundary of 1\ the cohomology is a proper subspace of /l(n+, L(I1)) <29/\ b_. For 9 = sl2 and sl3 it has been computed in [25].
3.3.
The BV-algebra structure of BV[g]
The space HV[g] carries a natural structure of a graded, graded cOIllIllutative algebra with the product"·" induced froIII the product on the underlying complex C(G). Indeed, an element (ji E c(n,m)(G) is of the form (jia, ...amCq ...Qn E
where 10) = Ibe) <29 luw). The product of two such elements, f/t E C(q,p)(G), is thus given by
(ji
E(G).
(3.11)
E c(n,m)( G) and
(:l.12) It is easy to verify that (3.1:3) from which it follows immediately that the product passes to the cohomology. Obviously, it is graded commutative according to the total ghost number. Consider the operator
.1 0 = -ba(/l/;
+ /l;W) + Uabcbabbcc,
(3.14)
where lI;w = - !r", iJ ba u"wj3. One verifies that [d, .10 ] = 0, and thus .1 0 is a welldefin<'d operator on BV[g]. Moreover, it follows from the explicit form of .1 0 that it is a second order derivation on E(G) ® F"w. (For example the first term in .1 0 is a product of first order derivations /l~ + Il: w on E( G) C$I F"W and ba on Fbc and thus this term is a second order derivation on the tensor product.) Finally, since -.1 0 can be viewed as the differential of the b- cohomology in E( G) (9 F"w, we have .16 = 0. This proves the following result.
BV-Algebras of W-Strings
297
Lemma 3. 2 ,10 is a BV-operator on BV[g].
It is natural to ask whether ,10 is a unique BV-operator on BV[g]. This turns out not to be the case as the following theorem shows. Theorem 3. 3 [25} Let 9 be a simple, simply laced Lie algebra and e : Q x Q ---+ {± 1} its asymmetry function with respect to the chosen Chevalley basis. 4 Define
L
,1' =
e L(o,a;)aOb-Obi
oELl+ i=\
Then ,1 t = ,10
+ t,1'
-
L
e(a,(3)a o + f3 b- o b- f3
•
(3.15)
o,f3ELl+
is a one parameter family of BV-operators on BV[g].
Remarks: I. It has been shown in [25] that ,1 t exhaust all 9 EB ~ invariant BV-operators on BV[g] for 9 = Sl2 and S13'
Consider the scaling a O -+ .\a o , W o -+ .\-\wo under which d -+ .\d. Then ,10 -+ ,10, ,1' -+ ,\,1' while the cohomology classes must scale homogeneously. This shows that all BY-algebra structures on BV[g] for t =J 0 are equivalent, and it is sufficient to consider just two BY-operators - ,10 and ,1\ = ,10 + ,1'.
2.
The role of the two inequivalent BY-structures introduced by ,10 and ,1\ on BV[g] is as follows. The algebra of polyderivations, '.ll[g], embeds as a subalgebra into BV[g],
(3.16) One finds that '.ll[g] C BV[g] is invariant under the action of ,10 (but not ,11)' In fact, ,1 s == -,1 u is nothing else but the BY-operator corresponding to the SchoutenNijenhuis bracket on '.ll[g]. The second BV-structure allows identification of two BY-algebras (n[g], " bo) and (BV[g], . ,,1\) in Conjecture (1). In fact the isomorphism n[g] ~ BV[g] holds as an isomorphism of 9 GJ ~ modules, if we identify the ~ generators as -ipf = -wu(ni ). The identification of the BV-operators is simply bo = ,1\. 3.4.
Example: the singlet subaIgebras for
9 = S(3
1 will now describe two algebras, no[sl:d and 13Vo [sI3], spanned by the elements in n[sl:!] and BV[sl:!], respectively, that transform as singlets under s13' Clearly, both subspaces are closed with respect to the product and the action of BY-operators, and, according to Conjecture (1), should be isomorphic as BY-algebras. 4Recall that «a, 13) = !o{J ~. <>.,13, -r E tl, see e.g. [44].
K. Pilch
298
By an explicit computation of the relevant cohomologies [18,25] one finds that singlet representations arise only for a finite number of (ud 2 weights, i.e., Liouville momenta. At each Liouville momentum one then finds a single quartet of states. The result can be read off from Table 1 in appendix A and is summarized in Figure 1.
0
0
0
0
0 0 0 0
0
• 2
0
0 0 0 0 0
0
• • 4
4
0
0
• • 4
0 0 0 0 0
• 6
0
0
4
5
0
0
0
0
0
0
0 0
• 4
0 0 0
0
Figure 1. The points on the fib-weight lattice lattice correspond to the shifted Liouville momenta -iA L +2p, or, equivalently, shifted (u)2- weights. The dots represent quartets of states and the labels give the ghost numbers of the prime state in each quartet. The boundary of the fundamental
Weyl chamber is outlined in thick lines.
Consider now clements at weight (0,0) in both algebras. In .iJo[sI3] the quartet of states consists of the unit operator, l(z), at ghost number zero, a pair of operators, C[21(z) and C[3!(Z), at ghost number one and a single operator, C[231(z) = (C[2]C[3!)(Z) at ghost number two. The explicit expressions for those operators are quite complicated and may be found in appendix B. Under the action of the BY-operator, bo, the quartet splits into two doublets, namely
(3.17)
!(
One can check that the operators C± = C[i! ± C[3]) generate, through the bracket, the (ud 2 algebra of the Liouville momentuIIl operator, _ipL. In BVo[sI3] one finds a similar quartet consisting of
(3.18) Here Ll\(CjC2)=2c\-2c2, (3.19) 2 and (-CI,-C2) generate the action of (ud . Thus, setting bo = Ll j , we have the correspondence (3.20)
L1\ci=-2,
i=I,2,
BY-Algebras of W-Strings
299
It is now an interesting question to determine how 1JO[S[3] and BVo [sI3] are generated as BY-algebras. This can be answered by studying explicitly the products and the action of BY-operators in both algebras. The result is Theorem 3.4 1. The BV-algebra 1JO[S[3] is generated by the elements, 1, C±, and Wi, i = 1,2, where WI and W2, given in appendix B, are opemtors at ghost number 1 at weights (0, -ad and (0, -(2). 2. Similarly, the BV-algebra BVo[sI3] is genemted by 1, Ci, and
(70<"
i
= 1,2.
3. The isomorphism between the two algebms is given by the identification (3.20) and Wi <--------> (7"', i = 1,2. (3.21 )
4.
Conclusions
Conjecture (1) has becn found to bc consistent with all cxplicit computations of the cohomologies. In particular, I havc argucd that it holds for the singlet algebras. It may be worth noting that the isomorphism 1J[g] ~ BV[g] holds at the level of cohomologics, and cannot be lifted to the underlying complexes. This follows from a simple counting of dimcnsions of both spaces. Onc may hope to prove this conjecture by showing that both algebras arisc, upon a suitablc rcduction, from a same structure. Such an expectation is supportcd by the fact that thc BRST current of W[g] gravity can be constructed by a Drinfel'd-Sokolov reduction of a super Kac-Moody algebra [23,24]. However, since it is not evcn known at this point how to use this reduction to determine the cohomology, it is rather hard to imagine that this path would yield an easy proof of the conjecture. In some sense one may consider the results presented here quite remarkablc, as while trying to discover ncw structurcs hidden in cohomologies of infinite dimensional nonlincar algebras onc ended up studying an old problem of ordinary cohomology in its classical form. Hopefully, with a new insight. Acknowledgments I would like to thank Jim McCarthy and Peter Houwkncgt for the collaboration. A partial support for this work was provided by the U.S. Dcpartment of Energy Contract #DE-FG03-84ER-40168. Appendix A.
The cohomology H(W3 ,C)
In this appendix I sUlllmarize the result of the cohomology computation for the W 3 string. As discussed in scction 2, one is interested in computing the space 1I(W3 ,C), where C is the complex (2.3). Let me point out some aspccts of the result that will simplify its presentation. First, H(W3 ,C) is naturally a direct sum of irreducible 9 ED ~ modules. Secondly,
300
K. Pilch
as a vector space, it has a natural quartet decomposition, such that each quartet contains states at ghost numbers n, n + I, n + I and n + 2. The states with the lowest ghost numbers in the quartets are called the "prime states," and the corresponding subspace in the cohomology is denoted by H pr (W3 ,C). Finally, the cohomology lies in cones of 9 EB ~ modules, each cone parametrized by a pair of weights, (11, A'), A E P+, A' E P, at its tip and an element, w, of the Weyl group, W, of 09[3. More specifically, a cone (A, A')w is the direct sum of modnles with the highest weights (A + ,x, A' + w-I,X), ,x E P+. ~ote that the second weight gives the value of the Liouville momentum, _ipL. Let AI and A2 be the fundamental weights of 5(3. The result for the cohomology is given by the following theorem.
Theorem A.I [18}. The cohomology /l(W 3 , q:) is isomorphic, as an 5(:\8 (ud 2 module, to the direct sum of quartets of irreducible 09(3 @ (UI)2 modules with the highest weights in a set of disjoint cones {(A, A') + (,x, w-I,X) I,X E 1\(09(3), (A, A') E Sw} labeled by wE W(09(3); i.e., 11;'r(W3 ,q:) ~
when the sets
S~
EB
EB
EB
wEW(,.)
(A,A')ES~
'\EP+
(L(A +,X) Q9 iCA'+w- 1 .\)
•
(A.I)
(tips of the cones) are listed in Table 1, and H"
~
H"pr
ill W
/l,,-I pr
C:l \.LJ
11"-1 pT
ill \.LJ
11,,-2 . pr
(A.2)
I refer the reader to [J 8] for a more detailed discussion of this cohomology. Here let me only demonstrate how the base affine space of 09[3 arises from the above result. Consider the cohomology at ghost number zero. It consist of a single cone with tip (0,0) and w = I, i.e., it is concentrated in the fundamental Weyl chamber. At the lowest weights one finds the unit operator, I(z), and, at Liouville momenta pL = iA 1 and iA 2 a triplet, Xi, and an anti-triplet, xi, i = 1,2,3. Since there is no cohomology of ghost number zero with Liouville momentum pL = i(A 1 + A2 ), and transforming as a singlet under 5(3, the following relations must hold Xi' xi
=
o.
(A.3)
One may show [18] that the products of the Xi and Xi span the entire cohomology at ghost number zero. Then the constraint (A.3) shows that the ground ring R[o9(3] ~ c[Xi, x i ]/ (Xi' xi) is isomorphic with the algebra of functions on the base affine A(SL(3)).
Appendix B.
Some explicit operators in fJO[5(3]
An operator O(z) E fJO[5(3] is of the form O(z) =
p[8>L,i,C[Jl,b[jl, ...
]VO._ iA L(Z) ,
(8.1)
BY-Algebras of W-Strings
301
where P[ . .. ] is a polynomial in the fields iJ
Appendix B.l.
The generators of the identity quartet
-4 (iJc[2] + Db 12 ]iJc[3]e[3]) -
C[2]
-fi (fJIj>L,1 + y'30Ij>L,2)( e[2] + b[2]oe[3]c3])
_4b[2]j)2(yJ]c[3] -
-fi( J3iJ
+J2( J3a 2
+ iJ2rj>L,2)(P]
c l3]
-4y'3iJ2 e [3] -
I -
o
- ':!f(iJrj>L,liJrj>I,,1 - iJL,2)C[3] ,
.ft( y'3fJ
O
_y'3( J2fJ 2rj>L,1 + 30 2Ij>L,2)C[3] - 3~(Orj>L,1 + j3iJ<jJL,2)iJC[3] - V}(o<jJL,larjJL,l
Appendix B.2.
t/J I
(
+ 2y'30<jJL,liJIj>L,2 + O<jJL,2iJ<jJL,2)c[3].
The generators t/J1 and t/J2
6b[2] (;l2]iJ2 C[3] c[3] - 6b[2]Dc[2]8c[3J c [3] + 12y'3b[2]82e[3]iJc[3]e[3] +3y'6fJIj>L,1 e[2] OC[3] + 9iJ<jJL, 1 8<jJL,1 8c[3]c[3] + 6j38<jJL,1 8<jJL,2 8C[3]C[3J _3y'6iJ¢>L,18c[2]C[3J + 9J28rjJL,liJ 2c[3]e[3] + 3J28¢>L,2 c [2]8c[3] +3fJq.,L,28Ij>L,2iJc[3]C[3] _ 3J2iJ<jJL,28e[2]c[3]
+ 3y'68<jJL,2iJ2c[3]e[3]
+68b[2]e[2J oc[3]e[3] + 6fJc[2]ef2] - 6y'38c[2J8e[3] +
3 y'68 2q.,L,1 e[2J e[3]
+
-9J28 2
t/J2 = (_6b[2]c[2]iJ 2c[:I]e[:I] + 6b[2]iJc[2]8el3]c[3] + I 2y'3b[2]8 2e[3]iJe[3] c[3J
+6y'3e[2]a 2c[3J + 6J2iJq.,L,2 e[2]8c[3] - 12o<jJL,2 o <jJL,28e[3]e[3J - 21iJ 2c[3]8c[3] _83c[3]C[3] _ 6ob[2]c[2]oe[3]c[3] - 6oe[2]e[2] - 6y'3oe[2]iJe[3J +6J28 2
302
K. Pilch
° 1
1 1 7"1 7"2
2
1 7"[ 7"2 7"12 7"21
3
1 1'1 1'2 1']2 1'2 [ 7"3
4
1'] 1'2 7"[2 1'21 1'3
5
(A 2,A] - A2), (A] + A2,0), (A],-A I (0, -2A] + A2 ) (0, Al - 2A 2 )
+ A2)
(2A 2, - A2), (0, - A] - A2), (2A I , - /h) (A], -2A]), (A 2 , -3A[ + A2), (0, -4A I + 2A 2) (A 2 , -2A 2), (AI, A] - 3A 2), (0,2A] - 4A 2 ) (0, -3A 2) (0, -3A]) (A] + A2, -AI - A2) (A 2 , -2A I - A2), (A], -4A] + A2 ), (A 2 , -5/h + 2A 2 ) (AI, -A[ - 2A 2 ), (A 2 , Al - 4A 2), (AI, 2A I - 5A 2) (A 2, -AI - 3A 2 ), (0, A] - 5A 2), (A 2 , -5A 2) (AI, -3A] - A2 ), (0, -5A] + A2 ), (AI, -5Ad (0, -2A] - 2A 2)
(0, -4A] - A2) (0, -A] - 4A 2 ) (A 2 , -2A 1 - 4A 2 ), (AI, -A] - 5A 2 ), (0, -6A 2) (A], -4A I - 2A 2), (A 2 , -5A I - A2), (0, -6Atl (0, -3A[ - 3A 2), (2A I1 -4A] - 3A2), (2A 2, -3A 1 - 4A 2 )
1'3
(0, -2A] - 5A 2 ) (0, -5A] - 2A 2) (A], -5A] - 3A 2), (A] + A2 , -4A] - 4A 2 ), (A 2 , -3A] - 5A 2)
1'3
(0, -4A I - 4A 2 )
7"12 7"2]
6
(A, A') (0,0)
Table 1. The sets
S::,
BY-Algebras of W-Strings
303
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