erutceL Notes scisyhP in detidE yb .H Araki, ,otoyK .J ,srelhE ,nehcnJiM .K Hepp, hcirJiZ .R ,nhahneppiK ,nehcnJiM H.A. ,rellJimnedieW ,grebledieH .J ,sseW ehurslraK dna .J ,ztrattiZ n16K gniganaM Editor: W. Beiglb6ck 162 lamrofnoC Groups dna detaleR seirtemmyS lacisyhP stluseR dna lacitamehtaM dnuorgkcaB Proceedings of a Symposium Held at the Arnold Sommerfeld Institute for Mathematical Physics (ASI) Technical University of Clausthal, Germany August 12-14, 1985 Edited by A.O. Barut and H.-D. Doebner galreV-regnirpS Berlin Heidelberg NewYork London Paris Tokyo Editors A.O. Barut Department of Physics, University of Colorado at Boulder Boulder, Colorado 80309, USA H.-D. Doebner Arnold Sommerfeld Institute for Mathematical Physics TechnicalU niversity Clausthal D-3392 ClausthaI-Zellerfeld, FRG ISBN 3-540-17163-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-17163-0 Springer-Verlag NewYork Berlin Heidelberg sihT work si tcejbus Ot .thgirypoc llA sthgir era ,devreser rehtehw of part or whole the eht lairetam si yllacificeps ,denrecnoc esoht of ,noitalsnart ,gnitnirper esu-er of ,snoitartsulli ,gnitsacdaorb noitcudorper yb gniypocotohp enihcam or ralimis ,snaem dna egarots ni data .sknab rednU of 54 § eht namreG thgirypoC waL copies where era edam than other for etavirp ,esu fee a si elbayap to tfahcsllesegsgnutrewreV" Wort", .hcinuM © galreV-regnirpS Berlin grebledieH 6891 detnirP ni ynamreG :gnitnirP suahkcurD Beilz, ;.rtsgreB/hcabsmeH :gnidnibkooB .J reff,~hcS OHG, tdatsn~rG 2153/3140-543210 PREFACE This volume contains contributions presented at an International Symposium on Conformal Groups and Conformal Structures held in August 1985 at the Arnold Sommerfeld Institute for Mathematical Physics in Clausthal. We hope that the wide range of subjects treated here will give a picture of the present status of the importance of the conformal groups, other related groups and associated mathematical structures (such as superconformal algebra, Kac-Moody algebras), and spin struc- tures. Symmetry, with group theory and algebras as its mathematical model, has always played a crucial and significant role in the develop- ment of physical theories. One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincar~ group. It opens the door to appli- cations far beyond the standard kinematical framework provided by the local symmetries of flat space-time. It is stimulating to recognise the progress which has occurred in the last 15 years by comparing these proceedings with those of a similar con- ference held in 1970 (A.O. Barut, W.E. Brittin: De Sitter and Conformal Groups and Their Applications, Colorado University Press 1971). The emphasis ihas changed and numerous new fields have appeared which are mathematically and physically associated with the conformal group. The great interest shown in this conference and the material presented in this vol~ne indicates that the field centred around conformal symmetry is very much alive and active. The material is organised into six chapters: I. Symmetries and Dynamics If. Classical and Quantum Field Theory III. Conformal Structures IV. Conformal Spinors V. Lie Groups, -Algebras and Superalgebras VI. Infinite-Dimensional Lie Algebras The papers range from direct physical appiications~ (e.g.p. Magnollay and Dj. ~ija~ki) to the presentation of mathematical methods and results (e.g.V.G. Kac) with likely future influence on particle phyiscs. We have also included articles with a bias towards fundamental questions using syn~setry tO reinforce parts of the foundations of physics and of space-time structure (e.g.C.F.v. Weizs~cker and also P. Budinich). VI Some of the developments during xecent years, and hence some of the contributions, have utilized conformal symmetry in combination with e.g. differential geometric and algebraic structures, as in string theory (e.g.Y. Ne'eman). There are also research reports based on applications of groups related to the conformal group (e.g. SL(4,R)). The extended lectures by I.T. Todorov on "Infinite Dimensional Lie Algebras in Con- formal QFT Models" aims to give new results combined with a review as an introduction to an important and fast-growing subject. Furthermore, some articles present reviews in a new and updated context. We have also inCluded the material of some of the invited speakers who did not have the opportunity to present it at the conference. To give this volume special value to postgraduate students and to physicists and mathematicians who want to enter the field, we asked for contributions which contain some introductory and review sections. ACKNOWLEDGEMENTS We wish to express our gratitude to the following organisations and persons for generous financial support and other assistance making the conference symposium and this volume possible: Der Nieders~chsische Minister fur Wissenschaft und Kunst, Hannover - - Alexander von Humboldt-Stiftung, Bonn - Technische Universit~t Clausthal, specifically Rector Prof. Dr. K. Leschonski Dr. V.K. Dobrev (Sofia/Clausthal) and Dr. W. Heidenreich (C1austhal were scientific secretaries of the symposium. Dr. W. Heidenreich assisted furthermore in the preparation of the proceedings. We want to thank Springer-Verlag, Heidelberg, for their kind assistance in matters of publication. Last but not least we want to thank the members, co-workers and especially the students of the Arnold Sommerfeld Institute and the Institute of Theoretical Physics A who made the symposium run so smoothly and efficiently. H.-D. Doebner A.O. Barut TABLE OF CONTENTS me SYMMETRIES AND DYNAMICS A. O. BARUT From Heisenberg Algebra to Conformal Dynamical Group ............................. 3 DJ. SIJACKI SL (4,R) Dynamical Symmetry for Hadrons .... 22 P. MAGNOLLAY A New Quantum Relativistic Oscillator and the Hadron Mass Spectrum ................... 34 A. INOMATA Path Integral Realization of a Dynamical R. WILSON Group ...................................... 42 M. IOSIFESCU Polynomial Identities Associated with H. SCUTARU Dynamical Symmetries ....................... 48 TH. GORNITZ De-Sitter Representations and the Particle C. F. V. WEIZS~CKER Concept, Studied in an Ur-Theoretical Cosmological Model ......................... 63 II. CLASSICAL AND QUANTUM FIELD THEORY D. BUCHHOLZ The Structure of Local Algebras in Quantum Field Theory ...... ......................... 79 M. F. SOHNIUS Does Supergravity Allow a Positive Cosmological Constant? ...................... 91 W. F. HEIDENREICH Photons and Gravitons in Conformal Field Theory ..................................... 101 B. W. XU On Conformally Covariant Energy Momentum Tensor and Vacuum Solutions ................ 111 C. N. KOZAMEH The Holonomy Operator in Yang-Mills Theory. 121 III. CONFORMAL STRUCTURES B. G. SCHMIDT Conformal Geodesics ........................ 135 J. D. HENNIG Second Order Cenformal Structures .... ...... 138 H. FRIEDRICH The Conformal Structure of Einstein's Field Equations .................................. 152 C. DUVAL Nonrelativistic Conformal Symmetries and Bargmann Structures ........................ 162 IV IV. CONFORMAL SPINORS M. LORENTE Wave Equations for Conformal Multispinors... 185 P. BUDINICH Global Conformal Transformations of Spinor L. DABROWSKI Fields ...................................... 195 H. R. PETRY P. BUDINICH Pure Spinors for Conformal Extensions of 205 Space-Time .................................. J. RYAN Complex Clifford Analysis over the Lie Ball. 216 Vo LIE GROUPS, -ALGEBRAS AND SUPERALGEBRAS R. A. HERB Plancherel Theorem for the Universal Cover J. A. WOLF of the Conformal Group ...................... 227 G. v. DIJK Harmonic Analysis on Rank One Symmetric Spaces ...................................... 244 H. P. JAKOBSEN A Spin-Off from Highest Weight Repre- sentations; Conformal Covariants, in Particular for 0(3,2) ....................... 253 E. ANGELOPOULOS Tensor Calculus in Enveloping Algebras .... .. 266 R. LENCZEWSKI Representations of the Lorentz Algebra on B. GRUBER the Space of its Universal Enveloping Algebra ................................ • .... 280 V. K. DOBREV Reducible Representations of the Extended V. B. PETKOVA Conformal Superalgebra and Invariant Differential Operators ...................... 291 V. K. DOBREV All Positive Energy Unitary Irreducible V. B. PETKOVA Representations of the Extended Conformal Superalgebra .............. .................. 300 VI. INFINITE-DIMENSIONAL LIE ALGEBRAS Y. NE'EMAN The Two-Dimensional Quantum Conformal Group, Strings and Lattices ........................ 311 V. RITTENBERG Finite-Size Scaling and Irreducible Repre- sentations of Virasoro Algebras ............. 328 V. G. KAC Unitarizable Highest Weight Representations M. WAKIMOTO of the Virasoro, Neveu-Schwarz and Ramond Algebras .................................... 345 J. MICKELSSON Structure of Kac-Moody Groups ............... 372 D. T. STOYANOV Infinite Dimensional Lie Algebras Connected with the Four-Dimensional Laplace Operator.. 379 I. T. TODOROV Extended Lecture: Infinite Dimensional Lie Algebras in Conformal QFT Models ........ ~ ............... 387 FROM HEISENBERG ALGEBRA TO CONFORMAL DYNAMICAL GROUP .A .O Barut Department of Physics Campus Box 390 University of Colorado Boulder, CO. 80309-0390 ABSTRACT The basic algebraic structures in the quantum theory of the electron, from Heisenberg algebra, kinematic algebra, Galilean, and Poincar~ groups, to the internal and external conformal algebras are outlined. The universal role of the conformal dynamical group from electron, H-atom, hadrons, to periodic table is discussed. I. Introduction The postulates of quantum theory can be expressed most concisely as the representation theory of the symmetry groups and dynamical groups of physical systems. And the analytical methods and specific calculations in quantum theory are performed most economically in terms of the representations of the Lie and envelopping algebras and their matrix elements. In these notes I give an outline of the developments of the group theoretical ideas and methods mainly for the electron, but of course also applicable for other quantum systems. With an audience of both mathematicians and physicists in mind, I hope this presentation will be elementary and self-consistent, although some may find the text to be a bit too mathematical, others to concise in physics. II. The Heisenberg Algebra h n and Kinematical Algebra k n The algebraic quantum theory goes back to the initial work of Heisenberg, and the Born-Jordan-Heisenberg formulation of quantum mechanics. For a mechanical Hamiltonian system of n-degrees of freedom with n generalized coordinates qi, and n conjugate momenta Pi, i = ,i ,2 .... n, we have the Heisenberg algebra h n defined by the commutation relations: qi' qj = 0 ' Pi' Pj = 0 ; ,i j = 1,2 .... n h : (I) n qi' Pj = i~ ~ij J ' Ji qi = 0, ,J pj = 0 Here we have introduced, for purpose of later generalization, an operator J which in h n has been choosen to be the identity operator. This can be done as long as, as is well known, p's and q's are not finite-dimensional matrices. Originally Heisenberg introduced Pi, qi as matrices in the energy basis of the quantum system. With the advent of transformation theory and Hilbert space formulation, eqs. (i) are general operator relations independent of basis. The Heisenberg algebra h n can be extended to a kinematical algebra k n with the inclusion of SO(n)-rotation elements £ij = - £ji. The additional commutation relations to eqs. (I) are qi' £jk = i~ (~ik jq - ~ij qk ) Pi' £jk = i~ (~ik Pj - 6ij Pk ) k': (2) n £ij' ~£ = i~ (~ik £j£ + 6j£ £ik - ~jk £i£ - ~i£ £Jk ) £ij ,J = 0 1 The dimension of k n = h n + k n' is ~n+l)(n+2) , the same as that of the Lie algebra of SO(n+2) or SO(n,2). Any representation of h n can be extended to a representation of k n by the following realization of £ij: £ij = qi Pk - qk Pi (3) derived from the physical meaning of £ij as the components of orbital angular momentum. In this case k n is just a derived algebra from hn, a Lie algebra in the enveloping algebra of h n. For this type of representations of kn, the representations of k n remain irreducible for the subalgebra hn; conversely representations of h n are automatically extended to the representations of k n. But there are other representations of k n. For example we can set Aij = qj Pi - qi Pj + Sij (4) where Sij are the spin operators. We can then enlarge our dynamical system by the inclusion of the commutation relation of Sij , Sij , Sk£ , or just keep the algebra kn, independent of the realizations (3) or (4), and consider all its representations. Sofar the kinematic algebra k n describes the quantum system at a fixed time t. They can be realized also as differential operators acting on a time-dependent wave function ~(q,t) (SchrDdinger representation), or they can be given a time-dependence q = q(t), p = p(t), acting or a time-independent Hilbert space (Heisenberg representation). Since the Hamiltonian system is characterized by a Hamiltonian H and the time evolution of the system by a unitary operator U(t-t0) = e-i~H(t-t0 ,) we have a quantum dynamical system of 2n-dimensions: = ! ,H jq pj = { i H, Pjl , j = 1, 2 .... n (5) Because we are interested in the generalization of the operator J in eq. (1), it is important to note that if one postulates quantum mechanics first by eqs. (5), instead of eqs. (i), the most general Heisenberg commutation relations compatible with (5) are of the form I qi' Pj = iI ~ij F (6) where F can be a function of the Hamiltonian. A nonrelativistic quantum system must also show the symmetry under Galilean transformations of space and time if it is a system existing in space-time. For this purpose we introduce the total momentum of the system ~. If qi are the + cartesian coordlnates, then ~ = El + .... + Pn, otherwise ~ is related to Pl, qi in a more complicated way. Similarly, the system will have a total angular momentum ~, also a function of p's and q's. The Introdution of the generators of velocity (or boost) transformations is more subtle. They have explicit time-dependence in addition to their time evolution = ~ ~j = I (tP i - mj qj) , (7) J J for Cartesian coordinates qj. The ten operators P0 = H, ~, ~ and ~ are the generators of the Galilean group G . The representations of the Galilean group G cannot completely characterize our dynamical system of 2n degrees of freedom; the system is composite, it has a lot of internal degrees of freedom; the representa- tion of G will be highly reducible. Irreducible representations of symmetry group apply to elementary systems. 2 In the purely geometric definition of the Galilean algebra we have '~ Pi = 0 (8) But in the quantum mechanical realization (7) we have 3 M~ j) , ek = ih m (j) ~ik (9) or, more generally, M i , ek = lh~ 6ik (9') where~ is a mass operator. This is another instance, llke eqs. (i) and (6), where we obtain new operators J, F,~ in generalizing the simple commutation relations. The mathematical interpretation of (9) instead of (8) is that quantum