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Symmetry Methods in Physics in Memory of Professor Ya. A.Smorodinsky International Workshop SYMMETRY METHODS IN PHYSICS IN MEMORY OF PROFESSOR YA. A. SMORODINSKY Volume 2 Edited by A.N.Sissakian G.S.Pogosyan S.I.Vinitsky Dubna 1994 NOTICE PLEASE BE AWARE THAT THIS IS THE BEST REPRODUCTION POSSIBLE BASED UPON THE ORIGINAL DOCUMENT RECEIVED JOINT INSTITUTE FOR NUCLEAR RESEARCH International Workshop SYMMETRY METHODS IN PHYSICS IN MEMORY OF PROFESSOR YA.A.SMORODINSKY Dubna, Russia July 6-10 1993 Edited by A.N.Sissakian G.S.Pogosyan S.I.Vinitsky Volume 2 Dubna 1994 Published by Publishing Department Joint Institute For Nuclear Research Joliot Curie, 6 141980, Dubna, Moscow Region Russia PROCEEDINGS OF THE INTERNATIONAL WORKSHOP ON SYMMETRY METHODS IN PHYSICS IN MEMORY OF PROFESSOR YA. A.SMORODINSKY Photographs: By Yu.A.Tumanov © Joint Institute For Nuclear Research, Dubna, 1994 CONTENTS Volume II A. J. Macfarlane Generalised oscillator systems and their parabosonic interpretation 319 L. G. Mardoyan, A. N. Sissakian, V. M. Ter-Antonyan, T. A. Ghatrchian Generalization of the Rayleigh formula to the model with ring-shaped poten- tials 326 S. Mashkevich Symmetries and quantum mechanical spectra of anyons 332 R. M. Mir-Kasimov The Snyders space-time quantization, Poincare group deformations and ultra- violet divergences 337 S. S. Moskaliuk, Yu. F. Smirnov Using reduce system for calculation of integrity bases of Lie group invariants 346 F. Miiller-Hoissen, A. Dimakis Differential forms and gauge theory on discrete sets and lattices 351 M. A. Mukhtarov On integrability of generalized toda lattice in quantum domain and self-duality equations for arbitrary semisimple algebra 358 A. F. Nikiforov Classification of q-polynomials as polynomial solutions of hypergeometric type difference equations 361 A. P. Nersessian BV-geometry on Kahlerian supermanifolds 369 A. V. Nesterov, A. G. Kosinov Generating invariants in many-cluster microscopic problems of nuclear physics 375 J. Nyiri Boson vacuum polarization in the field of a supercritical charge 378 V. S. Otchik On the two Coulomb centres problem in a spherical geometry 384 iv V. Papoyan, G. Haroutyunian The conform connect/on between equations of GR and the Jordan-Brans- Dicke theory 389 V. Pervushin, V. Papoyan Quantum evolution of the Universe 396 A. D. Popov Symplcctic manifolds with symmetry and weakly G-invariont complex struc- tures 404 Z. Popowicz " SUSY Lax pair in the Gelfsnd-Dickey hierarchy 409 P. N. Pyatov, A. P. Isaev Towards the classification of the differential calculi on g(,(n) 1,15 Y. P. Rybakov Maximally invariant configurations in the SU($) Skyrme model 4^3 V. L. Safonov Symmetry and statistics of the elementary collective excitations in solids ... R. M. Santilli Application of isosymmetrics/Q-operator-deformations to the cold fusion of elementary particles 433 V. I. Sanyuk Algebraic and analytical features of (3+1) dimensional topological solitons . 443 P. Schaller Loop space and W-type algebras 450 W. Scherer, H. D. Doebner A nonlinear Schrddinger equation and some of its solutions 454 A. Schirrmacher Varieties on quantized spacetime 463 M. B. Sheftel' Symmetries, recursions and linearization for two-component systems of hy- drodynamic type 471 N. A. Smirnova, Yu. P. Smirnov, Level clustering in the vibrational-rotational spectra of the icosahedral Hamil- tonian 475 V Yu. F. Smirnov, A. Del Sol Mesa Orthogonal polynomials of discrete variable associated with quantum algebras su,(2) and su,(l, 1) 479 A. I. Solomon, R. J. McDermott General deformations of bosons and their coherent states 487 A. N. Sissakian, I. L. Solovtsov, O. Yu. Shevchenko Method of variational perturbation theory 494 Б. Sorace Inhomogeneous quantum groups in physics 501 V. Spiridonov Infinite soliton systems, quantum algebras, and Pain/eve equations 512 I. A. B. Strachan Infinite dimensional Lie algebras and the geometry of integrable systems ... 519 S. I. Sukhoruchkin Relationship in baryon mass spectrum as a possible reason of the fine structure of nuclear excitation 538 S. I. Sukhoruchkin Manifestation of electrodynamics parameters in energies of nuclear states and in particle masses .... 536 A. A. Suzko Geometric nonadiabatic phases and supersymmetry 544 V. M. Ter-Antonyan, A. N. Sissakian Matrix of finite translations in oscillator basis 55S N. "V. Tho, V. I. Kuvshinov Geometry of group manifold and properties of chiral fields in vector parametrization of groups 556 L. Vinet, P. Letourneau Dynamical polynomial algebras in quantum mechanics 563 A. A. Vladimirov On quasitriangular Hopf algebras related to the Borel subalgebra of ah 574 R. F. Wehrhahn, A. O. Barut Geometric motion on the conformal group and its symmetry scattering 577 vi P. Winternitz, G. Rideau Spherical Functions for the quantum group su(2), 58J V. I.Yukalov, E. P. Yukalova, A. A. Shanenko Influence of colour symmetry on string tension 592 B. N. Zakhariev Fragments of reminiscences and exactly solvable nonrelalivistic quantum mod- els 595 В. M. Zupnik Quantum deformations for the diagonal R-matrices 599 319 GENERALISED OSCILLATOR SYSTEMS AND THEIR PARABOSONIC INTERPRETATION A. J. Macfarlane Department of Applied Mathematics & Theoretical Physics University of Cambridge, Cambridge CB3 9EW, U.K. 1. Introduction In this work, we consider the Fock space descriptions of and some interesting properties of various bosonic oscillator systems. All are based on a single creation-annihilation pair. All are represented in a suitable Fock space of states |n) of form (a*)"|0), n = 0,1,2 • • with a ground state |0) such that o|0) = 0 and a number operator N such that N\n) = n|n). we pass from the quantum harmonic oscillator with its well-known commutation relations [a,a'] = 1. to the study of a generalisation of it that we have elsewhere [1] called modification. This gives rise to the Calogero-Vasiliev oscillator [2],[3],[4] governed by [0,0'] = l+2i/A", (1) where i/gR, and К = (—)N obeys K = K\ K2 = 1, аК + К а = 0, а* К + Ка* = 0. Below we review the construction of its Fock space !FU, for which 2e > —1 emerges as a condition sufficient to ensure the hermiticity properties implied by use of the dagger in a', and also the sti(l,1) and озр( 1|2) properties of A notable result, believed new here, is then explained. It states that for v = (p — l)/2, Tv coincides with the Fock space of a single paraboson of order p = 1,2* • •• An important consequence of this observation is that Cor Fock space work, one may use the bilinear commutation rule (1), with 1v — p+ 1, instead of the more awkward trilinear commutation rules [5] that characterises paraboson systems of order p. Then we go on to consider the q-deformation of the Calogero-Vasiliev oscillator. Setting out from the Fock space description of this, it is easy to follow the familiar method of replacing round brackets suitably by square brackets of the type [*: r] = w<Tr). [*;!) = [*]• (2) This leads to a system involving a pair b and ft1 governed by 44« _ 9±<i+2Wf)6t6 = [1 + 2(3) in which 9 and i- € R, and К and N have the meanings indicated above. While (3) might seem unduly complicated, it does describe a very natural q-deformation of the Calogero-Vasiliev oscillator and calculations in the Fock space F4U in which it is represented are easy enough to do. Our Fock space methods allow us to derive a variety of interesting simple formulas that hold true in these can be used to show clearly its relationship to the various simpler systems that can be reached from it as special or limiting; ctses. We show how interpolates between 320 the spaces, defined for v = (p — l)/2, that describe q-deformed parabosons of order p — 1,2 • • •• Wc discuss also the relationship of T4V to representations of su(l, 1), and osp(l|2),. Wo discuss next some relationship between the content of the present paper and previous work. Firstly we mention our recent paper [1] in which a different approach was followed to the combination of q-deformation and (/-modification. There, rather than effecting a q-deformation of (1) as here, we considered tile simplest modification of the now familiar commutation relation [6] pi.I»] t ( aa - qa]a = q~N, (4) namely ae' — fja'a = g~N(l + 2fK). (5) The consequences of (5) - its Fock space, su(l, 1), and osp( 1|2), properties, ar. 1 hidden super- symmetry - show similarities with features of the present work, but have no parabose aspects. We ttesd the term Calogero-Vasiliev oscillator in relation to (1), because the latter author introduced it first explicitly [2] and gave its Fock space representation, and because it plays a crucial role in the modern progress in understanding [3],[4],[9] of the integrable many body model discovered by tlie former author [10]. Previous treatment of q-analogues of parabosc (and also of parafermi) oscillators appears in [11]. This paper also discsses their su(l,l), or sp(2R), and oap(1|2), properties, but not the relationship to ^-modification. We remark here that our approach gives rise to various interesting formulas not given explicitly in [11]. Wc mention here two works on general deformation schemcs, including parabose and para- fcrmi oscillators, namely the paper [12], which surveys earlier work, and the paper [13]. There is also the article [14] useful especially for su(l, 1), questions.Thc paper [15] contains information of q-''eformation of parabosc and parafermi operators, and mentions the relation- ship of osp(l|2)? representations to the Fock space of parabose oscillators. Finally wc note a matter which attracted debate at the International Workshop at which the work described in this paper was presented. The matter concerns the understanding of systems of the type in focus here within the context of I.ie-admissibie algebras.
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