The unitary irreducible representations of SL(2, R) in all subgroup reductions a) Debabrata Basu b) and Kurt Bernardo Wolf lnstituto de Investigaciones en Matematicas Aplicadas yen Sistemas (IIMAS). Universidad Nacional Aut6noma de Mexico. Apdo. Postal 20-726 Mexico 20. D. F.. Mexico (Received 23 December 1980; accepted for publication 5 1 une 1981) We use the canonical transform realization of SL(2, R ) in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group. Explicit results are given for all mixed bases and subgroup reductions. These provide the full multiparameter set of integral transforms and series expansions associated to SL(2, R ). P ACS numbers: 02.20.Df 1. INTRODUCTION The complete classification of the Unitary Irreducible called canonical transforms. IX.2X It is unique in that the asso­ representations (UIRs) of the three-dimensional Lorentz ciated Lie algebra is an algebra of second-order differential group SO(2, 1) and of its twofold covering group SL(2, R ) operators on a dense common domain in these Hilbert were given by Bargmann in his classic 1947 article,1 where spaces. The action is thus distinct from-although unitarily one can find the UIR matrix elements-rows and columns equivalent20.21 to-the SL(2, R) action as a Lie transforma­ classified by the UIRs of the compact subgroup SO(2)-in tion group on coset spaces, of the Lie-Bargmann multiplier L2 representations29 on the unit circle or disk. explicit form. This group, its covering groups SO(2, 1)~ The canonical transform realization has provided a de­ 1 00 ---- l gree of uniformity in the treatment of the discrete series '! of SUr 1, 1)~Sp(2, R );:::; SL(2, R ) ~ SL(2, R ) and its represen- UIRs on the one hand and the continuous series21 of UIRs 2 tations were further studied by Barut and Fronsdal, Pu­ on the other. In this article it has enabled us to evaluate, in a 5 kanski/ Sally, 1r.,4 and in a book by Lang. straightforward and unified way, the UIR matrix elements The study of group representations in different bases is and integral kernels of finite SL(2, R ) elements. In contrast of interest both from the mathematical and the physical with some of the previous investigations, this approach deals point of view. The intimate connections between the repre­ with the general SL(2, R ) group element, rather than with sentations of Lie groups and the special functions of math­ specific one-parameter subgroups. Although Bargmann's ematical physics have long been recognized and treated in results on UIRs ofSL(2, R ) in the compact subgroup basis30 6 textbooks. In physics, subgroup reductions corresponding are well known, it is also true that other continuous noncom­ to different bases of the Lorentz and other groups lead to pact and mixed-basis reductions have so far not received various ways to correlate or interpret data, as in the descrip­ uniform consideration2.9.lo.12,31-13 and are scattered in the tion of the high-energy scattering dynamics,7 which requires literature. The discrete series ofUIRs in all subgroup reduc­ the reduction SO(2, 1):J SOt 1, 1) among others. This interest tions was undertaken by Boyer and W 01(,4 using canonical coincided with the investigations ofMukunda,R-11 Barut,2.12 transforms. We repeat their results here since the journal is Lindblad and Nagel, 13 and others, who analyzed this chain not generally available and the article contains some errata. in some detail and computed the generalized representation The mixed-basis matrix elements of the continuous series matrices (or integral kernels) of one-parameter subgroups were treated by Kalnins,31 who gave expressions for one­ and found the coupling coefficients. parameter subgroups in terms of Whittaker and Laguerre In the study of the role of canonical transformations in functions of the second kind. 35 All our expressions are given quantum mechanics, the work of Moshinsky and l4 15 in terms of confluent and Gauss hypergeometric functions, Quesne . started from linear transformations between co­ and have uniformity of notation, normalization, and phase ordinate and momentum observables and lead to the oscilla­ conventions. The purpose of this paper is to give a compre­ tor (metaplectic) representation ofSp(2, R ). In contrast to the hensive derivation and listing of all subgroup reductions. realizations given by Bargmann 1 and by Gel'fand et ai., 16 in The plan of the article is as follows. In Sec. 2 we display which the group acts as a Lie transformation group on func­ the needed formulas from the theory of canonical transforms tions of a coset manifold, the group actions in the construc­ for the general method of construction and, since we want to tions of Moshinsky, 14.15.17 Seligman, Wolf, IR-23 Burdet, Per­ describe all VIR matrix elements and integral kernels, we rin and Perroud,24 and present in the work of others, 25-27 is organize the notation properly in due accordance with Barg­ an integral transform realization ofSL(2, R ) on y2(R ) Hil­ mann's conventions. In Sec. 3 and 4 we give the results for bert spaces. This group of integral transforms has been the discrete and continuous (nonexceptional and exception­ al) representation series. The first subsection of each lists the subgroup-adapted basis functions, the second treats the a)Work performed under financial assistance from Consejo Nacional de Ciencia y Tecnologia (CONACYTI Project ICCBIND 790370. mixed-basis expressions, while the third subsection treats "'On leave from Department of Physics, Indian Institute of Technology. the subgroup reductions, i.e., the cases when the row and Kharagpur 721302. India. column variables refer to the same subgroup. These are ex- 189 J. Math. Phys. 23(2). February 1982 0022·2488/82/020189-17$02.50 © 1982 American Institute of Physics 189 Downloaded 28 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions pressed as Gauss or confluent hypergeometric functions 3o 37 and, alternatively, as cylinder and Whittaker functions • of the three independent SL(2, R ) parameters. Certain cases where the symbol SA 'EA stands for summation in the case of of interest are pointed out in a further subsection. Compari­ proper, and integration in the case of generalized, bases, The son with alternative derivations available in the literature is unitarity and irreducibility properties ofD follow from simi­ pointed out whenever we are aware of such results. lar requirements for the action (2.1) on JY. The representation matrix elements for the compact The reasons for which this straightforward program of­ subgroup chain were obtained by Bargmann as solutions to ten fails to provide a definite result have to do more with 38 differential equations with boundary conditions imposed knowing the "best" choice of basis functions! ifJ). (r) l AEA and by the group identity. We come to the evaluation of an inte­ the problem of explicit computation of the integral in (2.2b), gral as the last step to the same end. We make use of a meth­ than with matters of principle. The bases are usually chosen od by Majumdar and Basu32 on hypergeometric series Mel­ as the eigenvectors of one or more operators in the Lie alge­ lin expansions to solve three of the six chains in each series. bra-so that subgroup reductions result-while the space In the special case of the continuous series in the compact JY' is an !f 2(X) space on a coset manifold X = G / H (or subgroup reduction, such an integral (a Gaussian of imagi­ H \. G) with some convenient subgroup He G. A closely re­ nary width times two Whittaker functions, one with a res­ lated approach to part (ii) of evaluation of (2.2b) calls for caled argument) is not available in the literature. Through (ii') finding these functions for various one-parameter Bargmann's result this is evaluated. subgroups of G as solutions of differential equations ob­ In Sec. 5 we point out that the six different mixed-basis tained from the subgroup generators, subject to the bound­ and subgroup-reduced representation matrix elements con­ ary conditions D(e) = 1 at the group identity eEG. stitute six families of SL(2, R ) integral and discrete trans­ The group G which we consider here is SL(2, R ): forms, as well as series expansions, of which the set of ca­ nonical transforms is but one. The Appendix summarizes {g = e!)la,b,c,dER, det g= I}. (2.3) some information about the groups SU(1, 1), SL(2, R ), and Starting with Bargmann I a number of authors have imple­ their UIRs as classified by Bargmann. Throughout this arti­ mented the program (i)-(ii) or (i)-(ii'), using for the support­ cle Z and R stand for the set of integers and real numbers. ing space X the coset space provided by the I wasawa decom­ Boldfaced symbols indicate vectors or matrices. For brevity, position NA \.NAK = SI (i,e., the circle) and Bargmann's we shall speak ofUIR matrix elements encompassing both multiplier action. 29 This is unitary in !f2(SI) for the continu­ the ordinary and generalized (i.e., integral transform kernel) 29 ous non exceptional representation series ; for the continu­ cases. 2 ous exceptional and discrete series it is 'y n c(SI) and As a general observation, we should remark that the 39 0 y2 n o(Sd with non local measures .4 n C and n D, The lat­ canonical transform realization ofSL(2, R ) can be regarded ter is equivalent20 to a space of analytic functions on the unit as a complementary alternative to Bargmann's treatment of 29 the same group.