Appen41z A The symplectic groups, their parametrization and cover ABSTRACT: In this appendix we summarize the parametrization and some of the pro- perties of connectivity and covering of the sympleetic groups. This material was developed by Valentin Bargmann in his early work on the three-dimensional Lorentz group and on Hilbert spaces of analytic functions, and has been shown to be particularly relevant for Lie optics. Wave optics --at least in its paraxial approximation-- seems to work with the double cover of the symplectic group of geometric first-order optics. This is strongly reminiscent of the double cover which the spin group affords over the classical rotation group, and makes necessary a closer acquaintance with the metaplectic groups. A.1 Rank one: SL(Z,~), SpCZ,m ), '~U(1,1), SO(e,1), and sp(e,~) Lie groups of rank one present several accidental homomorphisms. Among the compact groups, the three-dimensional rotation group S0(3) and the two-dimensional special unitary group 8U(£) are probably the most famous pair of homomorphic groups, the latter covering the former twice and allowing the description of phenomena such as spin. Non-compact groups of rank one present a fourfold such homomorphism: the group of 2 X 2 real matrices SL(2~), is isomorphic to the two-dimensional sympleetic group Sp(2,~) and to the two-dimensional pseudo-unitary group SU(1,1), and covers twice the three- dimensional pseudo-orthogonal group S0(2~1). These groups are themselves infinitely connected, and possess a common universal cover Sp(2,~). A particularly relevant group for Lie optics is the me~aptectic group, Mp(~,~); it covers Sp(2p~) twice. We start with SL(2,~) and relate to it all other homomorphic groups. A.1.1 SL(~j~) We denote by SL(2,~) the set of 2 X 2 real, unimodular matrices (a b) detg= ad-bc= 1, (1) g:~- c d with the group product defined as ordinary matrix multiplication. It is a three-parameter noneompaet semisimple groups connected and infinitely-connected. The latter facts are not obvious, and will be further elaborated in Section A.2, below. 228 Appendix A. A.l.2 Sp(2,Pk) The set of 2 X 2 matrices (1) is, at the same time, a set of 8yrapleetic1 matrices, i.e., they satisfy gMs~c~gT ~--~Ms~c~), Msp(e) := (O1 O1), (2) where gT is the transpose of g and Ms,(~) is the symplectic metric matrix. The unimodularity condition in (1) implies the validity of (2), as may be verified through elementary algebra. The set of matrices satisfying (2) constitute the group, denoted Sp(~,~), of two-dimensional real symplectic matrices. A.L3 su(~,O The above two groups are isomorphic to a third one, the group of complex #I÷2" unimodular pseudo-unitary matrices, denoted SU(1,1), whose elements u satisfy uM~cl,1)ut = M~(,,1), M~(~,,) := (O 1 gX)' (3) where u t := u T* is the ad]oint (transpose, complex conjugate) of u, and ME(~,~)the pseudo-euclidean metric matrix. It is easy to show that the most general matrix u E SU(1,1) has the form u --- I~l 2 -I~12 = 1. (4) The elements of the SL(~,~) = Sp(~,~) and SU(1,1) groups realized as 2 × 2 matrices, are related through a similarity transformation, an outer isomorphism by the complex unitary matrix 1 (w~-1 w~'l~ = Wt_l, w, :=e ''~/4 = ~-2(1 +i). W:=~\_~4 ~4 ] (5) We display this isomorphism explicitly in terms of the group parameters as (Re (~ + Z) -~m (~ - a)~ (8~) (o 2) ,(, o, fl* -----u(g)=W-lgW= 2~,(a-d)-i(b+c) (a+d)+i(b-c) " Since W-ZMs~(~)W = -iM~(1,o, (2) for g is equivalent to (3) for u. Of course, any other matrix W I = g0W or Wu0, for fixed go E Sp(Z,~) or u0 E SU(I,1), may be used to define equivalent isomorphisms between the two groups. What makes (5) particularly convenient for us is that it establishes the appropriate link between the s~andard realizations of the groups. ISYMPLECTIC (simplektik), adjsefive and subsianties,first appeaxance: 1839. [Adaptation from Greek cv~Trker~'~da, formed on qOv SYM- +Ifkg~e~u~ TO TWINE, PLAIT, WEAVE.] A. ~djective Epithet of a bone of the suspensorium in the skull of fishes, between the hyomandihular and the quadrate bones. B. 8ubst~ntlve The symplectic bone. --The Oxford Universal Dictionary on HistoricalPHndpJes, third ed., 1955. The use of this name for the Cartan C-famUy of semisimple groups is due to Hermann Weyl, in The theory of groups and quantum mechanics, 2nd ed. (Dover, New York, 1930). The symplectic groups, their paxametrization and cover 229 A.I.4 so(2j) For every real 2 × 2 matrix g in (1), we construct the real 3 × 3 matrix {~(a ~+b 2+e 2 +d 2) ~(a 2 - b2 + c 2 - d 2) -cd- ab~ r(~) := |½(~2 + b~ - e ~ - ~) (7) ~(~ - b2 - c~ + ~) e~- ~b I k -bd -- ac bd- ac ad + be / It has the properties rM~,,~jr T = M~.,~, ME~,e~ := -1 , (8a) 0 ;1) det I" = 1, (8b) r(g~)r(g2) = r(g,g~), r0) = i, r(g -~) = r(g) -1. (8c) The matrices r satisfying (8a) are called "1÷2" pseudo-orthogonal matrices, and the statement of unimodularity in (8b) reduces the matrices under consideration to a set connected to the identity. They form a group denoted 80(~,I). The rows and columns numbering the elements Fg,~ range here over it, v = 0, t, 2; the value 0 corresponds to the first row, i.e., the time-like coordinate. Since r(g) is quadratic in the parameters of g, and r(g) ~--- r(-g), (7) defines a 2:1 homomor- phism between Sp(2,~) and 80(2,1), with kernel {-1,1} ~--- 32C8p(2,~). The 80(2,t)matrix cor- responding to any pair of given SU(1,f) matrices may be obtained through (B) and (7). It is /%12 + r~l ~ 2Re a]~* 2Imafl* r(u)={ 2Reafl Re (~2 + 82) Im (~2 - ~)|, \ -2 Im aft --Im(a 2+82 ) Re(a 2-f12)J and conversely, (0b) A.2 Connectivity The connectivity of a three-dimensional manifold is, when multiple, a challenge to the mind. If we are to picture this intuitively, we must build up a proper representation of the manifold and follow one-parameter lines. If these lines turn out to be circles, then the covering of these by the real line becomes plausible within the group. As is so often the case, the complex plane is useful and 8U(1,1) is, among the groups of last section, the easiest to analyze. 280 Appendix A. /~/gu~'e, The SU(1,1) ~-- Sp(2,~) group manifold is three-dimensional, connected and infinitely connected. It is ~a pierced by a one-sheeted equi- lateral hyperboloid. Three represen- tatives of one-parameter subgroups are drawn: uo(r)(circle), uz(r) (one branch of a hyperbola), and ue:~(r) (straight lines), lying on the fir -~ 0 hyperboloid. A.2.1 The connectivity of SU(I,1) The connectivity properties of Sp(~,~) and its isomorphic groups is seen best in terms of the SU(1,1) parameters (4). We write the real and imaginary parts of a and fl as a ----- aR + in[ and fl -~ fir + iflt, so the unimodularity condition reads a~ + a~ - fl~ = 1 + fl~ _> 1. For fixed fit, the remaining three parameters are constrained to a one-sheeted revolution hyperboloid with its circular waist in the a-plane. As we let fit range over ~, we fill twice a region of ~a-space which is bounded by the fli ---- 0 equilateral revolution hyperboloid. See the figure above. Topologically, the group manifold of Sp(2,~) is @1 X !1¢2, the circle times the cartesian plane. The points (aR, ai, fiR) in ~ which do not belong to the group manifold as described above, form a simply connected tubular region. Any plane for which this region has an elliptic section is an infinitely connected space (the plane minus the unit disk), and hence so is a foliation of ~3 by such planes. One-parameter subgroups of 81I(1,1) are lines in ~a which must pass through the group identity: the point (aR = 1, a[ -~ O, fir -----0) on the circular waist. The class representatives of the three nonequivalent subgroup are: a R ~ COS 2 ' Ellip~ic: u0(r), r E ~ (rood 4~-) given by at ~- sin ½r, fn = O. aR ~ cosh it, Hyperbolic: u2(r), r E !l¢ given by at ---- 0, fir -~ - sinh iv. aR ~ l~ Parabolic: up:l:(r), r ~ !l¢ given by at ~-- ½r, f, = ±½~. The elliptic subgroup is the circular waist in the figure, the hyperbolic subgroup is one of the two branches of the equilateral hyperbola on the az ---- 0 plane, while the parabolic ones are the straight lines contained in the hyperboloid which pass through the identity parallel to the at-fR plane. The only subgroup class with a nontrivial covering is thus the elliptic one. The symplectic groups, their parametri~ation and cover 231 A.2.2 The covering group of SU(I,1) Following Bargmann's pivotal work: in reference [1], we now present the simply-connected universal covering group of 8U(1,1), making use of the (maximal) compact subgroup U(1) = 80(~). This corresponds to the Iwasawa decomposition which factorizes globally a noncompact semisimple group into its maximal compact subgroup, times a solvable subgroup. We write u E SU(1,1) in (4), ia the form a 0 X (ft.