non-buective quadratic transformations and theory of angular ...

LYCEN 8942 November 1989

NON-BUECTIVE QUADRATIC TRANSFORMATIONS AND THEORY OF ANGULAR MOMENTUM

M. HAGE HASSAN1'2 and M. KIBLER2 1

Université Libanaise Faculté des Sciences, Section 1 Hadath, Beyrouth, Lebanon 2

Institut de Physique Nucléaire de Lyon IN2P3 - CNRS/Université Claude Bernard 43, Boulevard du 11 Novembre 1918 69622 Villeurbanne Cedex, France 1.

Paper to appear in the Proceedings of the V International Symposium on Selected Topics in Statistical Mechanics held in Dubna, USSR (22-24 August 1989), as a tribute to Academician Nikolaj Nikolaevich Bogolubov on the occasion of bis 80th birthday. To be published by World Scientific: Singapore.

cal me gei nu ext sh 1U Ja

cai R4 m be

NON-BIJECTIVE QUADRATIC TRANSFORMATIONS AND THEORY OF ANGULAR MOMENTUM*

M. HAGE HASSAN and M. KIBLER Institut de Physique Nucléaire de Lyon IN2P3 - CNRS/Université Claude Bernard 69622 Villeurbanne Cedex, France

extension are

The Kustaanheimo-Stiefel map and it

derived in the context of Fock-Bargmann-Schwinger calculus with special emphasis on angular momentum theory.

1, Introduction and Preliminaries In recent years, non-bijective canonical transformations generalizing the socalled Kustaanheimo-Stiefel (KS) transformation1'2) have been the object of numerous studies. 3 " 20 ' In particular, the algebra of quaternions 1 ' 4 ' 5 ' 7 ' 9 ' 10 ) and, more generally, the Cayley-Dickson algebras of (ordinary and hyperbolic) hypercomplex numbers8'16* proved to be useful for deriving quadratic transformations which extend the Levi-Civita1) and the KS transformations. In addition, it has been shown19) that the use of Cayley-Dickson algebras also allow to generate nonquadratic transformations as, e.g., the Fock21) (stereographic) transformation. It is the aim of this work to generate the (R 4 —* R3 ) KS transformation and its R8 —• R5 compact extension in the framework of Fock-Bargmann-Schwinger22"24) calculus. In this approach, the vector fields associated to the R8 —* R5 and R4 —+ R3 transformations acquire a significance in terms of angular momenta. A basic ingredient of this work is the generating function * for the rotation matrix elements TP(U)mm' of the group SU{2). To establish the notation, let * be the function denned by (cf. Réf. 23) U =

~Z2

Dedicated to Professor N. N. Bogolubov on the occasion of lus 8Oth birthday.

The matrix V1I2U, r = Z1Z1 + z2z2, belongs to 517(2) and * can be expanded as * i m («i, a2 ) rj V* (U)mm> $jm' (h , b2 )

, a 2 ,0 1 ,6 2 ; Z1, Z2 ) = J ]

(2)

j'mm'

where the function ^) in K4. Since P^2^'(l/)min' = 0 for m = O, it is natural to consider the series

o\

>iM)

or

(7)

«m'

especially in view of its importance for generating basis functions of SO(Z). The expression Sj, can be derived by integrating in the Fock-Bargmann space T2 the function \f, in the form given by (2), multiplied by a convenient factor. As a matter of fact, it is immediate to show that (8)

»

18)

Now, we pass to the form (1) for the function $ and easily obtain e(2lbl+Zïb3)(-Iï6l+2lfr3)

(9)

As an intermediate result, from Eqs. (7)-(9) we have ^m'(6i,& 2 )

(10)

It is known that the generating function for the 50(3) D 50(2) solid harmonics (H) V

tTV

is given by (cf. Réf. 24)

with f" = (xi,x 2 ,x 3 ). Introducing (11) into (12) and identifying the so-obtained relation to (10), we finally get + ZiZ2

X2 = —i(ziz 2 — ZiZ2)

X3 = Z1Z1 - Z2Z2

(13)

The latter relations are a simple rewriting of three of the relations defining the KS transformation. Indeed, by putting Zi = Ui + iu2

Z2 = U3 + IU 4

(14)

(13) can be rewritten in the Cartesian coordinates (ui,u 2 ,U 37 U 4 ) of R4 as Xi = 2(U]U 3 + U 2 U 4 )

x 2 = 2 ( - U i U 4 + U2U3)

with the evident property that r = (X)t=i x i )

x 3 = Uj + u 2 — u\ — u 4 (15) =

Z)o=i u a -

The R* -+ R3 differentiate map defined by (15) corresponds to the transformation introduced by Kustaanheimo and Stiefel1* up to permutations on the Xi's and the u a 's. The compatibility between (4), (6) and (13) gives to (15) its specific form owing to (14). In the terminology of Lambert and Kibler18* and of Kibler and Winteraitz17*, the tranformation (15) can be identified to an Hurwitz transformation of type (c') associated to an anti-involution of the Cayley-Dickson algebra A(—1,— 1), which is isomorphic to the algebra of Hamilton quaternions.

Should we have chosen to work with the series Sa (with an evident definition) instead of Sj,, we would have been left with the transformation X2 = 2(W1W4 + U2U3)

x\ =

x'3 = u\ + u\ - u\ - u\

(16)

which corresponds, in the notation of Ref. 16, to the Hurwitz transformation associated to the anti-involution Ji of A{—1,— 1) through the relation x' = A(UJe1 u, up to a re-labeling in x' and u. Equations (4) and (16) can be combined to yield (x'j = T sin 9 cos tp, x'2 = r sin 0 sin if>, x'3 = r cos B) and thus the transformation (16) is inherent to an approach the starting point of which is Pv x'{ — 0 (i = 1,2,3) for the choice of coordinates ( s ' ] , ! ^ ^ ) hi K,4. Of course, the transformation (16) is equivalent to the transformation (15). Both transformations are associated to the Hopf fibration on spheres S3ZS1 — S2. 2.2. The R* —+ R3 vector fields. The next step is to examine, still in the framework of Schwinger-Bargmann calculus, the implications of (15) and (16) in the language of angular momentum theory. The basic relations are (see Ref. 25)

where K = K+, Kx or K- and L = L+, Lz or L~ are the images in T^ of K = K+, Kz or K- and L = L+, Lx or L- in the variables (01,02) and (61,62), respectively. The operators K = ( K + , K z , K _ ) and L = (L+,LZ,L-) generate the lie algebra of the group 50(4). Let us simply recall that Vi(U)mmi is an eigenvector of the operators Kz and Lz with the eigenvalues m and m!, respectively. The FockBargmann representations of K and L are given by K+ = -ax — Oa2

Kz = - I O1 2 V Oa1

a2 — ) Ua2J

K- =~a2 -^~ Oa1

f

. .

°

and act on the space T2 %T2. Let us consider the image K of K. From direct calculations, we have

Therefore, from (17) we obtain the three spherical components of K

O

il

K+ = J, \

»

')

dz2

Oz1J

K_.

+

*é *é-*ê)

(20)

With the help of (14) and of the well-known relations A

=

(l/2)(0,-i02)

^ - = (l/2)(ft-ifl|)

(21)

where da stands for j£-, we can derive from (20) the expressions of K+, Kz and K- in terms of the coordinates ua (ct = 1,2,3,4). Finally, by introducing the Cartesian components Ki (i = 1,2,3), we get the vector fields

K1 = - ( i K2 = -(i/2)(-u 3 O 1 + u4O2 + U103 - U2d4 ) K3 = +(i/2)(+u 2 di -U1O2+U4O3-U3O4)

(22)

defined in the real symplectic Lie algebra sp(8, E). Similarly, from L^l — L* $, we would obtain the three Cartesian components of L in the algebra sp(8, R). This would lead to the commutation relations [Kk ,Kt]=i

ek£m Km

[Lk ,L1] = i cklm Lm

[Kh ,Le] = 0

(23)

(for fc,£,m = 1, 2 or 3) indicating that the set {Kj,Lj : j = 1,2,3} spans the Lie algebra so(4) hi an su(2) 0 su(2) basis, as expected from the angular momentum theory developed in an 50(4) presentation. At this stage, it should be noted that the vector operators K and L are angular momenta associated to the coordinates ( X ' I , ^ » 3 ^ ) an x 3 ) or for X = (2Ji)L3 and / = / ( z j , x 2 , x 3 ) , where / is a (one-fold) differentiable function. (Equation (27) was the starting point of our derivation of the transformations (15) and (16).) The operator X generates an so(2) subalgebra, with so(2) = SO(2)K for X = (2/i)K3 and 3o(2) = so(2)i, for X = (2/i)Zr3, of the algebra ap(8,R). The corresponding lie algebra under constraints cent^ ,- ^(so(2))/so(2) is so(4,2) ~ sti(2,2) (see Refs. 6 and 17). To close this section, note that the one-form u; (see Ref. 16) associated to the vector-field X can be recovered from the property w[ 27-^1 = 1. 3. The R8 -» R5 Case To deal with R 8 , we start from two copies of R4 and choose the coordinates as follows. Let us consider two particles, say, 1 and 2. Let (V>,0,y) be, in this section, the angular coordinates of the collective motion and ( x i , x 2 , . . . ,X5) the five remaining coordinates necessary to completely describe, in terms of the R4 ©R4 space, the position of the two particles. In this case, Eq. (5) reads Equations (28) may be taken in the form IVJ -Ci

l\,2 X i

A»3 Xi

F

tl

N

b U

^l

X, £ty . . . )O)

\*")

where K1, K2 and K3 denote the (Cartesian) coordinates of the total vector ope() i I^ V~/-i\ I £r/n\ £ il__ x _f i t . i 1 ! .1 _ _ /ITVL A Tpl-t\ rator K = K(I) + ^ ( 2 ) for the system of the two particles. (The operators () and K(2) correspond to the operator K for the particles 1 and 2, respectively.) We call S0(3)JC the l i e group with infinitesimal generators K1, Ki and K3. The function *«c=*l*2

^'

(30)

1

where $ i and Jf2 stand for generating functions of type