The moyal representation for spin

ANNALS

OF PHYSICS

190,

107-148

(1989)

The Moyal

Representation

JOSEPH C. VARILLY Escuelu

AND Jo&

for Spin

M. GRACIA-BOND~A

de Matemcitica, Universidad de Costa San Jo&, Cosra Rica

Rica,

Received June 6, 1988

The phase-space approach to spin is developed from two basic principles, SU(Z)-covariance and traciality, as a theory of Wigner functions on the sphere. The twisted product of phasespace functions is related to group convolution on SU(2) by means of a Fourier transform theory on the coadjoint orbits, which yields the Plancherel-Parseval formula. Coherent spin states provide an alternative route to the same phase-space description of spin. The Wigner functions for spin states and transitions are exhibited up to j=2. It is shown that for Hamiltonians such as arise from time-dependent magnetic fields, the quantum spin dynamics is given entirely by the classical motion on the sphere. The Majorana formula becomes transparent in the Moyal representation. cl 1989 Academic Press. Inc.

1.

INTRODUCTION

The phase-space formulation of Quantum Mechanics is almost 40 years old. It was Moyal [l] who, by noticing that Wigner’s recipe [2] for associating a function on phase-space to a density operator on Hilbert space was essentially the inverse of Weyl’s correspondence rule [3], opened the way to formally representing (nonrelativistic) Quantum Mechanics as a statistical theory on classical phase space. This theory was given a more general mathematical framework, as well as an autonomous reconstruction based on the concept of twisted product, in two remarkable papers by Bayen et al. [4]. While the Moyal formalism has been applied successfully more and more often, its extension to spinning particles has remained undeveloped. It is certainly possible, as has indeed been done several times, to introduce spinors whose components are Wigner functions. The spirit of the Moyal formalism, however, requires that one replace the operators of the conventional theory by functions on the classical phase-space, that is, on the sphere S*. In this way one sticks as closely as possible to classical ideas, and a more pictorial description is achieved. We will review, in the final section, several attempts to “quantize” the sphere. Surprisingly, however, the most adequate solution to the problem was presented more than 30 years ago ! The paper [ 51 by Stratonovich contains the broad outline of the theory which we will expound here. It seems that Stratonovich’s paper was little read and less understood. And so the “conventional wisdom” had that a 107 Ot?Q3-4916/89 $7.50 CopyrIght Q 1989 by Academic Press. Inc All rights of reproduction in any form reserved

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Moyal-like version of the quantum theory of spin could not, or at least did not, exist. That belief can now be dispelled. We present here the Moyal representation for spin in a fully explicit mathematical setting. Crucial to this endeavour is what we call the traciaI condition (see Sections 2.1 and 2.4) assuring that quantummechanical expectations can be calculated by taking integrals over the sphere. It turns out, in a way both surprising and pleasing, that this very condition is necessary and suffices to obtain a completion of the Kirillov theory for compact Lie groups, because it yields a concrete Fourier transform theory in the arena of the orbits of the coadjoint action, including an inversion theorem and a PlancherefParseval formula. We uncover here in a new context and in a novel way the deep link between Harmonic Analysis and Quantum Mechanics. We could not refrain from a summary of this mathematical development here, both because of its intrinsic interest and elegance and because it will presumably lead to physical applications. The paper has two rather differentiated parts. Sections 2 to 4 build the mathematical apparatus. There follows a long section devoted to illustrative applications, where the physical transparency of the formalism is exploited to throw light on the results and methods of quantum spin theory. Section 2 constructs the Weyl correspondence between operators on Hilbert space and functions over the sphere from a basic set of assumptions; the basic mathematical toolkit of the formalism is derived there. Section 3 contains the new Fourier theory for the group SU(2). Section 4 recasts the Moyal representation in terms of coherent states. (We felt it convenient to include this section as it has often been argued that these states provide a “royal road” to quantization on the sphere; however, it can be skipped without prejudice to later developments.) Topics treated in Section 5 include (a) a review of classical spin dynamics, (b) description of the Moyal spin states and twisted exponentials over the sphere, (c) quantum spin dynamics in the spirit of [4], (d) a theorem relating classical and quantum spin dynamics through the Lie algebra of distinguished Hamiltonians for spin, (e) the Majorana formula revisited, (f) a comment on dissipative dynamics, and (g) the minimal coupling recipe for spin-4 particles. In the subsection on the Majorana formula, one sees how the “spectroscopic” (or “quantum”) and the “dynamical” (or “classical”) views on nuclear magnetic resonance phenomena-consult the beautiful historical and conceptual article by Rigden [6]-are neatly reconciled in the Moyal representation. Section 6 contains a brief set of notes about previous literature which deals more or less directly with the theme of phase-space representations of spin.

2. THE STRATONOVICH-WEYL

CORRESPONDENCE

In the ordinary formulation of Quantum Mechanics, the j-spin is represented by operators acting on a (Zj+ 1)-dimensional vector space C*j+‘, where

THE

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REPRESENTATION

FOR

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SPIN

Jo {t, 1, 3, 2, ...}. An orthonormal basis is given by the eigenvectors Ijm), where J2 ]jm) =j(j+ 1) Ijm), J= Ijm) =m ljm), on setting the Planck constant fi= 1. C2jf1 carries an irreducible representation nj of N(2), whose matrix elements are conventionally [7] given as gi,(g):= (jml nj(g) Ijn). The phase-space is the sphere S2, equipped with the SU(2)-invariant symplectic form dd A sin 0 dtl; this familiar fact may be seen as an application of the Kostant-Kirillov-Souriau theorem [S, 91 for the group SU(2), since its coadjoint orbits are spheres (plus a point orbit for the trivial case j = 0) and SU(2) acts on each orbit by rotations of the sphere. We denote points of S2 by n = (0, 4) in spherical coordinates; dn:= sin 8 d8 d# is the surface area measure. Elements of SU(2) will generically be denoted by g, and the natural action of SU(2) on S* is written as g .n. Iffis a function on the sphere, its translate f g is given by f”(n):=f(g-’ . n). The Haar measure dg on SU(2) is normalized so that f sU(2) dg = 1. For most facts about conventional spin angular momentum theory in Quantum Mechanics, we shall refer to the treatise [7]. 2.1. Th‘e Stratonovich

Postulates

By a “Stratonovich-Weyl correspondence” for the j-spin, we mean a rule associating to each o.perator A on the Hilbert space CZ’+’ a function W, on the phase space S*, which satisfies the properties [S] (0)

Linearity:

A H W, is one-to-one linear map.

(i)

Reality:

(ii)

Standardization:

(iii)

Traciality:

(iv)

Covariance:

W,,(n)=

W,(n), A* being the adjoint of A. ((2j+

1)/4rc) fs2 W,(n) dn = tr A.

(2.la) (2.lb)

((2j+ 1)/4x) js2 W,(n) W,(n) dn = tr(AB). W,.,

= ( W,)g for ge N(2),

where g.A:=

(2.lc) xi(g) Anj(g)-‘. (2.ld)

As Stratonovich points out [S], the linearity and the traciality conditions are directly connected with the statistical interpretation of the theory. The tracial property (2.1~) asserts that the statistical averaging of the phase-space observable WA and the ordinary quantum rule for averaging the operator A should yield the same result. Here we might think of B as a density matrix; of course, by linearity, (2.1~) will be valid for arbitrary operators B. The reality condition (2.la) says that W, is a real-valued function if and only if A is a selfadjoint operator. Condition (2.lb) is simply a normalization: note that, together with (2.lc), it says that the identity operator I corresponds to the constant function 1, as is natural. The covariance property (2.ld) expresses the symmetry of the system in an explicit way. We shall call W, the Stratonovich-Weyl symbol associated to A.

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2.2. The Stratonovich-

By linearity,

AND

Weyl Operator

GRACIA-BONDiA

Kernel

we may write W,(n)

= tr(A A’(n)),

(2.2)

where A’ is an operator-valued function on S2, which we shall call the Stratonovich-Weyl operator kernel. Now the tracial property tells us that the Weyl rule (2.2) may be inverted to give A,2j+l

J W,(n)

4n

A’(n) dn.

52

(2.3)

In other words, the direct and inverse correspondences A,’ W, may be implemented with the same operator kernel. It remains to show that such a kernel exists. The Stratonovich postulates for A + W, translate to the following properties of A’: (i)

A’(n)* = A’(n) for all n E S2.

(2.4a)

(ii)

((2j-t 1)/47c) Is2 A’(n) dn=I.

(2.4b)

(iii)

((2j+ 1)/4n) js2 tr(Aj(m)

(2.4~)

(iv)

A-‘(g.n)=xj(g)

A’(n)) Aj(n) dn = A’(m).

Aj(n)r,(g)-‘.

(2.4d )

The equations (2.4a), (2.4b), (2.4d) come from substituting (2.2) in (2.la), (2.lb), (2.ld). To check (2.4c), note that for any operator A on CJi+‘, W,(m) = tr(A’(m)A)

=T

s tr(Aj(m) s

A’(n)) W,(n) dn

=T

Js2 tr(Aj(m)

A’(n)) tr(Aj(n)A)

dn,

(2.5)

and (2.4~) follows on eliminating A from this equation. We introduce the matrix elements of A j(n) with respect to the standard basis: A’(n) =:

i Z!,(n) Ijr)(js(. r,s= -j

The functions Z;, are linearly independent, The covariance condition (2.4d) gives ZQg.n)=

(jr

A’(g.n)

since AH

I@> = Ml

(2.6)

W, is one-to-one.

nj(g)A’(n)nji(g)*

=p q$mj (jrl nj(g) I&) z&#KjqI

lb>

n,(g)* Ijs) (2.7)

THE

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111

The product of two 9 functions is computed in the representation theory of SU(2) by using ClebschhGordan coefficients. From [7, (3.135) (3.190)] one can obtain

x q‘r,,-,k).

(2.8)

(Here (i, & 1 &) denotes the ordinary Clebsch-Gordan coefficient (not the Wigner 3j-symbol), sometimes written as Ci:gl,,, [7] or as (j,j,m,m, 1 jm).) Substituting (2.8) in (2.7) we get

If we now define y,(n):=

the orthogonality

2 t-l)‘-” p= -j

(

i

Z;.,+,(n), ’ I -m >

e,‘-m

relations for the Clebsch-Gordan

B,Jg.n)=

J$ (-l)j-” p= -j

j (P

P

k=O

i %Xd n= -k

n=

coefficients give ’ ZL.,+,(g.n) I -m )

.i j I 1>(j

=

x

j -p-m

(2.9)

-p-m

-m

p

~k,Cn)

-k

This is precisely the covariance relation for the usual spherical harmonics Y,m. Indeed, if n = (6, q5), we can write g, = exp( - $Jz) exp( - iU,) E SU(2), and then Y,,dn)=9Lo(g,) [7], so Ylm(gqn)=Cf,= -,9:,(g) Y,,,(n). Now, if C(1, k) is the (21+ 1) x (2k + 1) matrix whose (m, n)-entry is c:,, = ((2f+ 1)/4?r) I22 y,,,,(n) Y,*,(n) dn, then a,(g)* C(I, k) nk(g) has (m, n)-entry ((21+ 1)/4n)jsz Y,(g.n) Y&(g.n)dn=& by the SU(2)-invariance of dn: so C(I, k) intertwines the representations zk and x,. By Schur’s lemma, ctn = A{ 8,k 6,” for some constants A{. 595/190/1-a

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Since the spherical harmonics form an orthonormal that p,m = Ij Y,, . Thus z;&l)=(-l)‘-’

c (I ,YQ

=(.-l)j-r

‘,

1,i,>

C aiI (jl ,yo

basis for L2(S2), we conclude

L(n)

is / ,i,>

YI,s-rO).

(2.10)

From the reality property (2.4a), which shows that 2;: = Zir, it follows that the A{ are real. Let us denote by Z$ the Hilbert space of spherical harmonics of order <2j, which has the orthonormal basis { Y!,,, : 0 $16 Zj, - 16 m < Z}. Then (2.10), which is a consequence of the covariance property only, shows that the matrix elements of A’, and thus the Stratonovich-Weyl symbols W,, must lie in X2j. Indeed, (2.9) now says that these span the space %2j. Let us write Kj(m, n):= ((2j+ 1)/4n) tr(Aj(m) A’(n)). Then (2.5) becomes

Cdm) = Js2J%, n) WA(n)dn so that Kj(m, n) is the reproducing 2 i Y,Jm) I=Qm= -I

=- 2j+l

4n

2j+l

x(f

space X2j. Thus

Y%(n) = Kj(m, n)

i z&(m) r,s= -j a ’

c r.s=

=7

kernel for the Hilbert

c

czi 4% J J{;

-jI=Ol'=O

!,/

Zis*,‘(n)

Y,,-,(m)

,',)

isiris)

Y?.,-,(n)

2jS 1 2j =T

,Fo

(U'

i

m= --/

Y,m(m)

Y&(n)

(2.11)

which yields A{ = E{ dm with E{ = +_1. Finally, the standardization condition (2.4b) implies that ~4 = + 1. The remaining signs are undetermined: we have thus proved the following theorem. 1. The problem of finding a Stratonovich- Weyl correspondence for the has exactly 2” solutions, given by Eq. (2.10) with ,I& = dm, *JG@Zjfbr I= 1,2 ,..., 2j.

THEOREM

j-spin A{=

THE MOYAL

For definiteness, we will C;(,= -jZis(n) I~r>(@l, with

REPRESENTATION

choose

all

113

FOR SPIN

E,= + 1,

so

that

d’(n) :=

(2.12) and in particular (2.13) where the P, are the Legendre polynomials. We see in Section 4 that this is a reasonable choice in view of the physical interpretation, as was indeed seen by Stratonovich in particular cases [5]. 2.3. Some Formulae for the Basis Functions

We may now write (2.14) where the matrices C{i E M,,

r(C) have elements

(jrlC~Ijs)=(-l)j-’

(:

!,y j -‘m>

The operators C{, are known: they are (normalized, operators; see [7, (3.342)]. Now

irreducible

tensor) Wigner

Cii, Y&(n) Y,.Jn) dn c:~m~=Js*~om$~, A’(n) Y,,.(n) dn and on comparing

with (2.3), we see that the Stratonovich-Weyl symbol of the A Wigner operator C’im is Jmi Y,,,,. s a trivial consequence of the tracial in the property, we check that the Wigner operators are orthonormal Hilbert-Schmidt sense: tr(CjzCCj.,.) = ii

YL(n) Y,,,,,.(n) dn = ~3,~6,,..

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In the present context, the symbols .Zis play an even more fundamental role than the spherical harmonics. From the definition (2.6) and from (2.2), they are the Stratonovich-Weyl symbols of the transition operators between spin states: Zk(n) =

wljs>
Thus (2.lb) and (2.1~) show that 2j+47l 1 s2Zl,(n)dn=~,, -1 2j+ 1 Z&(n) .Z{Jn) dn = 6,, a,,. 47c I sz

(2.16)

The functions Zif form a most convenient orthogonal basis of L*(S*). We can now compute some other Stratonovich-Weyl symbols. That of the J,operator CA = _ j m ljm) (jml is 8’: = CA = _ j mZi,(n). Using the Clebsch-Gordan orthogonality relations, together with the formula

we compute

=$5%=)(3P,(

cos e)) = &Tij

cos 0.

An analogous computation shows directly that CL = ~ j Zi,(n) = 1; moreover, if Wi, WI are the Stratonovich-Weyl symbols associated to the spin operators J, and J,, one finds, as expected, that WC:= dm

sin 9 cos 4,

W-i = Jm

sin 8 sin d.

2.4. The Twisted Products on the Sphere In the modern approach to phase-space Quantum Mechanics on the flat phasespace R*” [4, 10-131, the Weyl correspondence recedes into the background and the principal role is assumed by the twisted product of two symbols, which corresponds to the usual product of operators. This allows the formulation of an autonomous theory based on a calculus of functions on phase space, without mention of operators at all. This twisted product is determined by the condition that

THE MOYAL

REPRESENTATION

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FOR SPIN

W, x W, = W,, for any operators A, B. Given the Stratonovich-Weyl correspondence (for a particular value of j), we note that (2.2) and (2.3) combine to give

W,,(n)

= tr(d’(n)AB) =(y)2tr(d’(n)Js1

Thus the appropriate

d’(m)

W,(m)

dm ii d’(k)

dk

>

recipe for the twisted product of two functions f, h in X2j is

(fx h)(n):= is2Js2Lj(n, m, k) f(m) h(k) dm dk, where the trikernel

W,(k)

(2.17)

Lj is just

tr[Aj(n)

d’(m) d’(k)]

k -G(n) r,~.c= -j

-Q(m)

Z:‘,(k).

(2.18)

Observe that from (2.15) and (2.1 l), we have

s

s2 L’(n, m, k) dn = Kj(m, k)

which gives the tracial formula

for the twisted product in %2j:

js2 (f x h)(n) dn = Is2 f(n) h(n) dn.

(2.19)

This formula is extremely important. It is obviously a version of the tracial condition, which assures equivalence of the calculation of expected values in both quantum-mechanical formulations for spin. In the case of the Moyal product, the analogous formula permits one to extend the product to a large algebra of distributions [ 131; matters are simpler in the present case, since the function spaces are finite-dimensional and questions of vector-space duality are moot. But here, this formula is what allows one to develop the harmonic analysis for the compact group SU(2), as we see in the next section. From the covariance condition (2.4d) and (2.18), we infer that Lj(g-n,g.m,g.k)=L’(n,m,k)

(2.20)

which implies that the twisted product (2.17) is equivariant: (f xh)R=fgXhg,

for all

g E SU(2).

(2.21)

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The formula (2.20) says that the trikernel is a rotationally invariant function of its vector arguments. According to Weyl’s theorem [14], it will depend only on permutation-invariant combinations of the scalar products and the triple product that can be formed from these arguments. For instance, for j = 4 we have -

Ll’*(n, m, k) = $( 1 + 3(n - m + m * k + k - n) + 3 ,,h i[n, m, k]). Note that we have z;, x ziu = 6,, zj ru

(2.22)

which by (2.15) and (2.16) would give (2.19) again. It is furthermore ( Wi x Zk)(n) = rZis(n),

(Zj ,S x W!)(n) =

szjIS(n).

clear that (2.23)

3. FOURIER ANALYSIS ON SU(2)

An important consequence of the quantization formalism set out in the previous section is that it opens a direct route to harmonic analysis on the compact Lie group SU(2). The traditional approach to the Fourier theory for a noncommutative compact group G has been first, to identify the irreducible (finite-dimensional) unitary representations of G; second, for a given functionfon G, to identify a set of representative operators T] on the various representation spaces (the “Fourier coefficients”); third, to reconstruct f as far as possible from these operators (the “Fourier series”) by means of a Plancherel inversion formula. This generalizes the abelian case (for example, Fourier theory on the real line or on the circle) because the irreducible unitary representations are then the unitary characters. The abstract nature of operator Fourier theory, however, is an obstacle to formulating the analogues of many classical results. The idea now is to couple to this procedure a correspondence between operators and phase-space functions; then we may redefine the harmonic analysis of a compact group as a process which gives a Fourier transformation from functions on the group to functions on phase-space. In general, the appropriate phase spaces arise from the Kirillov orbit method [S]. This method associates to “most” representations of a Lie group G (roughly speaking, those representations which are likely to reappear in the Plancherel formula) an orbit of G by the coadjoint action on the dual g* of its Lie algebra. Since these orbits carry a natural symplectic manifold structure [9], they can be thought of as abstract “phase spaces.” For a nilpotent group such as the Heisenberg group, the correspondence between coadjoint orbits and representations is bijective [ 151. For a compact group like SU(2), only certain “integral” coadjoint orbits enter the picture. Wildberger [16] proposed using the orbit method to obtain an autonomous theory of Fourier analysis for compact groups, without passing through the machinery of representation theory. What is needed is the characterization of “integrality” of an orbit, and an appropriate Fourier kernel, which provides a bridge

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between the group and the integral orbits. For a covariant theory, several choices of Fourier kernel are possible. The one chosen by Wildberger appears in the next section under the guise of the Berezin “covariant symbol” or “Q-symbol.” However, this choice does not yield (a suitable version of) the tracial property, and so he was stopped short of a Plancherel-Parseval theorem. Let us see how all of this works for G= SU(2). Here g = 5u(2)= (X=i(x,cr,+x,o,+x,cr,):x,,x,,x, real) and g* is identified with g via (X, Y) = - ttr(XY). The coadjoint action is then g. X=g-‘Xg, and the orbits are the spheres det X= constant. For ;1> 0, the orbit containing iLo, is integral if and only if the character exp(i&,) H era8 of the isotropy group of ila, is well-defined [S, 3.151: so ,.I must be an integer. Write A=2j. The casej=O gives a point orbit, and the orbits for j = 4, 1, $, ... are spheres. Thus the integral orbit space, which we shall call G = SU(2) “, the dual space, is indexed by parameters (j, n) where i is a nonnegative half-integer and n E S2 for j > 0. 3.1. The Fourier Kernel

We define the Fourier kernel as the function E on SU(2) x SU(2) h given by E(g;j,

n):=

tr(n,(g) P(n))

(3.1)

(with the convention that E( g; 0) = 1 in the trivial case i = 0). Note that E( g; j, n) is the Stratonovich-Weyl symbol of rcj( g), for any j > 0. Explicitly, E(g; j, n) = A few notational

i Z!An) %tg). r,s= -,

(3.2)

conventions will be useful:

1. For each j> 0, we will

normalize

the integral

over S2 as d,n =

((2j+ 1)/4?r) dn.

2. For each g E SU(2), we will write eg for the function on SU(2) A given by e,(O) := 1, e,(j, n) := E( g; j, n). 3. E will denote the complex conjugate of the Fourier kernel: E(g; j, n):= Etg; j, n).

4. As usual, xi(g):= tr nj(g) denotes the character of the representation ?L,. 5. To avoid notational clutter, we will write G= W(2), G= SU(2) h hereinafter. The properties (2.4) of A’ have the following consequences for the Fourier kernel: (i)

E(g;j,n)=E(g-‘;j,n).

(3.3a)

(ii)

fs Ekj,

(3.3b)

(iii)

((2~’ + 1 )2/47r) JG E( g; j, m) E( g; j, n) dg = Kj(m, n).

(3.3c)

(iv)

E(g;j,h-‘.n)=E(hgh-‘;j,n).

(3.3d)

n) djn=Xjtgf.

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Equations (3.3a), (3.3b), and (3.3d) are clear from the definition (3.1). For (3.3c), we obtain from (2.11) and the orthogonality properties of the 9-functions: Qg;j,

m) E(g;j,

(2j+ 1)2

=T

n) 4

1 Z&(m)Z’,:(n) lG%i(g) %ih) dg

T,S.U,”

Zis(m) Zis*(n) = K’(m, n). i ,-.s= -,

=q

Note that (3.3~) gives a representation integral over the group:

for the Stratonovich-Weyl

kernel as an

(3.4)

3.2. The Fourier

Transformation

The Fourier transformation on G given by

is the mapping

9 from functions on G to functions

For an integrable function f on G, it is usual to define the corresponding on C2’+’ by T’:= j f(g) G

operator

n,(g)dg.

Thus (%f)(j,

n):=[

f(g)tr(nj(g)

dj(n))dg=tr(Tj!dj(n))=

W,;(n).

G

So the Fourier transform off is the Stratonovich-Weyl symbol of T/j. This is the exact analogue of the usual “Weyl correspondence” in the flat phase-space [W”‘, where the “ordinary” Fourier transform serves a similar purpose. It is now appropriate to consider the twisted product for functions defined on the whole of 6. We can lump together different values of j by declaring functions supported on different spheres in G to have twisted product zero. THEOREM 2. The Fourier transformation G with the twisted product on 6.

% intertwines

the group convolution

on

119

THE MOYALREPRESENTATIONFORSPIN

Proof:

First, if g,, g, E G, then

eglg2(j, n)=E(g,g2;j,

n)=fr(nj(gl)n,(g*)d’(n))

= WrrJcg,jn,cg2J(.Ln) = (WIT,tg,J x K,&(j, Second, if f, h E L*(G), then f*h is continuous S(f

* h)(j, n) = ( E(g; i, n)(f*Md

=

HG

G

n).

on G, and

&

G

=

4 = kg, x e,,U

E(glg2; L n)f(sl)

hk2) dg, dg2

. SJ G

(eg, x e,,)(j, n)f(gl)

hk2) dgl 42 = Pfx

sh)(j,

n).

(3.5)

G

3.3. Examples of Fourier Transforms

If 6, denotes the Dirac delta on G concentrated at g, then 9 6, = eg. In particular, when g = e is the identity, B 6, = 1. If (0) is the isolated point of 6, then 91 =&, since for j>O, (Fl)(j, n) = c;(,= -, z;s(n)

fG

%,k)

&= 0.

PROPOSITION. The Fourier transforms of the representative functions and characters of G = N(2) representations are given by pgk’

---

1

ZL,

mn 2k+ 1

9xk=6,----

1

’ 2k+f’

where we use the convention that Z:,,( j, n) = 0 for j # k. Proof.

Since the character xk of G is real, xk = Ck = _ k gk,,, = Cl = _ k g:,, : thus

(~~!ZiJ(j~

4 =

Furthermore, (~%,)(j,

n)

i r..T=-,

Wg)

9En:,(g) dg = 6,, &

Z;,(n),

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PROPOSITION. Define f”(g):=f(g-‘), (l,f)(t):=f(g-‘t) Fourier transform has the intertwining properties

9(f Proof

#)=Ff,

Since G is unimodular, Ff(A

9(&f)

for g, LEG. Then the

= eg x 9f.

(3.6)

we get

4 = [G E(g; J n)f(g)

dg

= I GE(g-‘;j,n)fX(g-‘)dg=(~(fX))(j,n). Also, by the invariance property of the Haar measure on G, F@,f)(l,n)=~

W;j,n)f(g-‘k)dk=l

G

G = sG

(e, x e,U

E(gt;An)f(t)dt

n) f(t) dt = (e, x Ff )(i, 4.

3.4. The Fourier Inversion Theorem 3. Iff

THEOREM

is a continuous function on G = W(2), f(g)=

2

then (3.7)

(2j+l)SS2E(g;i.n)(~f)(j,n)djn.

5 = 0

Proof From the Peter-Weyl theorem, f has a series expansion in the representative functions 9An : f (g) = Cg= o CL,, = -i ajmnBi,(g). From the properties of the Z;i, and the TB”,,, the case g = e of (3.7) is straightforward: f 2j = 0

1) I sz(Ff )(L n) djn

Uj+ =

I? (2j+ 1) 2j= 0

= f

2j= 0

=

W+

2~oW+l)

i

RIh S,,ZA(n) S, g;,(g) r,S= - j

1) i j %(g)f(g) r= -j G

f(g) dg din

dg

i j .Wr,-,(g)f(g)dg= r=pJ. G

5

2j=O

i r=

-j

a<,.-,=f(e).

THE MOYAL

REPRESENTATION

FOR

121

SPIN

Thus

= f

CG+

1)Is2(en-lx Vf))(j, n) din

2j = 0

= f 2j

= f 2j

(2j+l)J =0 =0

(2j+l)J

As an immediate THEOREM

4.

s2e,-O

s2

n)(Ff)(j,

E(g; j, n)(Ff)(i,

by the tracial formula

n) djn.

consequence, we obtain the Plancherel-Parseval

Iffy

L2(G),

formula:

then

fG lf(g)12dg=

f 2j

ProoJ:

4 djn

=0

(2j+ l)Jsi I@Xj,

Set h = f “*f; then h is continuous

n)12djn.

(3.8)

on G, and

=,io 1)Js2 (9h)(j, n)djn =,zo Gj+ 1)fs2 (mxgff(j, n)djn =2,zo1)Is2 FfU, nNFf)(j, n)d,n =,,zo Vj+ l)fs2 I(@If)(.L n)12 djn (2 +

W+

on using (3.7), (3.5), (3.6), and the tracial formula (2.19) once more. We see that the tracial property was decisive in the proof of the two theorems. Remarkably enough, it is the insistence on that condition which brings the “correct” Fourier kernel-the one which allows the analogues of the classical results-into the open. The preceding constructions can be similarly carried out on other compact connected Lie groups. For abelian groups, such as the circle group U(l), the set of integral orbits is identified with the set of unitary characters. We

122

V;zRILLY

AND

GRACIA-BONDiA

have thus exemplified the recipe for a concrete geometric theory generalizing classical harmonic analysis for noncommutative, connected compact Lie groups. The Stratonovich-Weyl correspondence is seen at the bridge between the Fourier transform theory presented here and the abstract, operatorial Fourier theory. We can now make a more precise statement as to the range of the Fourier transformation, We have that 9’: L*(G) -+ OG=,%~~=: 2~ is an isometry if the norm on X6 is defined by

Since elements of Xe may be regarded as functions on G in the obvious way, an orthonormal basis is given by all the functions (2j+ l)-“*Z~~ (with the convention that Z& = 1). Then it follows from the Schwartz inequality that the twisted product of two functions in 26 is again in se. Explicitly, let f = c,,,, ~;~(2j + 1) -1/2Zis, g= I,,,, b!r(2j+ l)-“2Z~~, where (aA), (his) are square summable sequences. Then by G.W,

fxg=f

_1 2j=O

i

2J+ ’ r.s.t= -j

ai,b{sZ&,

(We note that (3.9) opens the way to extending the twisted product (2.17) to distributions on the sphere, more or less along the lines indicated in [13]; since this is only a formal extension of the theory set out above, we will not pursue the matter, but we do note that in this way our formal computations, e.g., with delta functions, may be rigorously justified. Also, the analogue of the usual Moyal algebra [ 13, 171 is the (twisted) multiplier algebra of the space of smooth functions on the sphere.) Remark. In order to obtain our results quickly, we have used the Peter-Weyl theorem which is ready at hand. However, this is not strictly necessary, and with the benefit of hindsight we could have proven Theorems 3 and 4 without recourse to such a “big gun.” We emphasize that the connection with standard representation theory is two-way, and that the Fourier kernel E in some sense contains all the relevant representation-theoretic information for the group. Taking the set of integral orbits as the primary object, one could in principle tind E directly, by solving certain differential equations given by the Casimir elements of the Lie algebra of the group. This method would lead straight to the definition of twisted product on the coadjoint orbits. The unitary irreducible representations could then be extracted from E by a generalized Weyl rule, the “geometric quantization” techniques receding further into the background.

THE MOYAL

REPRESENTATION

123

FOR SPIN

4. THE COHERENT STATE EXPANSION An alternative treatment of the Moyal representation of spin which makes manifest use of the SU(2)-covariance is the approach via coherent spin states. These are the analogue for spin systems of the well-known coherent states for the harmonic oscillator [18, 193. A suitable starting state is chosen, such as the eigenstate Ijj) of J* and Jz, and other states are produced from it by applying a representation of SU(2) rather than a set of ladder operators. The resulting states from a continuous family, indexed by the points of the sphere. This last circumstance led some people to believe that suitable operator-symbol correspondences could be constructed by means of coherent spin states. It turns out that the more natural symbol definition, namely, the expected value of an operator on the set of coherent states, lacks the tracial property. From our standpoint, coherent states yield an alternative, sometimes convenient, expansion of the Stratonovich-Weyl kernel, different from (2.6) (2.14), or (3.4) but little else (Stratonovich’s discovery antedates the introduction of the coherent states). It is nevertheless instructive to see how this expansion comes about. 4.1. Definition of the Spin Coherent States We choose and fix j E { $, 1, $2, ... }. If n = (6, #), e -ibJre-ieJYE SU(2), and define the coherent state 1j, n) by

we write

g(B, 4) =

(4-l) It is clear that (j, nlj, n) = 1 from the unitarity closure relation

of zj(g(O, 4)). We also have the

This is an easy consequence of Schur’s Lemma. If lo) EC’@+ ‘, gESU(2), Js2 (j, n)(

j, nl Xi(g)

Iv) dn = Js2 Ii

n>(4 nj(g-‘)

Ii n, dn

=Is=n,(g) Ii m>(vli, m> dm =I nj(g) Ii, m)(j, mlu> dm S=

then

124

VARILLY

AND

GRACIA-BONDiA

by W(2)-invariance of the area measure dn. By irreducibility of the representation rcj, ((2j + 1)/47r) js2 lj, n) (j, n( dn = ciZ for some constant cj. Taking traces, we get (2j+ l)cj=y/

s2

(j,nIj,n)dn=(2j+l)

so that cj= 1. Using the integral formula for Legendre polynomials, (4.2)

we may expand I( j, m 1j, n ) I * as a finite Legendre series:

(4.3)

Now the addition

formula for the spherical harmonics is just 21+ 1

7

so the orthonormality

P,(m en) =

i Ydm) tn= -I

YtW

of the Y, gives the useful formula: 21+ 1 -1 P,(m*n)P,.(n*k)dn=S,,.P,(m*k). 4K 9

(4.4)

4.2. Berezin Symbols

Berezin [20] introduced symbol calculi for functions on the sphere by expanding an operator A on CV+’ in terms of coherent states; this may be done in at least two ways, which Berezin called covariant symbols (or Q-symbols) and contravariant symbols (or P-symbols), respectively. The definitions are

Q,(n):= CL4 A IL n>,

A=2j+lI z-

s2

where the considering satisfy the P-symbols

PAn)lj, n>(L nl dn,

(4Sa) (4.5b)

definition of P, is of course implicit. Since 7rj( g)l j, n ) = lj, g . n ), by the symbols associated to B= rrj(g) Axj(g)-‘, we see that QA and P, covariance condition (2.ld). It also follows from (4.5) that the Q- and satisfy the reality and standardization properties (2.la), (2.lb).

THE

MOYAL

REPRESENTATION

FOR

125

SPIN

However, the tracial property does not hold: in general, 2j+ 1 Q,Jn) Q,(n) dn Z tr(AB) #T 47l ss2

1 PA(n) L(n) s2

dn.

(4.6)

For instance, if A = B= JZ, then Q,(n) =jcos 0, P,(n) = (i+ 1) cos 0, and the three terms in (4.6) are in proportion i2:j(i+ l):(j+ 1)2. With either class of symbols, expected values cannot be calculated as phase-space averages. However, a solution may be obtained from the observation by Berezin [20] that 2j+ 1 Q,(n) PB(n) dn =c 1 P,(n) QJn) dn = tr(AB). 4rL i .q 4n s2 Thus what we need is a symbol which is just “halfway” between the Q- and the P-symbols. To find such an intermediate symbol, note first that from (4.5) and (4.3) we get

(4.7) We can now use the orthogonality

relation (4.4) to invert (4.7), and we obtain (4.8)

(Note that the P-symbols may now be computed explicitly formula with (4Sa).) Now let us provisionally define the symbol m, by mA(n)=go(i

by combining

~l:)-l~~s~P,(n.m)(j,mlAli,m)dm.

this

(4.9)

This may be understood as a “square root” of the formula (4.8). Indeed, it is clear from (4.4) that

and

QA(~)=;~ (:

A / j> yJs2

P,(m*n)

RA(n) dn.

126

VARILLY

AND

GRACIA-BONDiA

The reality and covariance properties (2. la), (2.ld) for immediate; the standardization property (2.1 b) follows by (4.4),

@-symbols

are

2j+ 1 4n I s2 @dn)dn=$$jsze,(m)dm=trA, and the tracial property also follows from (4.4) and (4.8): 2j+ 1 it,(n) 47c s9

mB(n) dn

=e!o(: 47c =- 2j+ 1 Q,(m) 4n 5s2

~~~)~2~fs2~s2Q~(m)~~(m.k)g,(k)dmdk PB(m) dm = tr(AB).

Thus the p-symbol satisfies the Stratonovich-Weyl conditions (2.1) and so is given by @,Jn) = C;is= _ j a,,z;i,(n), with 2is given by (2.10), for some choice of the signs E{ defined in Section 2.2. Furthermore, from (2.13), zj 21+1 z;(n) = C -E/ ,d+l

j

I j

(j

OjI>

P,(cos e)

and using (4.9), (4.1) and (4.2), we compute that

so that the @‘A of (4.9) are just the WA of before. This provides an interesting a posteriori justification for the choice of sj = + 1 in Section 2. Moreover, it yields formulae for the previously derived quantities in terms of coherent states. For example, by comparing (4.9) with (2.2) it is evident that the Stratonovich-Weyl operator kernel may be expanded as

THE MOYALREPRESENTATION

FOR SPIN

127

This equation, together with the reproducing kernel formula (4.4), gives an instrument for calculating Stratonovich-Weyl symbols in the coherent state framework. In summary, the coherent state formalism, while not intrinsically necessary to a phase-space theory of spin, gives an alternative route to its derivation and provides a useful computational tool. Let us take stock that we have now four expansions for the Stratonovich-Weyl kernel: (2.6) in terms of the basis functions, (2.14) in terms of the spherical harmonics, (3.4) in terms of the representation operators, and (4.10) in terms of the coherent spin states.

5. APPLICATIONS

5.1. Classical Hamiltonian Spin Dynamics

As remarked at the beginning, we know that S2 is a symplectic manifold with canonical coordinates (cos 8, 4). The corresponding Poisson bracket is written

(5.1)

where &jik is the completely antisymmetric Levi-Civita symbol and the ni are the components of n. The factor l/d,m in the definition is natural in view of (5.4) to follow: the classical vector model of a j-spin has “length” ,/~m. If H is a Hamiltonian function, (5.1) leads to the following equation for the components of n, f-

n A grad H

{H,nIp=

(with A denoting the vector product). If p is a function on phase-space, we think of p as a state, and its classical equation of motion is

f$ (n;t) = (P,H}dn; t),

with

p(n; 0) = p,(n).

Note that the Lie algebra su(2) of SU(2) is canonically represented by Hamiltonians that are real linear functions of the spin variables. Specifically, for - +ib. 0, -&b’ . Q belonging to su(2), the corresponding functions are b . W/ b’ *W’, SO (b . W’, b’ . W’}, = - (b A b’) . WI= - [b, b’, WI], as expected. We may say that this class of Hamiltonians forms a Lie algebra of “distinguished observable? [4]. Let gb(t) be the one-parameter subgroup of SU(2) generated by 595/190/l-9

128

VARILLY

AND

GRACIA-BONDiA

- )ib .a, that is, ga(t) = exp( -$tb .a), and consider #(“(II) = Po(gb(t)-’ .n). By differentiation we check that this is the solution of (5.2) with H= b . Wj. We still get by integration an element of the group if H, of the previously considered form, is time-dependent: (5.3) Thus the classical motion is given by the formula p = dg,(t,

to)-’ .n).

In other words, gH(t, to). n describes the instantaneous position of the “rotating coordinate system” for which the initial spin configuration remains the same. These classical considerations are immediately relevant for the quantum problem. 5.2. Spin Eigenstates The arena of the Moyal representation for the j-spin is the Hilbert algebra X2. with the inner product (“Hilbert form”) Js2f(n)g(n) din and the corresponding twisted product. Before embarking on dynamics, it is good to get used to twisted product computations. We flex our muscles with W-ix WJ;+ W{x W.{+ Wix Wi=j(j+ as we expect to find from the conventional formula (1

$i>=

(3m2-j(j+ bj

1))

formulation

with

b,‘=j(j+

1)

(5.4)

of spin. Indeed, using the

1)(2j-

1)(2j+ 3),

we obtain from (2.23) ( w!‘x W!)(n) =

i m2Zi,(n) m= -j

and similarly (Wi x W:)(n) = $(j+ 1) + +bj(sin2 0 cos2 4 - f), (Wi x W;)(n) = $j(j+ 1) + ;bj(s’ m2 8 sin2 4 -i), from which (5.4) follows at once.

THE

MOYAL

REPRESENTATION

FOR

129

SPIN

We can think of these functions as functions on 6, supported on a single sphere; then (5.4) refers to j(j+ 1) times the indicator function of the jth sphere. So all of the Zi,,, are joint eigenfunctions of this function and of Wi, with the expected eigenvalues. Thus we may see that the entire apparatus of spin calculations can be carried out with functions in the Hilbert algebra Xzj. That is to say, the usual matrix formulation of spin is not forced by the physics, but may be regarded as a convenient formal analogue of the phase-space calculus, due to the orthogonality properties (2.16) of the basis functions .Z!s. A quantum state or Wigner function for the j-spin is a function p E Xzj which is normalized by ((2j+ 1)/4x) ss2p(n) dn = 1 and is positioe in the sense that p = h* x h for some hczzj. We already know some states. Let us write the .Zi, explicitly for j = 4, 1, $, 2, (55a) z,.2+fi

’

~312 +1/2,

f1/2-

G2.2:

- .-

.2+fi~2&+9@cos~ 8

40

2

3&os2~~3~cos3~.

8

8

’ (5.5b)

’

(5.5c)

112-31 fi~3&?-4&cos, 560 20 +3ficos2H+&os30+3ficos4(7 56 - 4

ZL,,,:

-- ficos2(3.

O”‘6

16

’

8

’

7-fiT6fi+dcosB

35

10

+15fi -cos2efficos3B-28 Z& : 56+47fi

-~ 33fiCos2~+9ficos4~

280

28

(5.5d)

These states exhibit cylindrical symmetry, as is clear from (2.13). That much was known and discussed in [7, p. 4661, but in rather obscure terms, for lack of a visual representation. In the Figs. 1-16, we have depicted the basis states, first by sketching the graphs of each Zi, as a function of cos 8 in the interval - 14 cos 0 < 1, and second by drawing (in orthographic projection) the nodal circles where they vanish on the

130

VARILLY

AND

FIG.

1.

GRACIA-BONDiA

Z;&(cos

0).

FIG.

2.

Z,&(cos

0).

FIG.

3.

Zi,(cos

0).

FIG.

4.

Z;j;~,,I(cos

0).

THE MOYAL

REPRESENTATION

FIG.

5.

FOR SPIN

Z~~~,3,,(cos

0).

FIG.

6.

Z&(COS 0).

FIG.

7.

Z:,(COS

FIG.

8.

0).

Z&(COS 0).

131

132

VARILLY

FIG.

FIG.

9.

10.

AND

Nodal

Nodal

GRACIA-BONDiA

circles

for Z$,,,.

circles for Z&,.

THE MOYAL

FIG.

FIG.

REPRESENTATION

Il.

12.

Nodal

Nodal

FOR SPIN

circles for Zi,.

circles for Zfjj.I,2.

133

134

VARILLY AND GRACIA-BONDiA

FIG. 13. Nodal circles for Z&.

FIG. 14. Nodal circles for Z&.

THE MOYAL

REPRESENTATION

FOR

FIG.

15.

Nodal

circles

for Z&.

FIG.

16.

Nodal

circles

for Z&.

SPIN

135

136

VARILLY

AND

GRACIA-BONDiA

sphere. The fact that for j> f the W/ spin eigenstates are rotationally inequivalent in general-which has given trouble to some authors, see again the discussion in [7]-is evident here at the first glance. We conjecture that the number of nodal circles for the Zi, is always equal to 2j. It would appear that the “classical limit” comes about by peaking of the states near their expected value as j + co, while the other regions of phase-space are awash in mild oscillations. The peaking, however, is not extremely rapid. From the Cauchy-Schwartz inequality we have the uiform estimate for the states /p(n)\
l+fi max /p(n)1 6-e

1.366<&

2

j= 1: max [p(n)1 <

2+3,,‘?f$?i 6

j=i:

5+5J5+3&5+&5 20

max Ip(

d

j= 2: max Ip(

<

7+7fi+lOfi

35

1.414;

N 1.567 < fi

N 1.732;

2! 1.686 < Ji N 1.759 < fi

= 2;

N 2.236.

The Zi, are pure states, i.e., they satisfy p x p = p. (This condition holds if and only if p is not a convex combination of other states.) It is of interest to see how the Hilbert space structure on the set of (unnormalized) pure states may be recovered: the minimal one-sided ideals of #zj are of the form p x szj, where p is any (unnormalized) pure state, and they carry a Hilbert space structure by restricting the Hilbert form on zzj. By the classical Wedderburn theorem [21, 221, the space of linear operators on this ideal is isomorphic to the whole algebra Xzj! To say that the observables are the real part of a simple Hilbert algebra expresses the basic assumption underlying j-spin Quantum Mechanics in a more penetrating way than to say that the states form a Hilbert space. As is to be expected in a theory of Wigner functions, “positive” functions need not have nonnegative values everywhere. An interesting question, already much studied for the phase-space R2” [23-261, is: exactly which quantum states correspond to nonnegative Wigner functions? The answer is easy to see for the spin-i case. If a state p has density matrix +(I+ k * a), subject to the positivity condition IIkll < 1, then we get p(n) = t( 1 + ,/? k * n). We conclude that p is nonnegative on S2 if and only if Jlkl( < f fi. In particular, pure states (corresponding to Ilk11= 1) never have nonnegative Weyl symbols in the spin-f case; the nonnegativity appears only for sufficiently unpolarized states (such as thermal Gibbs states beyond a certain temperature: see Section 5.6). That nonnegativity of the Wigner functions should not be expected for the eigenfunctions, but only for mixed states, was pointed out already by Moyal [ 11; we refer to [26] for a discussion of this point in the phase space R2”.

THE MOYAL

REPRESENTATION

FOR

137

SPIN

In addition to the W$eigenstates, the nondiagonal Z!S which help to generate zzj deserve some attention. Following a terminology introduced by Dahl in the flatspace case [27], we shall call them generically “transitions.” As a consequence of the tracial identity, we get the following result. PROPOSITION.

The matrix element (jm’l

A )jm)

equals fsz Zi.,(n)

W,(n) djn.

(To check this, just compute (jm’( jsz W,(n) d’(n) din 1jm).) There follows a list of transitions for j = 4, 1, $, 2. They are normalized

by

2j+ 1 lZ~3/’ dn = 1. 471 s9 On account of (2.12), which yields Z{Jn) = Zp(n) = (- l)‘-“Z”,-,( to take r
-l/2,1/2’

=ifisin 2

i3f9;

Z& : k (1 + Js cos 0) sin 8ei@,

~312 l/2,3/2

-n), it sufices

Zi I. I : a Jlo

sin’ 8e2+;

’

~312

-l/2.1/2.

z

3~

-

l/2.3/2

’

cos2 8 + q

ZZ,,,:

(

3fi 3J14 T-1,cos2

Zi,,2:~fi(1+3cosf3)sin30e3i~,

8 sin’ 8ezu, > Zi,,:Q$sin40e4?

cos3 e) sin Beid,

138

VARILLY

AND

GRACIA-BONDiA

5.3. Twisted Exponentials The twisted exponential for any real element f of XIj is defined to be the solution of the “Schriidinger equation”:

(fx

Ej)(n;

t) = ii

Ej(n;

t);

Ej(n;

0) = 1.

(5.6)

If we think of f as a Hamiltonian, we can think of E’ as a propagator: its Stratonovich-Weyl transform is the unitary evolution operator. From (2.23), the twisted exponential or “evolution function” for Wi is given explicitly by

This is the family of functions on phase-space corresponding to the group of operators nj(exp( - iJz t)) in the conventional formalism. A basic precept of phase-space Quantum Mechanics is that the spectrum of a Hamiltonian may be obtained as the support of the Fourier transform (with respect to t) of its twisted exponential. Moreover, as pointed out by Bayen et al. [4], the eigenfunctions are also obtained by this method in a single stroke. Fouriertransforming 3(n; t), we obtain the spectral function IZ$n;A)=

i &A-m)ZA,(n) m= -j

which yields the expected spectrum for Wi and contains the spin Wigner eigenfunction for each value of m. For the spin-i case [28], we find from (5.5a) that Sji2(n,

For j= 1, 3/2,2, we similarly zf(n,

t) = 1 - i $

t) =

cos 5 - i J5

cos e sin 5.

obtain from (5.5b)-(5.5d) (5.7b)

cosesint-;(4-Ji6+3Ji6c0s28)(1-cost),

3,312(n, t) = cos2~+(&l)cos~sin2~-i

xsinicose-3ficosisin’f

(5.7a)

(

fi+f(3fi-6fi)sin2i

608~

e+

i J%

sin3 f cos3 8,

> (5.7c)

THE

MOYAL

REPRESENTATION

FOR

SPIN

139

Sz(n,r)=&+39fi-28+(56-8fi)cosr+(112-31fi)cos’t)

-;J-t14 6+cost)sin’icos’B--4fiicosisin3icos3B + 3 Jii

5.4. Quantum Hamiltonian

In the parameter precisely, equation

(5.7d)

sin4 5 cos4 0.

Spin Dynamics

Moyal representation the evolution of states is governed by a twofamily of internal automorphisms of the twisted product algebra. More if H(t) is the Hamiltonian, we consider the solution Eh(n; t, to) of an which generalizes (5.6): .a

H(t) x Eh( t, t,)(n) = I z Eh(n; t, to);

This is clearly a “unitary” state, we have

ZA(n; to, to) = 1.

(5.8)

function, i.e., Ei x EL = Sk x Ei = 1. If pO is an initial

p(n; t, to) = &(t,

to) x p. xE&t,

to)(n),

or, in differential form, using (5.8) and the fact that (f x g) = g x f,

ap

at=

-i[H,p].:=

-i(Hxp-pxH).

(5.9)

In the Quantum Mechanics of spinning particles, one usually finds Hamiltonians of the form H = yB( t) . Wj. This is so in the case of nuclear magnetic resonance, but also for spin-orbit coupling, spin-lattices, the Heisenberg ferromagnet, etc., where of course the dependence of B on time reflects the complicated change of the parameters of the overlying system of which the individual spin is a subsystem. For such Hamiltonians, one can get the explicit form of @I,(n; t, to) from the formulae (5.7) for Ei. In fact, little more than the covariance condition is involved here. Let us write H(t) = yB(t) . Wj = b(t) . Wj, to adapt the notation to that of Section 5.1. We have

%(n; t, to) = WgAt, toI; i, n), 59s/190/1-IO

(5.10)

140

VARILLY

AND

GRACIA-BONDiA

where gH(f, t,):= exp( -$a j;, b(t’) dt’) as in (5.3). Indeed, E(e; j, n) = 1, and

if$ (gdt, to); j, n) = tr((b(t). J) y(s,(t, to)) A%)) = H(t) x E(g,(t, to); j, n). Now, if I;, b(t’) dt’ j;, B(t’) dt’ IJ;, b(t’) dt’J = Is;, B(t’) dt’l =: “(t’ to) determines the instantaneous

axis of rotation,

we have

On using (3.3d), this yields

Hence we obtain Eh(n;

t,

to) =

exp ( im y /CB(r’)dt’j)zb,(.(‘,

k

m=

to).“).

(5.11)

-j

Here we are making a slight abuse of notation in writing Z~,(cos 0) instead of Zk,(n) [see (2.13)], With this convention, note that the property Zi,(-a-n)= Zj --m, --m (a * n) has been used in deriving (5.11). As a consequence of (5.10), we find that Ani t) = egH(r,ro)x P x egH(,,,o)-4n) = p(g,(t,

W’

4.

(5.12)

We summarize this discussion in the following theorem. THEOREM 5. If the dynamics of a j-spin is governed by a Hamiltonian of the form H = yB(t) *W’, then the evolution of a Moyal quantum state p. is given entirely by the classical motion n H gH(t, to) . n, where gH( t, to) = exp( - fiya . J:, B( t’) dt’).

We conclude that for distinguished Hamiltonians a stronger result than the Ehrenfest theorem holds. In differential form, from (5.2) and (5.9), the result reads as a proportionality of Poisson brackets and Moyal commutators,

CK PI x = i{H, P>P,

(5.13)

valid whenever H is a distinguished Hamiltonian. This is an important result in its own right. It is amusing to see, for instance, how the ordinary differential equations

THE

MOYAL

REPRESENTATION

FOR

141

SPIN

for spherical harmonics, which in the conventional context are the basis of orbital angular momentum theory, arise in our context as a consequence of the equivalence of classical and quantum motion. From (5.1) we obtain

and in particular

On the other hand, the Moyal brackets (5.9) are readily computed with the help of (2.9) and (2.12). From (2.9) we get

and thus [Y,,,

Y,J,

=J3y1)

~1,xLin

-(p-im

1 ip+L+n)(S

f, I,+L+n)

ig ip;im))zL,p+m+n’

For n = 1, the recurrence relations for the Clebsch-Gordan

coefficients give

[Y,,,Y,~l,=~~~p~-j(j(j-pm,ci+p+m+*~(~ -d?iPK~+P+qp~l

~~p+~+l))z~,p+m+~

=&J~p~pj;;;:

x(:,

,il

blp;irn)

jp+~+l)z~.p+m+l

142

VARILLY AND GRACIA-BONDiA

and similarly [Yl,-,,

Y,,J x = -

1)/8nj(j+

3(l+m)(Z-m+

1) Y/,+,.

Also,

=-$$Ti-)yim* In all cases, we find that

CYln, ~hl x = i( y,,, Yh>P (which in fact provides an alternative Theorem 5).

computational

proof of (5.13) and hence of

5.5. The Majorana Formula

The Majorana formula for transition probabilities in magnetic resonance phenomena can be given a surprisingly simple interpretation and derivation in the phase-space formalism. The transition probabilities are computed as the overlap of two distributions of probability, one of which precesses around the field. Formally, the details of the computation are entirely classical; the quantum “input” is the shape of the states. Let us have a spin system prepared in a Zi,,, eigenstate. If the magnetic field is “switched on” at the instant to, we compute the probability P,,,(t, to) of a transition to a .Z$,, eigenstate over the time interval [to, t]. The result is: MAJORANA

FORMULA.

P,Jt,

to)=

@$,JgH(f,

to))12.

Proof: From (5.12), the state of the system at time t is Zim(gH(t, to)-’ ‘II). Thus the required transition probability is given (in accordance with the rule (2.1~)) by the overlap integral

Pm.mf(t9 to)=T

Is2ZA,(g;l .n) 2$,,(n) dn

=- 2j+ 1

ZL(n) 47L i s2

ZLJgH

(5.14)

.n) dn

=ss2ZL(n)(e,,xZ$,,xG)(n)

djn

= 5s2 wLl

=

x e,,)(n)(ZL,.

x G)(n)

.i ss2r, s1= ~ ,’ -%,(n) %,,m(gH) -%,(n) %?gH)

= %h(gH)

%LkH)

= I~~~,(g”)I’

by (2.19)

d’n

d’n

by (3.2)

by (2.16).

THE MOYAL

REPRESENTATION

143

FOR SPIN

An even shorter method uses the Proposition in Section 5.2: the desired probability is the absolute value squared of the integral of the transition function multiplied by the propagator. By (3.2), this integral is just

= 5s2 ZA.,(n)

i G(n) r.,= -,,

9ir( gH) d,n = 9&+J

gH).

Alternatively, one may choose to expand (5.14) using (2.13), for a more explicit computation. If n, denotes the north-pointing unit vector, then cos 8 = n, . n and gH(t, to) . no = n(t, to) is the direction of the “instantaneous axis,” so we find, using (4.4), that

=- 2j+ 1 ZA,(n) 471 ss?

Z$,,J

g, . n) dn

which is precisely Meckler’s practical form of the Majorana formula [29]. There is no need to compound the general case out of the spin-4 case. Moreover, the roundabout arguments that Meckler had to employ to prove equivalence of his formula to the usual one are not needed here, as both results spring directly from (5.14). It is clear that in the Moyal representation Purcell’s quantum view and Bloch’s dynamical view of magnetic resonance phenomena [6] are definitely reconciled. 5.6. A Comment on Spin Relaxation In a previous article by Emch and one of us [30], it was argued that a Markovian relaxation of a f-spin must be governed by a Bloch equation of a specific kind. The argument is that the spin relaxation should be an evolution in a subsystem of some (infinite) conservative dynamical system: this requirement imposes certain symmetry and positivity conditions on the subdynamics of the spin. These positivity conditions yield the standard form of the Bloch equation for a Markovian dissipative dynamics, but have a very abstract character. We want to illustrate here how pictorial descriptions of spin (such as that afforded by the

144

VARILLY

AND

GRACIA-BONDiA

Moyal representation) can be useful, by showing the conditions established in [30] to be equivalent to an entropy growth principle. Let us first calculate the Weyl symbol R$ corresponding to a Gibbs state of a j-spin. The Weyl symbol of eeBJz is CL= ~, e-@“Zh,,,(n) = s=(n; -$). From (2.la), the partition function is Z(o):=

tr(ePDJ~)=~

js15,(n;

-$)

dn

and the Gibbs state is described by Rb = Z(b) -‘E-;(n; - $). For example, for j = 4,

;

cash - - & Rbi2(n) = i

1 + fi

P dn = 2 cash T, B cos 0 sinh -2 >

cos 8 tanh i

(5.15)

.

The entropy of the Gibbs state may be calculated as usual to be (5.16) More generally, any (mixed) state of a spin-4 system is rotationally e uivalent to a Gibbs state (5.15), i.e., can be written as p(n) = +( 1+ P 3 k - n) = f(1 +fi m * n tanh(/?/2)), regarding m E S* and fi as parameters. As a function on the sphere S2, p is cylindrically symmetric about the m-axis and attains its maximum value for n = m. Note that for tanh p/2 3 f fi, this Wigner function takes negative values on a small cap centered at n= -mm; p is nonnegative only if the entropy (5.16) is sufticiently large. Now suppose that the spin evolution is Markovian (i.e., the evolution is governed by first-order differential equations) and has RY* as its only stationary state. Then in [30] it is shown that the assumption of the existence of a larger conservative system forces the evolution equations to have the “standard” form,

-$ k,(t)-tanhi with o real and 0

k;(l)-tanhp

THE

MOYAL

REPRESENTATION

FOR

145

SPIN

If q(t) denotes the entropy of the state p(n; t), then from the considerations following (5.16), if and only if From (5.17) subject to the initial

condition

$ IlUtNl ~0.

k(0) = a, one finds

2l(o:+ul)+Z~oj-2~a,tanh~

>

.

The limiting case is that of llail = tanh (p/2), which is to say that the initial state already has the equilibrium entropy. In this case, the Ansatz of “no entropy decrease” yields the condition (cl - 1)~: - pu, tanh y + 1 tanh* which is equivalent to the condition 5.7. The Minimal

0

f

whenever

Z 0,

1~~16 tanh B 2

0

Coupling Recipe

In a nonrelativistic context, a phase-space formulation can be constructed for spatial and spin variables simultaneously, simply by multiplying the kernels of the Moyal llat phase-space formalism and those introduced in this paper. We will not go into this matter here; it will be dealt with in detail elsewhere. However, a remark should be made. From the work of Levy-Leblond [31] it has become known that the “correct” gyromagnetic ratio for an elementary spin-t particle is “predicted” by a Galilean-invariant theory. As the argument is completely algebraic, it can be reproduced in our context. We note that ww

i

x ww

i

= wy* .)

x Wf’2

=

.I

W”2

Z

x W’l2

W.J/* x W,f12 = - W,il* x Wil* = tiW’f*.T

2 m

w”qx

1

49

(and circular permutations).

As the spatial and spin variables commute, spin t can be written as H,,,=-(p.

=

the Hamiltonian

of a free particle of

(p* W1’2),

If we now introduce an electromagnetic interaction in accordance with the minimal coupling recipe p I-+ p - (e/c)A, and take into account the formulae [4] pix A,=pi--l,

i aA 2 8%

A,xpi=p,+--,

iaA. 2 %I

146

VARILLY AND GRACIA-BONDiA

of the usual flat-space twisted product theory, we easily calculate

where B = rot A is the magnetic field. The intrinsic magnetic moment particle emerges automatically.

6. NOTES

of the spin-t

ON THE LITERATURE

The problem of generalizing the Moyal framework to spin variables has been taken up before. As stated in the Introduction, the early approach sketched by Stratonovich [S], in connection with Moyal’s original paper [l], was the more satisfactory one. The work of the eminent Russian mathematician, however, was nearly forgotten. In the seventies, Berezin [20, 321 started a “quantization” program for homogeneous Kahler manifolds. As remarked in Section 4, Berezin’s “covariant” and “contravariant” symbols are directly related to the Q- and P-symbols which naturally occur in the coherent state representation. In fact, Berezin’s method can be suitably represented in general in terms of systems of coherent states defined on the corresponding manifolds (see [ 191). After the 1978 papers by Bayen ef al. [4], “quantization” of the sphere became a natural target for practitioners of *-product theory. In this respect, the paper by Fronsdal [ 111, while containing some imprecisions, is full of ideas. His concept of a *-representation (already present in [4]) is the germ of the definition of the Fourier kernel in Section 3. In that paper, Fronsdal gives a general expression for SU(2)-covariant *-representations over the sphere, including, of course, our Fourier kernel E. However, the one singled out in the end essentially boils down to the Q-symbol under another guise. Lucid summaries of the *-representation program are found in [33, 343. The paper by Wildberger [ 163 is akin in spirit to *-representation theory; his Fourier kernel is essentially the Q-symbol for general compact groups. The work by Moreno and Ortega-Navarro [35] draws both from Berezin and *-representation theory. They note that in the flat-space case the equivalent of the Stratonovich-Weyl symbol is a family of reflection operators (as was discovered by Grossmann [36] and Royer [37] a dozen years ago); so they write down a kind of Weyl correspondence whose kernel is the family of SU(2) elements which perform the reflections on the sphere. This empirical recipe fails, in the sense that it is not equivalent to the tracial property; moreover, it has the weird consequence of assigning purely imaginary symbols to hermitian operators. When they make

THE MOYAL REPRESENTATION FOR SPIN

147

practical calculations of twisted exponentials in [35], Moreno and Ortega-Navarro fall back on the Q-symbol again. In [38] Gilmore expounds what seems to be the same basic idea of Section 4. However, this paper is very short and no explicit calculations are given. Some people have thought of a discrete formulation of spin variables. In this context, the interesting papers by Wootters [39] and by Cohendet et al. [40] must be mentioned. Here, phase space is taken to be the direct product of two copies of the cyclic group Z,. A correspondence between operators on C” and functions on Z, x Z,, incorporating the covariance and traciality properties, can be constructed without much trouble for n odd (not just for n prime, as Wootters does). The symmetry group can bc taken to be the finite Heisenberg group based on E,. This and the fact that in a discrete representation all the points are essentially equivalent (in contrast to what we learned in Section 5.2) suggest that this “Quantum Mechanics with finitely many degrees of freedom” is a discrete version of ordinary Moyal theory, rather than a theory for spin. We note finally that expression (5Sa) is arrived at in [41] on the basis of an argument employing negative probabilities; other attempts at continuous descriptions of spin which bear some resemblance to our construction here are found in [4244]. In summary, Moyal theory now provides a self-contained formulation of nonrelativistic Quantum Mechanics, including both spin and spatial variables, comparable to the Schrijdinger and Feynman formulations in its scope. (Details of this encompassing formulation, including proof of full Galilean covariance of the theory, will be presented in another paper.) We note that, by using a suitable class of classical generating functions, a path integral expression for twisted exponentials was given in [45]. A similar construction could prove useful in the context of this paper. Our findings here also contribute to the search for a Poincare-invariant Moyal-like framework. Sizable obstacles remain, however, to giving a formulation of relativistic quantum theory on classical phase-space. ACKNOWLEDGMENTS We are grateful to Simone Gutt and Jose Carifiena for helpful conversations. This work began in the hospitable atmosphere of the BiBoS Research Centre at the University of Bielefeld, while one of us (J.M.G.-B.) was visiting; support for this visit from the Deutsche Akademische Austauschdienst is most gratefully acknowledged. We heartily thank Philippe Blanchard, who prodded us to go beyond our early ideas on the spin-half case. We acknowledge support from the Vicerrectoria de Investigation of the University of Costa Rica. REFERENCES 1. 2. 3. 4.

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