PHYSICAL REVIEW A VOLUME 36, NUMBER 8 OCTOBER 15, 1987

Gaussian-signer distributions in quantum mechanics and optics

R. Simon and E. C. G. SudarshanThe Institute of Mathematical Sciences, C I T.. C. ampus, Madras 600113, India

N. MukundaCentre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India

(Received 2 April 1987)Gaussian kernels representing operators on the Hilbert space &=L'(1R") are studied. Necessary

and sufhcient conditions on such a kernel in order that the corresponding operator be positivesemidefinite, corresponding to a density matrix (cross-spectral density) in quantum mechanics (op-tics), are derived. The Wigner distribution method is shown to be a convenient framework forcharacterizing Gaussian kernels and their unitary evolution under Sp(2n, 8) action. The nontrivialrole played by a phase term in the kernel is brought out. The entire analysis is presented in aform which is directly applicable to n-dimensional oscillator systems in quantum mechanics and toGaussian Schell-model partially coherent fields in optics.

I. INTRODUCTION

The Wigner distribution gives a description of statesin quantum mechanics closely resembling the statisticalphase-space description of classical systems, without,however, losing any of the specific features of quantummechanics. ' It works at the level of the density operatorrather than the state vector, so the superposition princi-ple of quantum mechanics is not manifest. It is relatedto the Weyl correspondence between classical andquantum-dynamical variables in a natural way. '

In recent times a closely analogous method based onthe so-called Wolf function has been introduced into sta-tistical optics. This development is part of the relative-ly recent understanding of the relationships between ra-diative transfer and radiometry on the one hand, andelectrodynamics and physical optics on the other, ' al-though the fact that the ray-optics wave-optics connec-tion is largely identical to the classical-mechanics quantum-mechanics connection has beenknown for a long time. The basic objects such asspecific intensity and radiance in radiative transfer andradiometry are formally analogous to the phase-spacedistribution in classical mechanics. The Wolf functionis defined as a partial Fourier transform, or Moyal trans-form, of the two-point cross-spectral density of the opti-cal field, just as the Wigner function is the Moyal trans-form of the configuration space density matrix in quan-turn mechanics. It tries to capture as many of the sim-ple properties of rays in geometrical optics as possible,without sacrificing the specifically wave optical featuresof interference and diffraction; this is achieved throughthe introduction of the concept of generalized rays.

The cross-spectral density in optics and the densitymatrix in quantum mechanics have very similar definingproperties. In a configuration-space description, bothcan be viewed as operators on a Hilbert space I (R") ofsquare integrable functions for suitable n, and both areHermitian positive semidefinite. The only difference is

that, while in the optical case we require a finite trace, inquantum mechanics the trace must be unity correspond-ing to normalization of probability. As a consequence ofthis formal similarity, many of the well-known proper-ties of the Wigner distribution pass over to correspond-ing properties of the Wolf function, with appropriatechanges in interpretation.

Let us recall some of the more notable results con-cerning the Wigner distribution in quantum mechanics.It is well known that it is not a true phase-space proba-bility distribution but only a "quasiprobability distribu-tion, " since for a general state it is not pointwise non-negative. In fact, the only pure states for which theWigner distribution is everywhere non-negative are thosefor which the Schrodinger wave function is Gaussian.Although the Wigner distribution is thus in generalindefinite, it turns out that the convolution of twoWigner distributions is always a non-negative functionon phase space. This resulting function is in general,however, not a Wigner distribution corresponding to anystate. Nevertheless, attempts have been made to inter-pret the convolution process as the coarse grainingeffected by the interaction between a quantum systemand a measuring apparatus, the input Wigner distribu-tions referring to system and apparatus, respectively.Another property of Wigner distributions is that theycannot be arbitrarily sharply peaked in phase space sincethey must after all reproduce the uncertainty relations.

The Wolf function has similar properties. It is a realfunction on the phase space appropriate for light, but fora general cross-spectral density it can take on both posi-tive and negative values. Thus it can only be thought ofas a quasi-intensity distribution of generalized rays oflight. The only fully coherent optical fields for whichthe Wolf function is non-negative everywhere are thosefor which the field amplitude is Gaussian. The positivesemidefiniteness of the associated Hilbert space operatoris reflected in properties of the Wigner distribution andthe Wolf function in identical ways. While the Wolf

36 3868 1987 The American Physical Society

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3869

function cannot also be arbitrarily sharply peaked in thespace of its arguments, the convolution properties ofWigner distributions mentioned above do not have an es-tablished measurement theoretical meaning in the opti-cal context.

The use of the Wolf function becomes practically con-venient in paraxial propagation problems. In this case,the family of Gaussian Schell-model fields generated bypartially coherent planar sources of a particular typehave received particular attention. One speaks of isotro-pic' or anisotropic" Gaussian Schell-model fields(IGSM or AGSM} according to whether or not there isinvariance with respect to rotations about the beam axis.In the IGSM case the cross-spectral density in any trans-verse plane has the form'

I (q;q')= A exp[ q Lq q' Lq' ,'(q q') M(q q')

+ (q q') K(q+q')] .2Here q and q' are two-dimensional position vectors inthe transverse plane, while L,M are real, L being posi-tive and M non-negative, and K a real parameter. Clear-ly, L ', M ', and K are, respectively, measures of theintensity width, the coherence length, and the phase cur-vature. This three-parameter family of IGSM fieldstransforms into itself not only under free propagationbut also under action by all axially symmetric first-orderoptical systems (FOS's). ' In wave optics an FOS actson the amplitude via the generalized Huyghens in-tegral, ' which as is known furnishes a unitary represen-tation of the group Sp(2,E) on L (E) (in the axially sym-metric case). In the language of generalized rays, thesame system acts on the ray parameters through the nu-merical Sp(2, E) ray-transfer matrix. We have analyzedelsewhere the transformation of IGSM fields by FOS's,using the generalized ray distribution or Wolf function.For such fields, this function is again Gaussian, beingdetermined by a 2&2 real, symmetric, positive definiteparameter matrix G; the positive semidefiniteness of thecross-spectral density (1.1} interpreted as an operatorkernel leads to detG (1. Using the local isomorphismsSp(2, E)=SL(2,E)=SO(2, 1), a geometrical picture hasbeen developed in which IGSM fields and FOS's can berepresented, respectively, as timelike vectors andLorentz transformations in a fictitious (2+ 1)-dimensional Minkowski space. This picture makes itparticularly easy to visualize many of the properties ofthis class of fields. In particular, it leads to a naturalgeneralization of Kogelnik's "abed law"' to the partial-ly coherent IGSM beams. '

When rotational invariance about the beam axis isgiven up, L and M become 2&(2 real symmetric ma-trices, L being positive definite and M positivesemidefinite, while K is a real matrix. We then have theten-parameter family of AGSM fields. ' One can alsoconsider FOS's without axial symmetry, which make upthe group Sp(4, E);' these now map the family of allAGSM fields onto itself. The (astigmatic) FOS acts by aunitary representation of Sp(4, E) on L (E ) in thewave-optic description, and by a numerical Sp(4, E) ray-

transfer matrix in the generalized ray description. TheWolf function is Gaussian again but now characterizedby a 4&4 real symmetric positive definite matrix G.Compared with the IGSM case, however, the implica-tions on G of the positive semidefiniteness of the kernel(1.1) are much more intricate. We also succeeded in de-veloping a geometrical picture based on the local iso-morphism Sp(4, E)=SO(3,2). In this picture, eachAGSM field is represented by an antisymmetric second-rank tensor, and each FOS by a de Sitter rotation, in a(3+2)-dimensional space. This analysis showed that thepossible AGSM fields are made up of two distinct fami-lies or types, corresponding to distinct kinds of orbitsunder the adjoint action in the Lie algebra of SO(3,2). '

As a converse to the process of transferring known re-sults on Wigner distributions to new results for the Wolffunction, we can also state the following: for everycross-spectral density allowed by general principles,there must exist a state of a suitable quantum-mechanical system such that the given cross-spectraldensity can be reinterpreted as the configuration-spacedensity matrix of the state. In this sense, the family ofIGSM cross-spectral densities becomes the density ma-trices of the thermal states of a one-dimensional harmon-ic oscillator, and their Sp(2, E) transforms. A similaridentification can be made for AGSM fields and states ofa two-dimensional oscillator. For oscillators with N de-grees of freedom we have to deal with the groupSp(2n, E). Their thermal states and all Sp(2n, E) trans-forms thereof, will have density matrices of the form(1.1) but with q and q' being n-component vectors, andL,M, K being suitable n Xn matrices.

Following this generalization of the kernel (1.1) to ndimensional vectors q, q', we see that operators onL (E") with such kernels play an important role both inoptics and in quantum mechanics. All such kernels lead,via the Moyal transform, to Gaussian phase-space distri-butions (Wolf or Wigner) characterized by a 2ndimensional real parameter matrix G. Given the Gauss-ian kernel (1.1), it is important to know the conditionson the n-dimensional real matrices L, M, and K whichwill ensure that this kernel is a bona fide cross-spectraldensity or density matrix. In the phase-space version, itis important to characterize the necessary and sufficientconditions on G so that the Gaussian phase-space distri-bution will be an allowed Wolf or Wigner function. 'The present paper addresses itself to a complete analysisof these questions. We present the work in such a wayas to make the results applicable in both contexts. Aparticularly delicate point which will be brought out isthe role of the "phase matrix" K, which becomes evidentonly for N) 2 and which was not identified in Ref. 15where we studied AGSM fields.

The contents of the paper are as follows. In Sec. II wederive some algebraic properties of the Moyal transformwhich give an essentially complete characterization ofthis transform. In particular, the unitary Sp(2n, E) ac-tion on L~(E") via an n-dimensional Huyghens integralappears extremely simple in terms of the Moyal trans-form, when one is considering operators on, rather thanwave functions in, L2(E"). The problem of finding the

3870 R. SIMON, E. C. G. SUDARSHAN, AND N. MUKUNDA 36

necessary and sufficient conditions for a Gaussianphase-space distribution to be a Wolf (Wigner) functionis formulated and solved in Secs. III and IV. We use thefact that under the Sp(2n, E) action the operator withthe Gaussian kernel (1.1) undergoes a unitary similaritytransformation which preserves the defining propertiesof a density operator or cross-spectral density. Hencethe Sp(2n, E) action can be used to take the associatedMoyal transform to a standard or normal form wherethe defining properties can be conveniently tested. Allthis is made possible by a basic theorem of Williamson'concerning normal forms of quadratic Hamiltonians.The complete set of necessary and sufficient conditions ispresented explicitly in terms of certain Sp(2n, E) invari-ants of the G matrix. In Sec. V. the important role of,and restrictions on, the phase matrix E which arise onlyfor N)2 are highlighted. This was missed in Ref. 15and this incompleteness is rectified here. It turns outthat only the antisymmetric part of E is subject todefinite restrictions, which explains why there are nosuch conditions for IGSM fields. The final section, Sec.VI, contains some concluding remarks.

[qP, l =i&[qq,]= [PP,]=0 . (2. 1)The Schrodinger realization of these operators uses theHilbert space &=L (E") of square integrable functionsof n real variables q,r =1,2, . . . , n:

it(q)I f d "q I P(q) I '& ~ (2.2)

(The symbol q, and similarly p later on, stands for then-tuple qi, q2, . . . , q. ) On this space the qact asoperators of multiplication while the Pact as operatorsof differentiation; in the familiar notation using an idealbasis for & made up of simultaneous eigenvectors I q)of the q, one has

& q'I q) =6'"'(q' q), oo & q& ~,

1((q)=&qI g&,

&q II", I 0&= z~ a

a

(2.3)

A linear operator I on & (we shall mainly be concernedwith bounded operators) is completely determined by itskernel in the Schrodinger representation

(2.4)

II. ALGEBRAIC PROPERTIESOF THE MOYAL TRANSFORM

We develop here some algebraic properties of theWig ner-Moyal transform, familiar from quantummechanics, with respect to the real symplectic groupSp(2n, lR). These properties amount essentially to acomplete characterization of this transform. Consideran irreducible set of Hermitian operators q,P,r=1,2, . . . , n obeying the familiar canonical commuta-tion relations

The commutation relations (2. 1) are invariant underthe group of real linear homogeneous symplectic trans-formations Sp(2n, E). To express these transformationsin matrix form, we arrange qand P into a column vec-tor Q with 2n Hermitian operator entries:

q(2.5)

Pn

Then Eq. (2.1) can be compactly written as[QQb]=ip,b, a, b =1,2, . . . , 2nP=l &2 X In xn

I 0P =P '= P. (2.6)

The group Sp(2n, E) consists of 2n-dimensional real ma-trices S obeying

S PS=P . (2.7)

Q,' =S,i, Qt, U(S) 'Q, U(S),U(Si)U(S2)=U(Sis2) . (2.9)

In the Schrodinger representation the effect of U (S) on awave function g(q) is given by a generalized Huyghensintegral which for general S involves a nonlocal kernel:

@'(q)= & q I U(s) I q & = J d "q'& q I U(s) I q' &g(q') .(2.10)

The related action on operators I is given by conjuga-tion,

I =U(S)-'rU(S) . (2.1 1)On account of the fundamental significance of the

group Sp(2n, lR) with respect to the commutation rela-tions (2.1) and (2.6), we can ask if there exists a way ofdescribing operators I such that the action (2.11) be-comes as simple as possible. ' A suggestion as to howthis may be achieved is obtained by examining theinfinitesimal elements of Sp(2n, E) and generators forU(S). An SHSp(2n, E) close to the identity can be ex-pressed as

[It is sometimes useful to note that if S ESp(2n, lR), thenboth S and S ' belong to Sp(2n, E) as well. ] For anyS HSp(2n, lR), if we define operators Q,' by

Q =SQ (2.8)it is immediate that Q,' obey the same hermiticity andcommutation relations as the Q, . Therefore, there is aunitary operator U(S) determined by S, giving a unitaryrepresentation of Sp(2n, E), and carrying Q, to Q,':

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3871

S=I+eJ,(PJ) =PJ .

(2.12)The inverse is

I (q;q') = I d "p W( ,'(q+q');p} exp[ i(q q') p] .The corresponding U(S) must difFer from unity by asymmetric quadratic expression in Q since on commuta-tion with Q the result has to be linear in Q. A straight-forward calculation shows that

Q'=Q. +&J.bQb=Q. + &[Q GQ, Q. ]= U(S) 'Q, U(s),

(2.17)It is now easy to verify that the transition from I to itsanticommutators or commutators with the q's and p's isrefiected in very simple changes in W(q;p). For exam-ple,

r--,' [q,r I =- r(q ,'q';q+-, 'q')

U(S) = I EQGQ,2(2.13) ~ql (q ,'q';q+ ,'q')

~ W(q;p)~qW(q;p) . (2.18)G=G = PJ .

The correspondence between commutators among J'sand among symmetric hermitian bilinears Q GQis given by

In this way, the Moyal transforms of the listed operatorsare related to the transform of I itself by

,r j qw(q;p),p, W(q;p

'Q'GQ,

,Q'G'Q = Q'G"Q

~ [q,f']~i W(q;p),a~Br

(2.19)

G = PJ, G'= PJ', G"= PJ",J"=[J,J'] .

(2.14)

Now the effect of an infinitesimal transformation U(S)on an operator I can be simplified to the form

r = U(S)-'r U(S) =r+ 'e[Q 'GQ, r]2

[p,f']~ i W'(q;p) .Bq,

This set of results can be compactly expressed if, to ac-company the column vector Q made up of operator en-tries, we define a c-number column vector Q as

=r+ eG.bIQ. [Qb r]I . (2.15) qn (2.20)This suggests that if we can describe operators I in sucha way that the processes of commutation and anticom-mutation with the basic q's and p's assume a very simpleform, then the action (2.11) of Sp(2n, E) will becomeequally simple. It is precisely this that is achieved bythe Moyal transform description of I, as we now show.

Given the operator I on &, its Moyal transform is afunction W(q;p) defined in terms of the kernel I (q;q')b 10, 11

W'(q;p)=(2n) "f d "q'I (q ,'q', q+ ,'q') exp(iq' p) . .(2.16)

Pn

Then the Moyal transform of I can be written as W(Q)and Eqs. (2.19) read

,

' [QI ] ~Q, W'(Q),[Q. r] iP.b

'0 bW(Q) . (2.21)

The change in W(Q) when I suffers the infinitesimalSp(2n, E) transformation (2.15) is now easily calculated,

r =r+ e[Q GQ, I ] - W'(Q) = W(Q)+ieG bQ, l pb,2 W(Q)a

= W(Q)+eQ, JC

W'( Q)

=W(e'Q) . (2.22)

(2.23)

Hence, upon integration, we obtain for finite elements on a one-parameter subgroup of Sp(2n, E), and so for any S,I '= exp Q GQ I exp Q GQ - W'(Q)=W(exp(J)Q},2 2r = U(s)-'r U(s) =- w'(Q) = w(sQ) .

3872 R. SIMON, E. C. CJ. SUDARSHAN, AND N. MUKUNDA 36

This is the characteristic property of the Moyal trans-form, with respect to Sp(2n, IR), that we were lookingfor.

It should be emphasized that the above results are val-id irrespective of whether or not I has any specific prop-erty such as hermiticity, positivity, or finite trace, andalso whether it is the density operator of a quantumstate or the cross-spectral density (at a fixed frequency)of some optical field. In case I has one or more of theseproperties, W(Q) will acquire corresponding additionalproperties as dictated by the defining integral transform(2.16).

TrI =(rr/2)" ~ (3.5)Next we turn to condition (B). To calculate W(Q) weneed the expression

I (q ,'q', q+ ,'q') =(detL )'~X exp[ 2q L q ,' q' (L +M )q'

i q'TKq] . (3.6)

This will hereafter be assumed. In anticipation of this,the kernel in Eq. (3.1) has been normalized to have afixed trace:

III. GAUSSIAN KERNELS AND THEIRMOYAL TRANSFORMS

We are interested in a special class of operators I ongf =L (R") whose Schrodinger kernels have the follow-ing Gaussian form:

(q~

I~

q'):I (q;q')=(detL)' exp q Lq q' Lq'

,'(q q') M(q q')+ (q q ) K(q+q ) . (3 1)2

Here L, M, and K are real n-dimensional matrices withL and M symmetric, while K is arbitrary. Thus thenumber of independent real parameters in this family ofGaussian kernels is n (2n +1), which is the same as thedimension of Sp(2n, H). The reality of L, M, K leads tothe hermiticity of I:

It follows that W(Q) will exist if and only if the matrixL +M is positive definite:

Property (B)~ L+Mg 0 . (3.7)

(3.8)The quadratic expression in the exponent is positivedefinite; it can be disentangled and written in terms of Q,bringing in a real symmetric positive definite 2n-dimensional matrix G:

This too will hereafter be assumed, i e., we assumewithout exception in the sequel that we are dealing withkernels (3.1) for which both L and L +M are positivedefinite matrices. Then W(Q) is

W(Q) =(2m) "~ [detL /det(L +M))'X exp[ 2q Lq

,'(p Kq) (L +M) '(p Kq)] .

I (q;q')*=I (q';q) ~= I' (3.2) 2q L q+ ,'(p Kq) (L +M) '(p Kq) = Q GQ,

=(detL)' f d"q exp( 2q Lq) & ~ . (3.3)

We seek additional conditions on the parameter matricesL,M, K so that I (q;q') could be the kernel of the densityoperator for some (pure or mixed) quantum state, or thecross-spectral density of some optical field. Given thatI is Hermitian, in the quantum-mechanical case the ad-ditional conditions are that it should be positivesemidefinite, and have unit trace; while in the opticalcontext, it is enough to have positive semidefinitenessand finite trace corresponding to finite total intensity.In analyzing these conditions we will make crucial use ofthe results of Sec. II, so we need to work with the Moyaltransform of I . Therefore, we wish to impose threeconditions on I which we list thus: ( A ) the trace of Imust be finite, (B) the Moyal transform W(Q) of I mustexist, and (C) as an operator on &, I must be positivesemidefinite. It is obvious that each of these propertiesis separately maintained under the Sp(2n, R) action(2.11). We shall see that while (A) and (B) are easy tohandle, (C) is quite subtle.

Let us first impose condition ( 3 ) on I . It is thatTrr= f d "q I (q;q)

CC T (3.9)

A =2L+,'KT(L+M) 'K-,

B =,'(L +M)

C = ,'KT(L+M)L =

,'(A CB 'C ),

M =,'(B ' A +CB 'C ),

K= a-'C'

(3.10a)

(3.10b)

It is also useful to express G in the form

0 2L 00

,'(L +M)-'K (3.11)

From this we see that the positive definiteness of L andL +M is equivalent to the positive definiteness of 6:

The n &(n blocks A, B,C and the original L,M, K can beexpressed in terms of each other:

We see that L must be positive definite, L &O, L+M&O= ~G&0. (3.12)Property ( 2 ) ~~L & 0 . - (3.4) In this statement, K does not appear at all. Similarly,

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3873

from Eq. (3.11) we see that detG is K independent, ScS. (3.22)detG =detL /det(L +M) & 0 . (3.13)

The Moyal transform can now be written as

W(Q)=(2m) "~ [detL/det(L+M)]' exp( Q GQ) .(3.14)

This analysis of the two properties (3), (B), as ap-plied to operators with kernels (3.1), leads us to define aset of operators 4' as consisting of precisely all those Ifor which both (A) and (B) hold:

Unlike 4, however, it will turn out that 4 is notpreserved by Sp(2n, E):

I CZ, U(S)-'I U(S)CZ. (3.23)Indeed, we will later construct a whole family of opera-tors in 4 which exhibit this behavior. All this happensbecause the definition of 4 captures properties ( A) and(B) but not all of (C).

The set of operators we are really after possess allthree properties (A), (B), and (C). Let us call this set

4=[I iL &0, L+M&0] . (3.15)Z =[I ~L &0, I. +M&0, 1 &oj . (3.24)

It is immediate that the Sp(2n, E) action (2.11) preservesIt is clear that 4+ is preserved by the Sp(2n, E) action,and is contained in 4:

res= & r = U(S)- r U(S) e4' . (3.16)The change in the Moyal transform can be read oft' fromEq. (2.23)

I ~ g+ ~U(S) 'I U(S) &4+,S+ c4cS . (3.25)

W'( Q):(2vr ) " [detL '/det(L '+M') ]'X exp( Q G'Q)

= W(SQ),G'=g TG&

.

(3.17)

[&+If'l@)l' (3.18)Going to the limit of ideal basis vectors

~

P) ~~

q),~

g) ~~

q'), this means~

r(q;q')~

'&1(q;q)r(q', q') . (3.19)In the optics context this ensures that the normalizeddegree of coherence is bounded by unity. Using Eq. (3.1)here, we see that if I is positive semidefinite, then

Towards understanding the implications of imposingproperty (C) on I, we first show that a consequence ofthis property is that the matrix M must be positivesemidefinite. For, if I is positive semidefinite, it has aunique positive semidefinite square root I ' . For anytwo vectors P, gC&, the Schwarz inequality applied tor'"y, I '"q

IV. ORBITS IN Sp(2n, RQAND POSITIVITY OF I

We have shown in Sec. II that conjugation of anoperator f', specified by I (q;q'), by U(S) for anySESp(2n, E) is refiected in a "point transformation" onthe arguments Q of its Moyal transform W(Q). Thusone has the "commutative diagram"

r= [1(q;q ) I W(Q)U(S) (4.1)

As already mentioned, the (over)complete set of matrixconditions characterizing elements I in 4+ is L & 0,L+M~0, M&0 and some other conditions in whichthe matrix K also plays a role. We will arrive at theseconditions by an indirect analysis involving the structureof orbits in the Lie algebra Sp(2n, E) under the adjointaction. It is because of the subtlety of these conditionsthat, as an intermediary step, we have defined above theset of operators S "halfway" between 4+ and S, eventhough it is not invariant under the Sp(2n, E) action.

i.e.,exp[ (q q') M(q q')](I, r'= U(S)-'I U(S) W'(Q)=W(SQ) .Property (C) = & M & 0 . (3.20)

Z=[r~

L &M0&O], (3.21)it is true that

However, this is only a necessary and not a sufficientcondition to secure property (C) for I: The positivesemidefiniteness of M does not exhaust all the conse-quences of the positive semidefiniteness of I, but somemore restrictions on L, M, and K remain. Nevertheless,the result (3.20) suffices to show that properties ( A) and(C) together imply property (B). If, based on (3.20), wedefine a set of operators 4 by

We now have to express the positive semidefiniteness ofI, for kernels of the family (3.1), as a set of conditionson the matrices L,M, K or equivalently on the symmetricmatrix G in W(Q). In doing this, the above diagramwill prove essential.

In Sec. II it was also shown that generator matrices Jin the defining representation of Sp(2n, E), and real sym-metric 2n )& 2n matrices G, stand in one-to-onecorrespondence according to Eq. (2.13). Therefore anyI in S,S, or 4+ determines uniquely, via the matrix Gappearing in W(Q), some J in the Lie algebra Sp(2n, E)of Sp(2n, E). Of course not all J arise in this processsince, among other things, G has to be positive definite.

3874 R. SIMON, E. C. G. SUDARSHAN, AND N. MUKUNDA 36

O'= PJ', G= PJ .For a given J, as S varies over all of Sp(2n, IR),J' tracesout the adjoint orbit of J; i.e. , the orbit 6(J) determinedby Jis

6(J)= I J' ~ J'=S 'JS, S ESp(2n, R) ) . (4.3)(With no fear of confusion, this orbit of J may bethought of as the orbit of G as well. ) Therefore, theproperty that 4' and 4'+ are preserved by Sp(2n, H),while 4 is not, can be conveyed in this way: The set ofall r E4 determines some complete set of orbits inSp(2n, H); similarly, the set of all I E 4+ determinessome complete set of orbits, in fact, a proper subset ofthe set determined by 4; finally, the set of J's determinedby the I 's in 4 does not make up a complete set of or-bits at all.

Given that one has to deal with adjoint orbits inSp(2n, IR), it is natural to try to find a convenientrepresentative point on each orbit. It is the fact that thematrices S belong to a symplectic, rather than to an or-thogonal or unitary group, that makes the situationsomewhat dificult to visualize geometrically. As a pre-liminary to tackling the case where G is positive definite[an Sp(2n, R) invariant property], we make some re-marks about the properties of J and G as they run overan orbit.

Let 6(Jo ) be the orbit of Jo E Sp(2n, IR ). AsJ =S 'JOS runs over 6(JO), its spectrum does notchange since it undergoes similarity transforrnations.However, there may be no point on 6(Jo ) at which J isdiagonal. In fact, if JUG(JO) is diagonal,

J =diag(A, A.z, . . . , A,2),then the symmetry of PJ requires

~n +1 ~1~ ~n +2 ~2~ ~ ~2n ~n

In that case G has the form

(4.4)

(4.5)

0

I

I

I

I

0II

I

I

0

0

(4.6)

which is indefinite. This means that J is never diagonalon any orbit relevant to the study of 4 or 4+.

In contrast to J, G undergoes the symmetric transfor-mation (4.2), so its spectrum does vary over 6(Jo ).However, it is easy to see that the "signature of the spec-trum of G" is invariant. Thus, if G and G' are deter-

The action of U(S) on I results in the change (3.17)in G, which is equivalent to J being transformed accord-ing to the adjoint representation of Sp(2n, IR)

G'=S GS ~=~J'=S 'JS,(4.2)

mined by two points J and J' on the same orbit 6(JO),and if G is positive definite, negative definite, indefinite,or has an eigenvalue zero, then G is likewise positivedefinite, negative definite, indefinite, or has an eigenvaluezero. In particular, either G is positive definite at allpoints of 6(Jo ), or G is not positive definite at any pointof 6(JO).

The strategy will now be the following. We firstcharacterize, and find representative points on, all thoseorbits 6 over which G is positive definite: These corre-spond to all I"s in 4 and will be suitably parametrized.The kernel I' I(q; q') determined by any of theserepresentative points is quite simple and it is quite easyto impose the condition that I ' ' be a positivesemidefinite operator on &. This condition will put re-strictions on the parameters labeling the orbits and willtherefore select the subset of orbits corresponding to I 'sin +. We will then develop a set of conditions directlyon a positive definite G to ensure that it "belong to 4+,"without having to go to the representative point on itsorbit. These are the conditions that have been referredto in Sec. III and in which the matrix It plays an essen-tial role.

The basic result we use is Williamson's theorem If Gis a real symmetric positive definite 2n )&2n matrix, thereis a point 6' ' on the orbit 6 of G such that GI ~ is diag-onal with, of course, positive diagonal elements:

K20

G (Oj

0

I

I

K2

(4.7)

)Kn)0. Kn

Thus the parameter space of the orbits comprising 4 isthe convex cone K1)K2) K, )0 in the n-dimensional

Two comments must be made regarding the form ofG' ': First, we have arranged G'+, n+G, ,' forr = 1,2, . . . , n. This is permissible because within a pair(q,p) we can scale qby a factor yand simultaneouslyscale pby the reciprocal factor y', and such scalingscarried out independently on each canonical pair are ele-ments of Sp(2n, R). Second, the interchange of onecanonical pair (qp) with another (q, ,p, ) is also anSp(2n, R) transformation so that we can arrange the ir'sin nonincreasing order. The parameters Ia.j are not ofcourse the eigenvalues of a general G on the orbit ofG '. Nevertheless, they do invariantly characterize theentire orbit: While J can never become diagonal on suchan orbit, at all points its spectrum consists of +iK1, +iK2,. . . , +i K. Therefore, the orbit determined by G ' ' ofEq. (4.7) can be written 6, , and we can say that,in an obvious sense,

(4.8)

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3875

~ space.We next ask which subset of the orbits in Eq. (4.8)

occur in 4'+ . At the representative point G ' on6. . . Eq. (4.7), the submatrices A' ', B~ ', andC' ' are

We already know that a necessary condition forI ' ' & 0 is that M' ' must be positive semidefinite. Thisimposes x(1 for each r, but since ~] is the largest ofthe K's it suffices to say

KiM'"&0

~, (1. (4.11)=B(0) (0)

Kn

It now happens that at the point G ' ' this condition isalso sufficient to ensure I ' '&0! For, using the matrices(4.10) we have, apart from a numerical factor

C(a] (4.9) I' '(q;q')=' exp g [~ (q+q' )Therefore, the matrices L' ', M' ', and K ' needed forI' '(q;q') are, from Eq. (3.10),

L = 2 diag(Ki, Kp, . . . , K~ )M =

, diag(ai K), Kp Kp, . . . , KK),(0) i . ] 1 1+ ,'(~'~)(qq')']

(4.12)K"j=o. (4.10) If

~

ir'j) H&, then

& O ~ I'"~

&) = f f d "q d "q'V*(q) F ' (q q')C(q')= f f d"q d"q'P*(q)exp ,' g (~ ' ~)(q,q') .P(q'),

r=] (4.13)n

P(q)= exp ,' g ~qg(q) .r=l

$+=(6, , 1)x., &x2) )~&0( . (4.14)We can state this result in the form of a theorem.

The necessary and sufficient conditiors for the Hermi-tian Gaussian kernel (3.1) with parameter matricesLM,K to represent a density operator in quantummechanics or a cross-spectral density in optics, orequivalently for the real Gaussian phase-space distribu-tion (3.8) or (3.14) to be a genuine Wigner distribution orWolf function, are

G &0, xi(1, (4.15)where the subrnatrices A, B,C in G are related to L,M, Kby Eq. (3.10) and where x, is the largest diagonal ele-ment in the normal form G' ' of G.

Obviously, P(q) is also square integrable since each~&0, and the translation-invariant kernel multiplying(t(q)*(t(q') has a positive semidefinite Fourier transformsince each (K'K))0. Consequently, &1(

~

I '"~

it ) isnon-negative for any P, i.e. , I ' ' is a positive semide-finite operator on A. Once one has shown that I ' ' cor-responding to G' ' is in 4+, the commutative diagram(4.1) ensures that I determined by any other G on theorbit of G' ' is also in A'+. Thus, out of all the orbits ap-pearing in (4.8) and making up 4', only those with v, (1are present in 4+ and correspond to positivesemidefinite I"s:

It is a consequence of conditions (4.15) that detG isbounded above by unity. For, any SHSp(2n, R)is uni-modular, so that detG is an Sp(2n, R) invariant. On theother hand, at the representative point G' we have

detG' '= Q K' (4.16)

Property (C): &v, (1 . (4.17)Naturally this is an Sp(2n, IR) invariant statement, sincethe normal form G' ' of a positive definite G as providedby Williamson s theorem, is unique. However, it is clearthat the statement (4.17) is not yet expressed in terms ofL,M, K or A, B,C in a directly testable way. It is there-fore of interest to ask whether, given G & 0, we can ex-press the content of Eq. (4.17) in terms of G directly,

It follows that for a11 fields contained in 4+, detG (1.In fact, detG=1 if and only if K=1, r =1,2, . . . , n. Inthis case, and only in this case, I (q;q') in Eq. (3.1) be-comes (essentially) a projection operator representing apure quantum state, or in the case of optics a fullycoherent field.

It can be appreciated at this stage that, while proper-ties (A) and (B) of Sec. III imposed on I are fullyequivalent to the positive definiteness of G [see Eqs. (3.4),(3.7), and (3.12)], a complete and faithful rendering ofproperty (C) is [given properties ( A) and (B)]

3876 R. SEMON, E. C. G. SUDARSHAN, AND N. MUKUNDA

without having to find its normal form. This can indeedby done, in terms of the polynomial Sp(2n, E) invariantsformed out of G, as we now show.

Since J =PG undergoes similarity transformations un-der Sp(2n, IR), we see that the traces of powers of PG areSp(2n, IR ) invariants. On an orbit with G & 0,Williamson's Theorem allows us to calculate these in-variants by going to the normal form G' ':

0, m oddTr( G) 2X( 1) g x', m even .

(4.18)

That is, the nonzero invariant traces are equal to thesums of various powers of a.. We define

i nSi Tr(PG) '= g a, l = 1,2, . . . , n2 r=1and in terms of S~,

C, = S, ,C2= 2(S2+C1S) ),C3 ,' (Si +CiS2 +C2S) )

(4.19)

1C=(S+CiS i+C2S2+ . +C iS, ) .n

(4.20)We now form the nth degree polynomial equation withC's as coefficients:

P(y):y" +C,y" '+C y" + +C,y+C=O .(4.21)

By Bocher's theorem, ' the roots of this equation arev,r =1,2, . . . , n. Since G is positive definite, we knowin advance again from Williamson's theorem that whenthe quantities SI and then C~ are formed from G and theequation (4.21) is set up, all the roots will be positivedefinite. Thus to impose the condition ~l &1 which isthe same as ~l & 1, we have to state the conditions underwhich all the roots of Eq. (4.21) are less than or equal tounity. Suppose

n (n 1)+(n 1)(n 2)Ci + +6C3+2C2& 0,n!+(n 1)!C,&0 .

Thus we have an explicitly Sp(2n, IR) invariant algo-rithm to impose the crucial condition ~l & 1 on a given Gwithout having to put it into its normal form: Startingwith a positive definite G, we compute the traces of thefirst n even powers of PG. We use these traces to formSi and from them the Ci via Eqs. (4.19) and (4.20). Wethen form the n linear combinations of the C~ appearingin (4.24). If all of these are non-negative, then ~, (1;otherwise, al & 1. At least the last of these inequalities isquite transparent: It merely says that Cl, which is thesum &l+&p+ +K, must not exceed n.

The above results derived for n canonical pairs of vari-ables are relevant as they stand in quantum mechanics.However, in the context of paraxial propagation prob-lems in optics, the cases of direct relevance are n =1 andn =2, corresponding, respectively, to isotropic and an-isotropic Gaussian Schell-model beams. We therefore il-lustrate the general results by specializing to the casesn =1,2.

Case of n = 1. The relevant symplectic group isSp(2, E) which is isomorphic to SL(2, IR) and is a twofoldcovering group of SO(2, 1).' The normal form of G is,given its positive definiteness,

0(4.25)

There is only one algebraic invariant, namely,

Si C, = ,' Tr(PG) =x (4.26)Operators I ES determine a one-parameter family of or-bits (ri ir & 0, in the three-dimensional Lie algebraSp(2, E); each such I has a Gaussian kernel (3.1) withn =1, a finite trace, and a finite Moyal transform. Thematrices J making up the orbit 0, are evidently

explicitly in terms of the C's these inequalities are1+C, +C, + . +C&0,n +(n 1)C&+(n 2)Cq+ . +2Cz+C i &0,

(4.24)

p ( 1 ) dP (y )d" 'P(y)

n 1)0. 6=Ii i' 'o pS ~ S E Sp(2, E) I . (4.27)

y=1 y=l

(4.22)

P(y) =(y i~i)(y x2) (y x) . (4.23)Thus we have proved that the inequalities (4.22), n innumber, are the necessary and sufficient conditions toensure that ~i&1, given the positive definiteness of Gand hence the validity of Williamson's theorem. Written

Then P (y) & 0 for all y & 1 [as is easily seen by making aTaylor expansion of P(y) about y =1], implying thatthere are no roots of (4.21) beyond y =1. Conversely, if(4.21) has no roots beyond y =1, the inequalities (4.22)follow from writing P (y) in the factored form

To determine which subset of the orbits 6, for a & 0 cor-respond to 4+, we have to impose the inequality in thefirst line of Eqs. (4.24). Thus we see that only 8 for~ & 1 arises in correspondence with the positivesemidefinite I s comprising 4+. In this simple case withn =1, since detG =a, the condition detG &1 alreadyensures a & 1 and so the positive semidefiniteness of I .

Case of n =2. In this case the symplectic group isSp(4, E) which is a twofold covering of SO(3,2). Basedon this fact, we have elsewhere made a complete listingof all orbits that arise in the ten-dimensional Lie algebraSp(4, E) as corresponding to real symmetric positivedefinite 4&4 matrices G. ' Those same results are im-mediately recovered from the present analysis. Two pa-

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3877

K)) Kz) 0 . (4.28)0 Kz

This G' ' determines a J' 'HSp(4, IR) whose orbit is6, , Our analysis based on SO(3,2) has shown that thecase ~i ~2 is to be distinguished from the case ~i )~zbecause the natures of the J's in the two cases are ratherdifferent. Thus we prefer to exhibit 4 as

4= I6, ,~

K, &K2&OI

(4.29)

rameters~&, ~2 are needed to label the relevant orbits,

and the normal form of a positive definite 6 isK) 0

Kz

The increase in complexity as compared to the casen =1 is seen in the following fact: While detG=~&~z isbounded above by unity as long as one observes the limit~& (1, the condition detG & 1 is no longer sufficient toensure that I be positive semidefinite. Therefore, theconditions for 6 to "belong to 4+" have to be written asthe pair of inequalities.

Tr(PG) (4,2 Tr(PG ) [Tr(PG) ] 4 Tr(PG ) (8, (4.33)

and these imply, but are not implied by, detG (1. EveryI for a system with n =2, which is Hermitian, has finitetrace, a finite Moyal transform, and is positivesemidefinite is given by a real positive definite 4&4 ma-trix G obeying the Sp(4, R)-invariant inequalities (4.33).

6:C)2K, Cz ~42 2

2 26 .'C] = K] K2~ C2 =K)K2 (4.30)

while the relevant inequalities areCi (2, Ci Cz (1 .

Therefore, the orbits correspondingsemidefinite I 's in 4+ are

(4.31)to positive

t+ I6..~

1&K&0}u [6.~

1&K, &KoI . (4.32)

and call these two families of orbits as types I and II, re-spectively. In order to find what part of S makes up4+, we must impose two inequalities corresponding tothe first two lines of (4.24). In types I and II, respective-ly, we have

V. ROI.K OF THK PHASE MATRIX E

It has been emphasized in earlier sections that whilethe matrix K determining the phase of the Gaussian ker-nel (3.1) plays no role in implementing properties ( A) on(8) of Sec. III on I, it does play an essential role whereproperty (C) is concerned. We now study this aspect inmore detail.

It is first important to recognize that the symmetricpart of K is again irrelevant as far as property (C), i.e.,the positive semidefiniteness of I, is concerned. At thelevel of the kernel (3.1) we see that when K is split intoits symmetric and antisymmetric parts, K, and K, , theformer contributes a phase factor separable in q and q':

K =K, +K, ,

exp (q q') K(q+q') = exp q K, q exp(iq K, q') exp q' K,. q'

Therefore, the K, -dependent part can be removed by a unitary transformation on I, resulting in a I for which K ispurely antisymmetric. In fact, this unitary transformation is U(S) for a certain SHSp(2n, IR) determined by K, . Fol-lowing the notation of Eq. (3.16) one has

I = U(S)I"U(S) ', U($)= exp q K, q

0S= (5.2)

I"(q;q')=(detL)'~ exp[ q Lq q' Lq' ,'(q q') M(q q')+iq K, q'] .This identification of SHSp(2n, lR) is seen to be correct by working at the level of the matrix G and expanding Eq.(3.11) to the form

0 I 0 21.o

0 I 0K, (5.3)

3878 R. SIMON, E. C. G. SUDARSHAN, AND N. MUKUNDA 36

The factors involving K, obey the condition (2.7) and sobelong to the defining representation of Sp(2n, lR), whilethe K, -dependent factors do not. Comparing Eq. (5.3)with Eqs. (3.16) and (3.17) leads to the S given in Eq.(5.2). We see now that I is positive semidefinite if I is,and conversely. Therefore, the set of n inequalities(4.24) which are equivalent to the condition a. , & 1, andwhich in turn express the condition that I be positivesemidefinite, can involve only K, and not K, . In passingwe note that U(S) in Eq. (5.2) corresponds to a lens ac-tion in the context of optics (n = 1,2), and to an impul-sive harmonic potential in the context of quantum evolu-tion.

Of the full set of n inequalities (4.24) that must be im-posed on G to secure property (C) for I, the last one,

n+C) )0,is relevant for all n ) 1; the previous one,

,' n (n 1)+(n 1)Ci +Cq )0,

(5.4)

(5.5)

=Tr(AB C )=TrL (L +M)

+ ,' TrK r(L +M) '(K Kr)(L +M) (5.6)Since both L and L+M are assumed to be positivedefinite, there is a unique symmetric real positive definitematrix (L +M) '~, and the second trace term can beseen to actually involve only K K; we can then putT.Ci into the compact form

Ci TrL(L+M) '+ ,' TrX %,X =(L +M) '"K, (L +M) (5.7)

Therefore, the weakest condition on K, is this:TrL (L +M) '+ ,' TrNrX & n (5.8)

For n =1, this is empty since K, vanishes but for n )2we have a definite restriction on K, which prevents itfrom being "too large. " As an example, if M =0, anynonzero K, immediately violates the condition (5.8).Thus for any K, howsoever small but nonzero, the Her-mitian operator I with the kernel

I (q;q')= exp( q Lq q' Lq'+iq K, q') (5.9)is certainly not positive semidefinite. Returning to thegeneral case, the remaining inequalities in (4.34) can beexpected to put further restrictions on K, beyond (5.8).

We take the opportunity now to point out and rectifyan incompleteness in the analysis of anisotropic Gauss-ian Schell-model fields presented in Ref. (15). In thatwork one is concerned with n =2, and the conditions

is relevant for all n )2, and so on. Each of the condi-tions (4.24) does involve K, , but we shall here bring outthe weakest restriction on K, contained in (5.4), the con-dition relevant for all n values. From Eqs. (4.19) and(4.20) defining Ci and Eqs. (3.9) and (3.10), we haveC, = ,' Tr(PG)

imposed on the matrix G were the following:

G)0,detG ~1,2M =8 ' 3 +CB 'C~&0 .

(5.10)

It was then stated that any such G defines an acceptableoptical field of the AGSM type. The incompletenessconsists in not recognizing that M )0 is only a necessarycondition for I to be positive semidefinite and not asufFicient one. The subtle role of K, the matrix deter-mining the phase of I (q;q'), was not realized in Ref.(15). Apart from this, however, the fact that allowedAGSM fields can be classified into two types, as shownin Eq. (4.32), with 0&i'& 1 in type I and 0&ir~ &ir, & 1 intype II, is exactly as was found in Ref. 15.

Finally, we turn to the promised demonstration of thefact that the set S of operators I defined by Eq. (3.21) isnot Sp(2n, R) invariant. For this we consider a positivedefinite matrix G with L &0, M =0, and K =K,&0.The corresponding operator I definitely belongs to 4but is not positive semidefinite since the condition (5.8) isviolated. Now by Williamson's theorem we can trans-form G to its normal form G' ' and it will then happenthat w 1. This implies by Eq. (4. 10) that the matrixM' ' corresponding to G ' has at least one negative ei-genvalue, that is, M' ' is not positive semidefinite.Therefore, I ' ' determined by G ' does not belong to S.At the same time I and I ' are unitarily related byU(S) for the S needed to take G to its normal form G' '.This establishes that 0' is not Sp(2n, IR) invariant.

VI. CONCLUDING REMARKS

Property (C) ~=~a.&1, r =1,2, . . . , n (6.1)In this form we do have n inequalities, and their expres-sion directly in terms of the Sp(2n, lR) invariants of G isin (4.24).

The analysis and results have both been presented insuch a way as to be readily applicable to both quantum

We have presented in this paper a systematic analysisof configuration-space Gaussian density kernels and theirassociated Moyal transforms which are also Gaussian.The former are parametrized by the matrices L,M, Kand the latter by the matrix G, and we have allowed the"number of degrees of freedom" 2n to be quite general.Given a Gaussian phase-space distribution characterizedby the matrix G, the necessary and sufficient conditionsfor it to be a Wigner distribution have been shown to bethat (i) the matrix G must be positive definite, and (ii)the traces of (PG) ' for 1 =1,2, . . . , n must obey the ninequalities (4.24). These conditions are manifestlySp(2n, R) invariant. In this connection we may pointout that it is deceptive to view the condition ~& (1, ap-pearing in Eqs. (4.15) and (4.17) as an expression ofproperty (C), as constituting "just one condition, " sinceit has to apply to the largest of the ~'s that appear in thenormal form G' ' of G. A more complete statement in-volving all the K's and not relying on the verbal instruc-tion to pick out the largest of them would be

36 GAUSSIAN-WIGNER DISTRIBUTIONS IN QUANTUM. . . 3879

2

W(q;p) =' exp tanh(co/2k~ T) mcoq +me@

This means that G in this case is the 2&(2 matrix(6.2)

mechanics and optics. It is useful to have a physical in-terpretation of the Sp(2n, E) invariant tc s appearing inthe normal form (4.7). In the optical GSM case, n =1,there is only one tc which coincides with (detG)'~, andit has been shown' that it is the well-known degree ofglobal coherence defined as the ratio of the transversecoherence length to the intensity width. In the AGSMcase, n =2, there are two tc's (which coincide for thetype-I fields), and a similar physical interpretation hasbeen established. In the context of quantum mechanicswe have already noted in the Introduction that theGaussian Wigner distributions correspond to thermalstates of harmonic oscillator systems, and their trans-forms under Sp(2n, E). For a one-dimensional oscillator(n =1) of frequency co the Wigner distribution of thethermal state at temperature T is given by the familiarGaussian expression

mmG = tanh( co /2kB T )

0(6.3)

so that

tc=(detG)' =tanh(co/2k~ T) . (6.4)Thus the invariant ~ is essentially the ratio of the nat-

ural frequency of the oscillator to the temperature, andthis is a symplectic invariant. For n &1 we note fromEq. (4.12) that in its normal form the configuration-spacekernel is separable in the n degrees of freedom, so thevarious ~ s give the n-invariant co/T ratios.

As a final remark we wish to point out that theanalysis of this paper can be profitably used to study theproblem of squeezed states in a multimode system.This is because the G matrix is the inverse of the vari-ance matrix and the squeezing operator is an Sp(2n, E)transformation. Thus the transformation law for G un-der Sp(2n, E) as given in this paper contains as a specialcase the transformation law of the variances undersqueezing. We plan to return to this problem else-where.

E. P. Wigner, Phys. Rev. 40, 749 (1932).~H. Weyl, The Theory of Groups and Quantum Mechanics

(Dover, New York, 1931), p. 274.J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949). For

recent reviews on Wigner distribution methods, see M. Hil-lery, R. F. O' Connell, M. O. Scully, and E. P. Wigner, Phys.Rep. 106, 121 (1984); N. L. Balazs and B. K. Jennings, ibid.i04, 347 (1984).

4E. C. G. Sudarshan, Phys. Lett. 73A, 269 (1979); Physica 96A,315 (1979); R. Simon, Pramana 20, 105 (1982).

~For the connection between radiometry and physical optics,see A. Walther, J. Opt. Soc. Am. 57, 639 (1967); E. W.Marchand and E. Wolf, ibid. 64, 1219 (1974); A. T. Friberg,ibid. 69, 192 (1979); E. Wolf ibid. 68, 6 (1978); H. Pedersen,Opt. Acta, 29, 877 (1982); J. T. Foley and E. Wolf, Opt.Cornrnun. 55, 236 (1985); G. S. Agarwal, J. T. Foley, and E.Wolf, ibid. 62, 67 (1987).

The relationship between radiative transfer theory and elec-trodynamics was studied in E. Wolf, Phys. Rev. D 13, 869(1976); M. S, Zubairy and E. Wolf, Opt. Commun. 20, 321(1977), E. C. G. Sudarshan, Phys. Rev. A 23, 2802 (1981); M.S. Zubairy, Opt. Commun. 37, 315 (1981); R. L. Fante, J.Opt. Soc. Am. 71, 460 (1981).

~R. L. Hudson, Rep. Math. Phys. 6, 249 (1974).R. Jagannathan, R. Simon, E. C. G. Sudarshan and R.

Vasudevan, Phys. Lett. A (to be published) ~H. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940); N. D.

Cartwright, Physica 83A, 210 (1976); R. F. O' Connell and E.P. Wigner, Phys. Lett. 83A, 145 (1981); A. K. Rajagopal,Phys. Rev. A 27, 558 (1983); C. L. Mehta and E. C. G. Su-darshan, Phys. Rev. B138, 274 (1965); Y. Kano, J. Math.Phys. 6, 1913 (1965).

oJ. T. Foley and M. S. Zubairy, Opt. Commun. 26, 297 (1978);E. Wolf and E. Collett, ibid. 25, 293 (1978); W. H. Carterand M. Bertolotti, J. Opt. Soc. Am. 68, 329 (1978); B. E. A.Saleh, Opt. Commun. 30, 135 (1979); F. Gori ibid. 34, 301

(1980); A. T. Friberg and R. J. Sudol, ibid. , 41, 383 (1982);A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 (1982); F.Gori, Opt. Commun. 46, 149 (1983); F. Gori and R. Grella,ibid. 49, 173 (1984).A restricted class of AGSM fields has been analyzed in Y. Liand E. Wolf, Opt. Lett. 7, 256 (1982); F. Gori, Opt. Com-mun. 46, 149 (1983); F. Gori and G. Gauttari, ibid. 48, 7(1983); R. Simon, J. Opt. 14, 92 (1985); P. De Santis, F.Gori, G. Gauttari, and C. Palma, Opt. Acta. 33, 315 (1986).Coherent anisotropic Gaussian beams have been studied inR. Simon, Opt. Commun. 42, 293 (1983); 55, 381 (1985).R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev.A 29, 3273 (1984).

'3W. Brouwer, E. L. O' Neill, and A. Walther, Appl. Opt. 2,1239 (1963); S. A. Collins, Jr., J. Opt. Soc. Arn. 60, 1168(1970); A. E. Siegman, IEEE J. Quant. Electron. QE-12 35(1976); M. Nazarathy and J, Shamir, J. Opt. Soc. Am. 72,356 (1982).H. Kogelnik, Bell Syst. Tech. Jour. 44, 455 (1965); H. Kogel-nik and T. Li, Proc. IEEE 54, 1312 (1966).R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev.A 31, 2419 (1985).

' E. C. G. Sudarshan, N. Mukunda, and R. Simon, Opt. Acta32, 855 (1985); H. Bacry and M. Cadilhac, Phys. Rev. A 23,2533 (1981).

' See R. G. Littlejohn, Phys. Rep. 138, 193 (1986), and in par-ticular the discussion following Eq. (8.15) therein.J. Williamson, Amer. J. Math. 58, 141 (1936); V. I. Arnold,Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978), Appendix 6.N. Mukunda, Pramana 11, 1 (1978).A complete classification of the orbits in some low-dimensional Lie algebras is given by N. Mukunda, R. Simon,and E. C. G. Sudarshan, Indian J. Pure Appl. Math (to bepublished).

2~L. A. Pipes and L. R. Harvill, Applied Mathematics for Fn

3880 R. SIMON, E. C. G. SUDARSHAN, AND N. MUKUNDA 36

gineers and Physicists, 3rd ed. (McGraw-Hill, New York,1970), p. 107.

E. Collett and E. Wolf, Opt. Commun. 32, 27 (1980).See, for example, M. Hillery, R. F. O' Connell, M. O. Scully,and E. P. Wigner, Phys. Rep. 106, 121 (1984), Eq. (2.115).

~4D. Stoler, Phys. Rev. D 1, 3217 (1970); H. P. Yuen, Phys.Rev. A 13, 2226 (1976); J. N. Hollenhorst, Phys. Rev. D 19,

1669 (1979); D. F. Walls, Nature 306, 141 (1983); R. E. Slush-er, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley,Phys. Rev. Lett. 55, 2409 (1985).

~5An analysis of the squeezed states along these lines for n =1has been carried out in R. Simon, in Symmetries in ScienceII, edited by B. Gruber (Plenum, New York, 1987).