THE UNIVERSITY OF MIC H I GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report RELATIONSHIPS AMONG GENERALIZED PHASE-SPACE DISTRIBUTIONS P. F. Zweifel G. C. Srnmerfield ORA Projects 01046 and 08964 supported byNATIONAL SCIENCE FOUNDATION GRANTS NOS. GK-1713 AND GK- 1709 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION May 1968 ANN ARBOR

RELATIONSHIPS AMONG GENERALIZED PHASE-SPACE DISTRIBUTIONS* by G. C. Summerfield and P. F. Zweifel Department of Nuclear Engineering The University of Michigan Ann Arbor, Michigan Abstract The generalized phase-space distributions, including the Wigner distribution, are presented in terms of expected values of generating operators. A generalization of the Weyl correspondence is obtained to provide expressions for generalized Wigner equivalents. Finally, rather simple relationships are obtained connecting the generalized phase-space distributions to the Wigner distribution; and similar relationships are obtained for the generalized Wigner equivalents. In particular, it appears that among the class considered, there is no reason to use any distribution other than the Wigner for performing any calculations. 1

I. INTRODUCTION In 1932 Wigner introduced a method of performing quantum-mechanical ensemble averages in terms of phase-space integrations over c-number variables. Since that time a number of extensions, modifications, discussions, derivations, applications, etc., have appeared in the literature. We shall 2 refer the reader to a review in which further references can be found. Actually, there exist an infinite number of quasi-distribution functions which can be used for the same purpose as the Wigner distribution function. In a recent paper5 Cohen described one method for generating such distributions, and showed how the Wigner function, the so-called "symmetric" function and the Born-Jordan function could be generated. He also obtained equations of motion (quantum Liouville equations) for these distribution functions. In the present paper we present a particularly simple and elegant manner for generating an infinite class of distribution functions which include, as special cases, the Wigner, symmetric, and Born-Jordan functions. Also we show that all of these various distributions can be obtained from the Wigner distribution by a rather trivial transformation. For the purposes of our later discussion, it is convenient for us to point out several general properties that all of these distributions have in common. We represent the 6N dimensional phase space by the 5N dimensional momentum and position vectorsr and p. A generalized phase-space distribution is a function of the variables r and p and time, f(r,p,t). These functions satisfy the following conditions. 2

Classical Limit The function fc(r,p,t) = lim f(r,p,t) (1) "0O must be the "correct" classical phase-space distribution. That is fc(rpt) must satisfy the Liouville equation. Marginal Distributions The integral of f over one of the variables r or p must give the correct distribution in the other variable. f dr f(r,p,t) = < 0(P-p) > (2) f dp f(r,p,t) = < 5(R-r) > (5) where R and P are the position and momentum operators. Generalized Wigner Equivalents For any given function A(R,P) of the position and momentum operators, we must be able to determine a generalized Wigner equivalent a(r,p) such that < F(R,P) > = f drdp f(r,p,t)a(r,p) (4) We might point out here that the distributions introduced by Cohen3 do not in general provide for a generalized Wigner equivalent. In particular for Cohen's distribution (6.2), an operator of the form F(@.R+T.P) does not have a generalized Wigner equivalent.

The most convenient way of finding generalized Wigner equivalents is by first finding the generalized Weyl correspondence. That is we find the operator Ag(G,T,R,P) for which the Wigner equivalent is ag(G,T,r,p) = ei( rT (5) Then if the operators Ag are complete, we can expand any operator as A(R,P) = f ddT a(,T)Ag(~,T,R,P).(6) (We shall consider the completeness of the Ag's when we specify the details of the distribution.) Clearly we can determine the Wigner equivalent of A(R,P) by knowing the Wigner equivalent of the right-hand side of (6), that is using (5). a(r,p) = I dGdT ca(,T)e (r+T (7) It is easily shown that the expected values of the following generating operator D(R,P,r,p) = 1; dT'dG' ei() Ag(7',R,P) (8) (( 2-,rc) 6 —— )N - (2Tr) will give a distribution for which (5) holds. fg(r,p,t) = < D(R,P,r,p) > (9) We will show that this distribution also satisfies the other conditions, we listed earlier. Our approach here is related to that followed by Cohen.3 4

II. THE DISTRIBUTIONS We can specify a distribution by writing the operators Ag(@,TR,P). We take generally Ag( G,T,R,P) = g(~.T)ei('R+TP) where g(x) has a series expansion about zero of the form 2n X g (2n)(0) g(x) = 1 + n () 2n) n=l (2n) g (0 (10) (11) Clearly we must take g to be an even function of lTQ'G to insure that D is hermit ian. The completeness of the operators ei(R P) is shown in Ref. 2. The Wigner distribution is obtained by taking g(x) = 1 Then fw(r,p,t) - -- c' ei(g'-.r+T') <ei('. R+T' P) (2Tt) 4 This form was obtained by Moyal. If we recall that 1 [B,A] eAe = e A+Be (12) (13) for [A,[B,A]] = [B,[B,A]] = 0, 5

and [0.R,7TP] = i@K'T (14) we can write (12) as _ 1 =dt'dci(G r+T.p)< e fw(r,p,t) = - 6 dT' <ge-'< e (2i). P., P iTs.'p iT P 2 ig'.R 2 e e i P iT. 2 2.1, Tf' dP 3N / ds' e < ~e ( 2 T * P i. (R 2 F(R-a (15) or alternatively we can write fw(r,p,t) R R iQ'. - ie' - 1 f d,'d' e-i(G.r+-'.p) < e 2 T'.P'2 I -- - di-'dG' e'<e e e ( 2 t)-)R R i@?'R iQ'l 1 -iQ'.r 2 2 f dg<e e (P-p)e >. (2q)3N (16) Using (15) and (16), it is a straightforward matter to derive Eqs. (5a) and (5b) in Ref. 2. It is clear that the generating operator for the generalized distribution is related to the generating operator for the Wigner distribution by commutators of R and P, since g(.'-) = g(-i[.R,T.P]). (17) As an example let us consider the symmetric distribution introduced by Margenau and Hill. As discussed by Cohen the appropriate g(x) for this case is g(x) = cos(x/2). 6

In this case the distribution is 1s~r~p~t) -i(Q'.r+r'.p) i('.R+T P) f,(r,p,t) 6N f dQ'dT' cos (iT'. 0'2) e < (2t) (18) When we note that cos(T1'T'/2) = (e2 [@R ] + e_ [' P, (19) 2 and use (13), we can write (18) as ig'-~R iT \P iT1*P iTG^R> 1r t = -— ^;dPT) ^ - -! e e +e e fs(r,p,t) = f d'dT' e rei ei P + e (2Tt) = < 5(R-r)S(P-p) + 5(P-p)6(R-r) > (20) 2 The remaining distributions commonly found in the literature can also be generated by an appropriate choice of g(x). III. CONNECTIONS AMONG THE DISTRIBUTIONS First let us show that the three properties of generalized phase-space distributions listed in Section I hold for the distributions generated by (8), (9), and (10). Of course our choice was made to provide a simple means of determining the generalized Wigner equivalents. Therefore we need not discuss this further. 2 i(G.R+T. P) To find the classical limit we note that <e > has a series expansion in n and 7

lim <e.fi( R+T P)> = Jdrtdp' fc(r',pte (gr (21);n+0 Also we note from (11) that lim g(0g) T) =1 (22) h=0 Then lim fg(r,p,t) = 6N f dT'd)dr'dp' fc(r',p',t) e( ( ) (P );-T=O (2T) fc(r,p,t). (23) Now let us consider the marginal distributions. 1' j =B 1-i(Q'.r+T'.p) g(h') ei(Q'.R+T'.P) J dr fg(r,p,t) 6N f dT'd'dr e g(hQ.T)< e (2=) ~~1,~-.iT''p iT'-P N dT'dG' a(@')e < e > where we have taken G' = 0 and noted that g(o) = 1. The remaining integrations give Eq. (2) for fg. It is obviously just as easy to show that Eq. (3) holds for fg. To establish the equivalence of the various distributions we explicitly insert (8) and (10) in (9) f (r,p,t) -'N dT'dQ' g(Q' -t)e-i('r+ T'P)< e i( R+TP)> (24) g (2i)(2 ) Using the property g(x) = g(-x) we note that 8

- i('.r+T'.p) -i(G'.r+t'.p) g(O'')e = g(Vr -Vp)e. (25) Recalling Eq. (12), we see that fg(r,p,t) = g(-7rVp)fw(r,p,t). (26) 6 A form somewhat similar to this was used byvon Roos to obtain a distribution function for a molecular gas. Now let us consider the generalized Wigner equivalent ag(r,p) = f dGQdr g(,T)ei( r+Tp) (27) where ag is obtained from A(R,P) = f dOdT ag(Q,T)g( T)e ( GR+TP) Since g = 1 for the Wigner distribution, we must have w(0~,T) = ag(oT)g(~i.@ ). (28) Applying (28) and (2%) in (27), we have aw(r,p) = g(-Vr.Vp)ag(r,p) (29) IV. DISCUSSION Clearly the generalized phase-space distributions and the generalized Wigner equivalents are different for different choices of g(x). However the important conclusions regarding these distributions must be concerned with their connections with experiments in terms of Eq. (4). Consider then 9

< F(R,P) > = J drdp fg(r,p,t)ag(r,p,t) (30) Using (26) we have < F(R,P) > = f drdp ag(r,p,t)g( r'Vp)fw(r,p,t) Integrating by parts gives < F(R,P) > = f drdp fw(r,p,t)g(Vr.Vp)ag(r,p,t), and using (29) < F(R,P) > = f drdp fw(r,p,t)aw(r,p,t). (31) It is not surprising that both (30) and (31) hold, since we constructed the generalized phase-space distributions to satisfy just these equations. However, the rather trivial connections among the various distributions does not seem to have been pointed out in the literature; and leads one to wonder why more than the Wigner distribution need be considered for any calculations. Using Eqs. (26) and (29), we can immediately relate the results already obtained for the Wigner distribution (as for example in Ref. 2) to the corresponding results for a generalized phase-space distribution. 10

Footnote and References *Work supported by the National Science Foundation. 1. E. Wigner, Phys. Rev. 40, 749 (1932). 2. K. Imre, et al., J. Math. Phys. 8, 1097 (1967). 3. L. Cohen, J. Math. Phys. 7, 781 (1966). 4. J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949). 5. H. Margenau and R. N. Hill, Progr. Theoret. Phys. (Kyoto) 26, 722 (1961). 6. 0. von Roos, J. Chem. Phys. 31, 1415 (1959). \.E 11