ELSEVIER

16 May 1994

PHYSICS LETTERS A

Physics Letters A 188 (1994) 107-109

Group-theoretical derivation of the Wigner distribution function

Hui Li Department of Physics, Tsinghua University, Beijing 100084, China

Received 9 December 1993; revised manuscripts received 8 March 1994; accepted for publication 14 March 1994 Communicated by J.P. Vigier

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A derivation of the Wigner distribution function, which is based on the simple postulate that the correspondence between the distribution function and the wave function be invariant under linear canonical transformations which form a Lie group, is given.

The quantum-mechanical distribution functions P( q, p), where q and p are position and momentum coordinates, have a wide application in a variety of problems in physics; and various definitions for Pap- pear in the literature [ 11. The question as to what conditions are necessary for providing a unique def- inition for P has been investigated by some authors [2,3]. However, the sets of conditions proposed in Refs. [ 2,3 ] seem to have a lack of unity and mathe- matical simplicity. On the other hand, it has been noted that the Wigner distribution function has good transformation properties under linear canonical transformations [ 41, not just being Galilei invariant. This leads to the idea that we may use this property to characterize the Wigner distribution function.

First let us recall that the group of linear canonical transformations consists of translations, rotations and squeezes in phase space [ 41. The infinitesimal gen- erators are

IV,=-i$, A$=-i$ L=-ji(q$ -p-$,

B,=ii(q$ -p-/-), &=ti(q-$ +P$). (1)

Using the commutation relations (setting fi = 1)

[fi,Q]=-i, [-Q,fi]=-i, [@2,fj]=-ifi,

[-jcj2,fi]=-ifj, [i(@+&j),Q]=-id,

[t(@++WhBl=ifi, (2)

we see that the quantum realization of the intinitesi- ma1 generators is

rs, =p, lV2 z-4, B,-L=$@,

B2 + 2= - QP, B”, = - 4 (gfi+DQ) . (3)

Note that [fir, f12] =i, while [Nr, N2] =O, so that the quantum Lie algebra is just a central extension of the classical Lie algebra, and the quantum realization is a ray representation [ 5 1.

Next, we will give the derivation. Retaining the postulate P( q, p) = (v/l &( q, p) ] v) , we add a new one, the correspondence between P and the wave function ) w) is invariant with respect to linear ca- nonical transformations. To express it more explic- itly, we have

e*P(q, p> = twl e-*fi(q, PIeHI w> , (4)

where N may be any infinitesimal generator, and the infinitesimal forms of the above expression are as follows

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H. LiIPhysicsLettersA l&3(1994) 107-109

(5)

(6)

(7)

(8)

[tW+&V,~(s,~)l= iq$-b$)@(q,p). ( (9)

Let us suppose that

fi(W)= J$-( 4,P,~l,~2)l~l)<X2l chdx,.

(10)

Then we obtain, from Eqs. ( 1 )- (5), the following equations,

-i -&-+-$+” ( )r 84

(q,p,xi,X2)=0, (11) 1 2

(12)

(13)

( la2 . a A-&+-- 1 2 ax’, -lP aq

- (4,P,X,,X2)=0, Jf

(14)

( i(q--xl) j$ +i(4-x2) $7

1 2

-p(xl-x2)-i (ap,x1,x2)=0. Jf (15)

From Eq. ( 11) , we infer that f( q, p, x1, x2 ) is a fimc- tion of q-x, and q-x,, or 2q-x,-x, and x1-x2. Eq. ( 12 ) requires that the p-dependence of f( q, p,

x1, x2) is of the form

fo(2q-xi -x2, x1 -x2) exp[ip(x, --x2)1 .

Substituting this form into Eq. ( 13 ), we obtain

(Xl -x2)aI-x, -x21_mq-XL -x2, Xl -x2)

=o. (16)

Set x’, =2q-x1 -x2, x; =x1 -x2, then we have

a -=- ax, a a a d=2L -=--_- ax, ax; ax; 3 aq ax; * (17)

It follows from Eqs. (14)-( 16) that

&&fo(x;,x+o,

( a xi ax; +x2 ax; ‘&l e(x;,x;)=o, zf

(18)

(19)

x;x;fo(x;, x$)=0. (20)

Eq. (18) issatisfiediff,(x’i,x;)=fi(x’,)+f2(~;). In momentum representation, Eq. (20) requires that

fo(pi, p2) =fi(~i) +f2(p2), so that

.Mx;, xi)

= & SI fo(pi,p2) exp(ip,x; fipzx;) dpi dp2

= &6(x;) If2(~2) exp(ip2x’z) dp2

+ &4(x;) s .ITP~) ew(iplx;) dpl . (21)

Therefore, we must have

fo~x;,~;~=c,~~~;~+~26~x;~. (22)

But the term c2S(x$) can produce only a constant term in A?( q, p 1, which can be safely disregarded. So we have

f(4, p, x1, x2) =WQ--x~ -x2) exp[ip(xl --x2) I .

(23)

Note that we have not used Eq. (19) since it is not independent. This can be seen from the following,

( a a ---7,x;& owl,xi) ax; ax, If

( a = x; - +x;--, ax; az +1 )f

o(x;,x;)=o. (24) 2

Finally, the normalization of P(q, p) requires that c= 1 /rc, so that

H. Li/PhysicsLettersA lSS(1994) 107-109 109

p(q,P) = k I ezipyv*(q+y)y(q-y) dy , (25) References

which is just the original form proposed by Wigner [6] with fi= 1.

We observe that the relation between the Wigner function and the set of translations and inversions of phase space has been discussed by some authors; and the operator @(q, p) used in this paper is just the parity operator proposed by Grossmann and Royer [ 7-91. As to the applications, we note that, if 10)=etiIy),wehave

=(v/le-~~(4,P)eLaIy/)=e”P,(q,p), (26)

where A is an infinitesimal generator. This kind of properties has been used in Refs. [ 4,10 ] to derive the Wigner functions for coherent and squeezed states.

[ I] M. Hillery, R.F. O’Connell, M.O. Scully and E.P. Wigner, Phys. Rep. 106 (1984) 121.

[2] E.P. Wigner, in: Perspectives in quantum theory, Eds. W. Yourgrau and A. van der Merwe (Dover, New York, 1979) p. 25.

[ 31 R.F. O’Connell and E.P. Wigner, Phys. Lett. A 83 ( 198 1) 145.

[4] D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A 37 (1988) 807.

[ 5 ] A.O. Barut and R. Raczka, Theory of group representations and applications (World Scientific, Singapore, 1986).

[ 61 E.P. Wigner, Phys. Rev. 40 ( 1932) 749. [ 71 A. Grossmann, Commun. Math. Phys. 48 ( 1976) 19 1. [ 81 A. Royer, Phys. Rev. A 15 (1977) 449. [9] J.P. Dahl, Phys. Ser. 25 (1982) 499.

[lo] R.F. O’Connell, Found. Phys. 13 (1983) 83.