Quantum mechanics in phase space

Physica 91A (1978) 99-112 © North-Holland Publishing Co.

QUANTUM M E C H A N I C S IN PHASE SPACE Ill. LINEAR TRANSFORMATIONS

J.G. KR(]GER

Seminarie voor Wiskundige Natuurkunde, Rijksuniversiteit Gent, B-9000 Gent, Belgium and A. POFFYN*

Dienst voor Theoretische en Wiskundige Natuurkunde, Rijksuniversitair Centrum Antwerpen, B-2020 Antwerpen, Belgium Received 9 August 1977 The correspondence of a linear canonical transformation in phase space with a linear unitary transformation in the Hilbert space L2(R) yields in a unique way the Wigner kernel. With this kernel function the unitary transformations corresponding to finite linear transformations are calculated in different representations. The results are applied to propagators derived from quadratic hamiltonians.

1. I n t r o d u c t i o n

This is the third paper of a series of articles concerning the formulation of quantum mechanics in phase spacel'~). The joint distribution function [ in phase space is related to the density matrix p by the integral transformation with kernel function K, depending on the rule of correspondence used:

f(p, q) = f K ( p , q I x, y)p(x, y) dx dy,

(1.1)

where p, q are the phase space variables and the density matrix p is written in coordinate representation. In the first paper t) several requirements which can be restrictive for the kernel function are studied, leading to the Wigner kernel in a unique way. The uniqueness of the Wigner kernel is, however, also a direct consequence of the correspondence between linear canonical transformations in phase space and linear transformations of operators acting in the Hilbert space LE(R) (hereafter just called Hilbert space). This correspondence is analysed in section 2 with the use of infinitesimal linear * Research work performed as an I.I.K.W.-fellow. 99

100

J.G. KRI~IGER AND A. POFFYN

transformations. Finite linear transformations in phase space, defined by quadratic generators, are treated in section 3. Their corresponding unitary transformations are derived, by which each alternative form of the generator in phase space can be connected to a different representation in Hilbert space. Point transformations cannot be derived from all alternative forms of the generators: their unitary transformations in Hilbert space are described by means of distributions and therefore must be treated separately (section 4). In section 5 the results of section 2 are applied to linear propagators derived from quadratic hamiltonians.

2. Uniqueness of the Wigner kernel A classical canonical transformation in the phase space variables p and q cannot always correspond to a unitary transformation in Hilbert space. The content of this statement is reflected by the inconsistency of two "corr e s p o n d e n c e " rules, one proposed by Dirac and the other by Von Neumann. (a) Considering two successive infinitesimal transformations in phase space and in Hilbert space, respectively, one is led to associate Poisson brackets with commutators3):

{a,b}

i A > -~[a,/~],

(2.1)

where ~ and /~ denote the operators corresponding to the classical quantities a and b. This is Dirac's rule which is shown to be self-contradictory4). (b) Proposing an isomorphism between classical quantities and operators, one is led to the rule 5) a + b

f(a)

' ~ +/~, , /(d),

(2.2) (2.3)

where f denotes an arbitrary function. This is Von Neumann's rule, which is also proposed as a necessary requirement for a stochastic interpretation of quantum mechanics. It has been shown that this rule is also inconsistent4). The inconsistency of those two rules illustrates very clearly the impossibility of realizing an isomorphism between canonical and unitary transformations (otherwise classical and quantum mechanics would be equivalent). But it is possible to require the mentioned association for a subclass of transformations for which Dirac's (and also Von Neumann's) rule is valid and which does not lead to contradictory results. As a candidate for such a class of transformations we study now linear transformations (generated by quadratic functions). Consider an infinitesimal linear canonical transformation of the conjugate variables p, q into the new variables p', q'. Such a transformation can be written a s 6)

,Sp = {g, p};

6q : {g, q},

(2.4)

QUANTUM MECHANICS IN PHASE SPACE, III

101

where g denotes the quadratic function g(p, q) =½p • A . p

+ p • 13 . q + ½q • G . q + D . p + E . q + F,

(2.5)

where the coefficients A, 13, C are tensors, D, E are vectors, and F a scalar, which may all be dependent on time. A linear unitary transformation of the conjugate operators 1~, 4 into the new conjugate operators/~', 4' can be put into the form 7) i

^

.

~$1~= - ~ [ g , P ] ;

i

64 = - ~ [ g , 41,

(2.6)

where ~ denotes a hermitian quadratic function of 1~ and 4. Requiring that (2.6) is the same linear connection as (2.4), with the conjugate variables replaced by their corresponding operators, we find that = g(/~, 4) + C,

(2.7)

where C is an arbitrary constant which can be taken to be zero; ~ is hermitian by replacing Piqj

' ½(picli + CliOi).

(2.8)

The fundamental relation fixing the correspondence rule is {g, f}

i , ' - ~ [g, t~],

(2.9)

with g and ~ the quadratic generators of the infinitesimal linear transformations (2.4) and (2.6), respectively. Using f = tr/~t~, the k e r n e l / ~ must be a solution of i {g,/~} = - ~ [/~, ~1.

(2.10)

Denoting the Fourier transform of any function [(p, q), after scaling of the variables 0 and ~', by

[(,,,)= f

(2.,1)

we express the Fourier transform of (2.10) by means of ~ and /~. For the linear and quadratic part g~ and g2 of g, we obtain respectively, {g~, Igi} = -~g,(O, - ~')/~(0, ~'),

(2.12)

{g2, I~} = {g2(O, -~'), K (O, ~')},

(2.13)

and

where the Poisson bracket on the right side of (2.13) is defined in the new variables e and ~-.

102

J.G. KRUGER AND A. POFFYN

The relation between the Fourier transform of the generalized kernel/~ and the Wigner kernel /~w is given by 1) r) = f0(o, ~-)/~w(O, ~-)=/0(0, ¢)exp [ h ( 0 . ~ + ~-./~)].

(2.14)

Substitution of (2.14) into (2.13) leads to:

{g2(O, --~), I~ ( 0, ~')} = fo(O, ~'){g2(0,-r),/~w(0, r)} + {g~_(O,-r), fo(O, ~')}/~w(0, ~').

(2.15)

Now, the Wigner kernel fulfils the following relations:

hgl(O,-~')/~w(O, ~')= -h[/~w(0, ~'), ~,],

(2.16)

i

{g2(O,-~'),/£w(0, ~')} = -~[Kw(O, ~'), ~2].

(2.17)

By using (2.12)-(2.17), the Fourier transform of the relation (2.10) reduces to

{gz(O, -,r), fo(O, ~')} = 0.

(2.18)

The absence of the linear part in (2.18) is obvious, since i ,/~, {g,,/~} = - ~ t ~t]

(2.19)

expresses the requirement of a Galilei-invariance%, fulfilled by (2.14). Choosing successively in g2 all but one of the terms equal to zero, we find that: of0_ of 0 _ o. oo o~-

(2.20)

By the normalization requirement of /0(0, 0 ) = 1, we finally obtain f0 = 1, which is the definition of the Wigner kernel corresponding to Weyl's rule. Conversely, for Weyl's rule an infinitesimal unitary transformation with generator ~ (not necessarily quadratic) corresponds to ~t [~'t~]

, ~ sin 2

" Op~ t

O-pc"

gf"

(2.21)

It is understood that OlOq~ and OlOpg represent the differentiations Ol&q and OIOp only acting on g, while Ol&qt and &/&Pf only act on f, and after the differentiations are carried out the indices g and f are omitted. If ~ is of second order in 1~ and 4, eq. (2.21) reduces to the inverse of (2.9). However, if is of higher order, higher order derivatives than the first acting on [ appear in (2.21). This shows that the class of transformations for which the rule of Dirac (and Von Neumann) remains valid, and for which a canonical transformation in phase space corresponds to a unitary one in Hilbert space, is

QUANTUM MECHANICS IN PHASE SPACE, III

103

necessarily restricted to linear transformations (and quadratic generators).

3. Finite linear transformations

Consider a unitary transformation of the density matrix t~ into t~'. t~': OgO +.

(3.1)

By means of the Wigner kernel the following integral transformation in phase space corresponds to eq. (3.1):

I'(P', q') = ( W(p', q'lP, q)f(P, q) dp dq.

(3.2)

J

The generating function of a canonical transformation can only be written in one of the four following forms6):

F,(q, q');

Fz(q, p');

F3(p, q');

F4(p, p').

The circumstances of the problem will dictate the form to be chosen. For example, if we are dealing with point transformations (which will be treated in section 4) q and q' are not independent and generating functions of the form F~(q, q') must be excluded. The three alternative generating functions F2, F3 and F4 can be defined in terms of F1 by F2(q, p ' ) = p ' " q ' + F l ( q , q'),

(3.3)

F3(p, q ' ) = - - p • q + F l ( q , q ' ) ,

(3.4)

F,(p, p') = p' . q' - p • q + F,(q, q').

(3.5)

Using transformation equations we obtain for the corresponding Green functions in phase space:

w.(,'.q'lp.q)-- dp' 8(p- OFt\)s/p , /

Wz(p', q' l p, q) =

dq'

{ _ a v % . / , aF2 8 ff aq / O ),

dp' W,(p', q' l p, q) =

dq'

OFl\

+ OFf~ {

-~p 8\q

+

Op'/"

Here 6[p -(OFJOq)] is the shorthand notation for N

dEll,

etc.,

(3.7) (3.8)

OF,\ / ,_OF,]

)81o

(3.6)

(3.9)

104

J.G. K R U G E R A N D A. P O F F Y N

while [dp'/dql, etc. are jacobians different from zero. The four Green functions can be compactly written as

dP' ( P - -O~F.) '* ( p , + O W. = -d-~16 F. ~ OQ']'

(3.10)

where the variables P, Q, P ' and Q' are replaced by those indicated in table I. We calculate the superoperator U ® / J + in a convenient representation, according to (2~h) N

f (YII((P, O)lx)w.(x'l~+(P ', O')lY')dP dO dP' dQ' = (x'lOlx)(YIO+lr'),

(3.11)

where the integration interval for all the variables is (-o~, oo). The variables X, Y, X' and Y' are replaced by those indicated in table II. Those variables are chosen such that we have always: exp [ - h P • (X -

( r I K ( P , Q)IX) = ~ 1

Y,]6[Q-~(X+Y)].

(3.12,

Inserting (3.12) into (3.11) yields: (X'I u l x ) ( Y I 9+1Y')

=~

f i raF.(½(x + r ) , ½(x' + r')) exp,- ~ [ O(½(X + r ) ) • (x - r)

J"

+ OF.(½(X + Y),~(X + Y')) o(½(x' + Y ' ) )

( X ' - Y') "

TABLE I Alternative forms of contact transformations n

1

2

O

q

q

P

p

P

Q' p'

q' p'

p' -q'

3

4

p

p

-q

-q

q' p'

p' -q'

TABLE II Alternative representations unitary transformations n

1

2

3

4

X X'

x x'

x k"

k~ x'

k~ k"

Y r'

y y'

y ~',

ky y'

ky k;

of

(3.13)

QUANTUM MECHANICS IN PHASE SPACE, III

105

where J, denotes the jacobian J" =

[OF,(½(X+ r ) ,

½(X'+ V')) 0(½(X+ Y)a(½(X'+ V')) # 0 .

(3.14)

As the transformation is linear, the corresponding functions F, are quadratic:

F,(Q,Q')=½Q . A . .

• O'

O+O-B.

+½O'. C.. Q'+D," O + g , . Q'+C.. (3.15)

It is sufficient to choose symmetric matrices for A, and C,. All the coefficients are independent of Q and Q' but may still be dependent on time. C, is arbitrary for a given transformation. The jacobian (3.14) reduces to J, = [det B,[.

(3.16)

Using the Taylor expansion of the function (3.15), we finally obtain < Vl U+I V'> = ~

J"

exp

{-h[F,(X,X')-F,(Y, Y')]}.

(3.17)

Hence

= ~

1

~

OF.

1/2

i

exp{--~tF,,(X,X')+

h.]},

(3.18)

where A,/h is an arbitrary coordinate independent phase which may be different for different representations and which can be absorbed in the arbitrary constant C,. For n = 1 (3.18) reduces to the well-known expression obtained by the path integral approachS). The other forms (n = 2, 3, 4) express the same result in different representations:

jrI/~ (x'lOlx)=~exp{-h[F,(x,x')+ X,]}

j~12

i

'

x'+

i ,)+ k . x + As]l dk j~/2 f exp{-g[F3(k,x

J

(2w---~ J -

j~/2 ff exp {- ~i [F4(k, k')+k (2=n)~-~

• x - k'

"x'+

A4]} dk dk', (3.19)

where the functions F:, F3 and F4 are related to Ft by means of eqs. (3.3), (3.4) and (3.5). All the integrals are based on the one-dimensional form f_+~7exp[i(px 2 + qx)] dx, which must be considered as the limit of s) +00

lira f e x p ( - . x 2) exp [i(px2+ -®

qx)] d x :

// Tr ~1/2

(i4

(i-p-i) exp - - s i g n p

)exp/k-~p-p/. iq2"~ (3.20)

106

J.G. KROGER AND A. POFFYN

For the one-dimensional case with F~(x, x') defined by F 6 x , x') =½axE+ b x x ' + ½cx'2 + d x + ex' + f,

(3.21)

we find (up to a multiple of 2~rh) ,X2- AI = ~rh sign c,

(3.22)

A3 - Al = -~rh sign a,

(3.23)

A4- A1 = llrh[sign a][1 + sign(ac - b2)].

(3.24)

Consequently, for the N-dimensional case the differences of the phases are always of the f o r m (x.

- ;~,.)/h

= zTrN ' ....

(3.25)

where N,,m are integers depending in a complicated way on the coefficients appearing in (3.15). It is seen f r o m eq. (3.25) that the differences of the phases are piecewise constant in time. We can fix the phase factors An/h in the vicinity of the identity transformation. For that purpose we require that the identity operator in phase space c o r r e s p o n d s to the identity operator /] = i. As the identity transformation is a special case of a point transformation, neither Fl(q, q') nor F4(p,p') can be used as a generating function. This difficulty can be solved by choosing for F~, e.g. 1

(3.26)

F~(q, q') = ~-~(q - q,)2,

and taking the limit or ~ 0 afterwards. As

'

I -i

1

lim . . . . . Nn exp(~ilrN) exp ~ - ~ ( x - x') 2 ~-.0 (~1rno')

~ 8 ( x - x'),

(3.27)

we have that Z~ = l r h ( - A N + 2k0,

(3.28)

where kl is an arbitrary integer. The generating function F2(q, p') corresponding to F~(q, q') is F2(q, p') = q . p, - ~trp l ,2.

(3.29)

By the requirement that the transformation /], derived from (3.29), with tr -- 0 is

:

i

which is just a mixed representation of the identity transformation, it follows f r o m (3.18) that A2 = 2~rhk2 (k2 integer). The generating function F3 ( p , q ' ) corresponding to (3.26) is given by F3(p, q') = - p

• q ' - ~o'p 2.

(3.31)

QUANTUM MECHANICS IN PHASE SPACE, III

107

T h e t r a n s f o r m a t i o n 0 derived f r o m (3.31) with tr = 0 is required to be

x')

1

f r o m which it results that A3 = 27rhk3 (k3 integer). T h e f u n c t i o n F4(p, p') c o r r e s p o n d i n g to (3.26) does not exist. H o w e v e r , to fix ,~4 in the vicinity of the identity t r a n s f o r m a t i o n we can start f r o m F4(p, p') = ~

1(

p - p,)2,

(3.33)

which is a n a l o g o u s to (3.26), and will give rise to analogous e x p r e s s i o n s for h4 and )k3 as f o r A1 and A2. Concluding, in the vicinity of the identity transf o r m a t i o n , we h a v e Ai = 7 r h ( - 1 N + 2k0;

)t 2 = 27rhk2,

(3.34)

A4 = ~rh(-¼N + 2k4);

A3 = 27rhk3,

(3.35)

w h e r e kn (n = 1, 2, 3, 4) are arbitrary integers. An e x a m p l e w h e r e all four g e n e r a t o r s exist and r e d u c e to the identity t r a n s f o r m a t i o n w h e n the p a r a m e t e r z = t' - t -->0 is Fl(q, q') -

into

2 sin toz

[ ( q 2 + q , 2 ) C o s t o t --

2qq'],

(3.36)

which is H a m i l t o n ' s principal f u n c t i o n (up to the sign) for the linear h a r m o n i c oscillator s) with m a s s m and angular f r e q u e n c y to. A~ can be c h o s e n to be constant. Following eq. (3.25) the other p h a s e s are then p i e c e w i s e constant. In the e x a m p l e of the linear h a r m o n i c oscillator the coefficients a and c c h a n g e sign for tot = ½7rl (I integer), while a c - b 2 < 0 . C h o o s i n g k, = 0 (n = 1, 2, 3, 4) for simplicity, it follows f r o m eq. (3.24) that A4 is also constant, but at the time instants tot = ½~rl the j u m p s of A2 and A3 in the increasing direction of z are: [A2] = [A3] = (-- 1)t+l-~'rrh.

(3.37)

4. Point transformations

If the variables q and q' are not i n d e p e n d e n t , f u n c t i o n s of the f o r m F,(q, q') must be excluded. T h e functions F2(q,p') and F3(p, q') g e n e r a t e point transf o r m a t i o n s if G2 = 0;

A3 = 0.

(4.1)

I n d e e d , the t r a n s f o r m a t i o n equations g e n e r a t e d b y F2(q, p') are q ' = q " B 2 Jr- E 2 ,

(4.2)

p'

(4.3)

= 821 •

(p - A2" q - DE),

108

J.G. KROGER AND A. POFFYN

B~-' denotes the inverse matrix of Br. F2(q, p') is still a quadratic function in q if A2 ~ 0, but is a linear function in p'. Hence, the second expression for (x'[ 0 I x ) in (3.19) yields (x'l UIx) = Idet B21'nS(x'- x . B2 - E2)

[i

xexp

~-(~x.A2.x+D2-x+C2+A2)

]

,

(4.4)

which replaces the first expression for (x'10Ix) in (3.19). Equation (4.4) can also be derived from F1 by a limiting procedure. Indeed, I

,

Fl(q, q') = ~-~(q - q • Iq2 -

E2) 2 +

½q • A2" q + D2" q + C~

(4.5)

generates (4.2) and (4.3) in the limit or-->0. Following the first expression in (3.19) we obtain (with C~ = C2+ X2- A~ - ~47rhN):

( x ' l O l x ) : exP(4~rN)(21rhcr)N,2[detB211/2exp[-~--~ ( x ' - x . B 2 -

[i

]

E2) 2]

x exp --~(ix • A2- x + D2 • x + C2+ A2) ,

(4.6)

which reduces to (4.4) in the limit o - ~ 0. An analogous reasoning is valid for

F4(p, p'). 5. Green

functions

The evolution equation for the density matrix can be written as

O(t'- t)p(x', y', t') = f G(x', y', t'[ x, y, t)p(x, y, t) dx dy.

(,.1)

Acting upon (5.1) with (ih 0 / 0 t ' - ~ ) , whereby ~ = H (~) 1 - i (~) H, we find as equation for the evolution operator G: ( i h ~ t , - ~ ) G ( x ' , y', t' [ x, y, t)= i h 6 ( x ' - x ) 8 ( y ' - y ) 8 ( t ' - t).

(,.2)

Together with the boundary condition of forward propagation in time:

G(x', y', t' [ x, y, t) = 0,

for t' < t,

(5.3)

eq. (,.2) defines the retarded Green function. The solution of eqs. (5.2) and (5.3) is

G(x', y', t' I x, y, t) = O(t'- t)Go(x', y', t' [ x, y, t),

(5.4)

where Go is the solution of the homogeneous equation, corresponding to (5.2), that fulfils the initial condition

Go(x', y', t' I x, y, t) = 6(x' - x)8(y' - y),

for t' = t.

(5.5)

QUANTUM MECHANICS IN PHASE SPACE, III

109

From the expansion of p in terms of eigenfunctions of the Schr6dinger equation,

p(x, y, t) = ~.. ai.i~i(x, t)rF~[(y, t),

(5.6)

t,]

it follows that Go can be expressed as

Go(X', y', t' l x, y, t) = Uo(x', t' l x, t)U~d(y ', t' l y, t),

(5.7)

where U0 is the solution of the homogeneous Schr6dinger equation, fulfilling the initial condition

Uo(x', t' ] x, t) = 8 ( x ' - x),

for t' = t,

(5.8)

and U~ is the complex conjugate function of U0 fulfilling the same initial requirement. By direct calculation or by means of transformation with the Wigner kernel, we obtain, as corresponding equations of (5.1) and (5.2) in phase space,

t'l

p, q, Of(p, q, t) dp dq,

(5.9)

(~t'- "~) W(p', ,I', t' l p, ~, t)= 8(p'- p),(q'- ,1),(t'- t).

(5.10)

O ( t ' - t)f(p', q', t') = f W(p', q',

The propagator in phase space is given by

W(p',q', t' Ip, q,

t) = f K(p',q' Ix',y')G(x',y',

t ' l x , y, t)

x K-~(x, y [ p, q) dx' dy' dx dy,

(5.11)

where K is the kernel appearing in (1.1), while K -t is the kernel of the corresponding inverse transformation. ~ is the quantum Liouville operator. The corresponding relations of (5.3) and (5.4) are

W(p',q',t'lp,

q,t)=O,

(5.12)

for t ' < t ,

and

W(p', q', t' l p, q, t) = o ( t ' - t) Wo(p', q', c lp, q, t),

(5.13)

where W0 is the solution of the homogeneous equation corresponding to (5.10), fulfilling the initial condition

Wo(p', q', t ' l p , q, t) = 8 ( p ' - p ) 6 ( q ' -

q),

for t'= t.

(5.14)

From (5.11) and (5.13) it follows:

Wo(p', q', t'lp, ~, t)= f K(p', ~'lx',y')Uo(x', t'lx, t)U~(y', t'ly, t) × K-~( x, Y I P, q) dx' dy' dx dy.

(5.15)

110

J.G. KROGER AND A. POFFYN

If U0 only depends on the differences x ' - x , t ' - t : U0(x', t' I x, t) = U~(x'- x, t ' - t),

(5.16)

as in the case of a free particle, for example, then it is easy to check that W0 has the form

Wo(p', q', t' l p, q, t) = 6(p'- p)W~(p', q ' - q, t ' - t).

(5.17)

The first factor on the right-hand side of eq. (5.17) expresses the conservation of momentum, while W~(p', q', t') corresponds to the Wigner equivalent of G0(x', y', t' I 0, 0, 0):

W~(p', q', t') = f K(p', q' ] x', y') U~(x', t') U~*(y', t') dx' dy'.

(5.18)

As an example of the property (5.16), consider the propagator for a free particle with mass rn ,0): [

m

U~(I', t ' ) : ~ }

\3/2ex /im[x'12\

P~--~-~7-}.

(5.19)

Inserting (5.19) into (5.18) and taking into account (5.17) and (5.13) yields finally:

W(p" q" t' [ p' q' t) = O(t'- t)B(P'- P ) 6 ( q ' - q - P ( t ' - t))

(5.20)

which is indeed the propagator for the Liouville equation of a free particle: 0_f_f+.p'. . o.f _ 0. Ot' m Oq'

(5.21)

The inverse of the relation (5.11), given by (3.11), can be used to calculate the propagator U (or G) if W is known. This method is particularly useful for the case of systems defined by a quadratic lagrangian or a quadratic hamiltonian. Indeed, for these systems the quantum Liouville equation reduces to the classical one

Of {H,f} = O. Ot'

(5.22)

As a consequence, the quantum propagator in phase space reduces to the classical one

OF, 6 ( p _ O F , ~ 6 ( p , +OF, ~ W,(p', q'lp, q) = aq Oq Oq / -~Tq,/,

(5.23)

where p, q and p', q' are the initial and the final coordinates at time t and t', respectively. The function -F,(q, q') appearing in eq. (3.6) is now Hamilton's principal function t'

-Fl(q, q') = I L(t) dt. t

(5.24)

Q U A N T U M M E C H A N I C S IN P H A S E S P A C E , III

Ill

The propagator (5.23) has a very clarifying meaning: it propagates any distribution function along the classical trajectory in phase space. Following section 3 the unitary transformation corresponding to (5.23) is 1 i (x'[lJ[x)=~exp(~TrN)

1 I/2 Oxa F,gx' exP[hF'(x'x')]' •

(5.25)

which is the well-known form obtained by the path integral approach. Conversely, to the propagator (5.25) with a quadratic generator corresponds the classical Green function (5.23) in phase space. The phase space description is particularly well suited for subsequent approximations of order h 2, h 4, h 6. . . . . to the classical limit. For a general hamiltonian the Liouville equation in the WKB approximation reduces to eq. (5.22). We have 0fwKB_ {H, fWKB}= at

~?(h2)fwKB •

(5.26)

Hence, (x'[/-)WKB[X)is given by eq. (5.25) and fulfils (/-~ - i h ~)/-)w~B = ~(h2)/-)WKB,

(5.27)

which coincides with the usual requirement for the WKB approximation11).

6. C o n c l u s i o n

A classical canonical transformation in the phase space variables p and q cannot always correspond to a unitary transformation in Hilbert space. However, one can impose an isomorphism between linear canonical transformations in phase space and linear transformations of the corresponding operators acting in Hilbert space. The relation between the distribution function f and the density matrix p is then uniquely determined by an integral transformation with the Wigner kernel, which is associated with the Weyl correspondence rule. The same rule is also selected out of all other correspondence rules by the analysis of paper I1). The isomorphism between the linear transformations means also that Dirac's rule (the correspondence between the Poisson bracket and the commutator multiplicated by is valid if one restricts oneself to quadratic generating functions. This restriction means that the generators are represented by second order differential operators in the coordinate representation, and that the corresponding transformations in phase space are induced by linear differential operators. As a consequence, for linear transformations the generators in phase space are the same as the classical ones. It is well-known that the classical canonical transformations can be derived from generating functions of four different types. It is shown that these four types can be associated with four representations of the corresponding unitary transformations in Hilbert space. Their explicit form is determined by transforming

-i/h)

112

J.G. KRUGER AND A. POFFYN

the Green functions for the finite classical linear canonical transformation by means of the Wigner kernel. For point transformations, some of the four alternative forms of the classical generating functions do not exist in this case. The corresponding unitary transformations are found by a limiting procedure. This gives an insight into the different forms of the unitary transformations corresponding to a point transformation or any other linear transformation. The results are applied to propagators derived from a quadratic hamiltonian. The Liouville equation and the corresponding propagator in phase space reduce to the classical ones. This has a very clarifying meaning: any distribution function is propagated along the classical trajectory in phase space. On the other hand, the classical propagator in phase space is associated with the result of the well-known Feynman path integral, which is derived now in different representations each corresponding to one of the alternative forms of the classical generating function. The techniques used in this and the foregoing articles are easy to handle because we restrict ourselves to linear transformations. In general, there is no straightforward method of treating transformations which go beyond this class in an exact way and one has then to recourse to perturbation expansions. An analogous situation occurs for the calculation of the Feynman path integral, where it is only easy to handle problems when the action is a quadratic function.*

Acknowledgements We express our gratitude to Professor Dr. C.C. Grosjean for encouragement and we wish to thank Professor Dr. P. Van Leuven for facilities.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

J. Kriager and A. Poffyn, Physica 85A (1976) 84. J. Krilger and A. Poffyn, Physica 87A (1977) 132. P.A.M. Dirac, Proc. Roy. Soc. (London) Al10 (1926) 561. H.J. Groenewold, Physica 12 (1946) 405. J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, 1955). H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 1965). P. Roman, Advanced Quantum Theory (Addison-Wesley, New York, 1965). R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, New York, 1959). J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, I%5). M.M. Mizrahi, J. Math. Phys. 4 (1977) 786. M.J. Goovaerts and C.C. Grosjean (to be published).

* For techniques going beyond this class of functions, see ref. 12.