Relativistic Wigner function as the expectation value of the PT operator

Volume 114A, number 8,9 PHYSICS LETTERS 17 March 1986

R E L A T I V I S T I C W I G N E R F U N C T I O N AS T H E E X P E C T A T I O N VALUE O F T H E P T O P E R A T O R

C. D E W D N E Y , P.R. H O L L A N D , A. K Y P R I A N I D I S and J.P. V I G I E R

C N R S / U A 769. Lahoratoire de Physique Thborique, lnstitut Henri Poincarb,

11 rue P. et M. Curie, 75231 Paris Cedex 05, France

Received 6 January 1986; accepted for publication 10 January 1986

We define for the covariant relativistic Schr6dinger equation a Hilbert space structure and generalize the Wigner Moyal formalism to this case. We show how as a result the work of Royer may be generalized to express the relativistic Wigner function as the expectation value of the P T operator, and give an interpretation for the appearance of negative densities in terms of antiparticles.

In a recent publication we considered the possibility of constructing a Hilbert space of states for relativistic quantum mechanics of scalar particles [ 1 ]. Furthermore we have shown that this attempt is in- evitably connected with a reformulation of relativistic quantum theory in terms of a proper time dependent formalism [2]. Operating in this new frame one can encompass difficulties of relativistic quantum mechanics without being obliged to pass over to a second quantized formalism. The aim of the present letter is to show the implications of this new formalism on the attempts to achieve a phase-space approach to relativistic quantum mechanics in terms of Wigner-Moyal type relativistic phase-space distributions [3].

A Hilbert space construction evidently needs a scalar product definition that yields a positive definite norm 114112 = (4, ~), a fact which is not satisfied by the Klein-Gordon inner product

(c~, 4) = f d 3x 0*(f00 - 2eA0)4 •

If one tries to restrict the H-space to positive frequen- cy contributions where IIqS+l[ 2 > 0 then other difficulties emerge, as e.g. the non-hermiticity of the position operator X = iVp, i.e. (X¢, 4) :~ (¢, 2 4 ) . Instead of confining ourselves in this frame, we have shown that we can advance an alternative approach deduced from a relativistic stochastic variational principle on the average stochastic action [4] of a relativistic Markov

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process [5]. The result is a covariant Schr6dinger equation with a proper time dependence and a non- conserved mass (a mass shell with finite width)

(~/i)04(x, r)/Or = ~h2D4(x, r), x ~ M 4 , (1)

where the Klein-Gordon equation results as a proper time stationary solution with a mass-shell constraint by means of the ansatz [6] ,1

4(x, r) = 4(x) exp [ - i ( m 2 c 2 / 2 h ) r ] . (2)

Since this equation possesses the Schr6dinger form, r playing the role of an external evolution parameter [ 1], it is quite natural to define the scalar product by analogy with the Schr6dinger theory:

(,P14> = f ¢*(x, r)4(x, r) d4x. (3)

This form obviously yields a positive definite norm of the state vector (~bl¢) = 114112 > 0. Problems related to the convergence of the four-integration, and hence normalizability are treated by a kind of generalized "normalization in a box" condition [ 1,7]. With the conditions of positivity and normalizability for the norm of the state vector we have been able to define a Hilbert space of states since we have shown that the norm of the state vector is r-invariant [ 1]. Finally using this underlying H-space structure we have asso-

~-1 We have put a mass constant here to one.

0.3750601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


ciated the quantum relativistic operator algebra with the algebra of observables in phase space [4], reveal- ing thus the essential symplectic structure underlying the Hilbert space [ 1 ].

The essential point now is that we can proceed to define the following operator structure in our formalism [4]:

(1) A relativistic space-time position operator ~u = xU in the coordinate representation, which is associated with the infinitesimal canonical generator of phase changes

QU = fpx. d4x = (~, ~uff) .

(2) A relativistic space-time momentum operator Du = (ti/i)~/~x u in the same representation associated with the infinitesimal canonical generator of space- time translations

pu = f pa.s d4x = (¢,/3uff).

Using now our stochastic observable formalism, commutation relations between/3 and ~ can be derived. We simply exemplify the general relation [4]

1 f f dx dx' gg*(x)[.4,/}](x,x')~(x'),

(4)

associating Poisson brackets with quantum commuta- tors for .~ = QU and q3 = pv and correspondingly = ~u and/} = ~v. In fact we have:

(Qu,p~)= f [ 6Qu SPY 5Q u 6pv.1 6o(x) 8S(x) 5s(x) ~p(x)J dx,

(5)

where 5/5p is a functional derivative. A straightforward calculation using integration by parts and the vanishing of p(x) at the limits of space-time as well as the normalization condition yields

(Qu,P v) = 5uv f p d4x = 6uv . (6)

Comparing this result with the right-hand side of eq. (4) we deduce the commutation relation between/3 and~ [qu 'bv] =ih5 v , (7)

which establishes in the frame of this formalism a

fourth uncertainty relation Aq0AP0 ~> ~ (energy- time uncertainty) as an independent relation on the same footing as the three-position-momentum uncer- tainties in agreement with the postulates of a "canonical quantization" procedure [8].

In complete analogy with Schr6dinger QM we can choose the coordinate representation where//u is diag- onal and use the complete set of eigenstates as a basis of the Hilbert space• The following relations can be deduced:

(a) eigenstates-eigenvalues of position

?lulq ') = q 'U lq ' ) , (q" l? lU lq ') = q '~(~ q , , , q , ,

(b) o r thonorma l i t y

(q lq ' ) = 6 (4) , q,q ,

(c) completeness

f lq) dq (q[ = 1,

(d) representation of/3u • t t PP t ( q " l h U l q ') = (~/1)(~) f i )q . ) (q Iq ) .

The state vector I~k) can be specified in this representation by its components with respect to the basis vectors Iq), specified above. We thus have

~(q ) = (q l~ )

and using (c) we can write

f ( ~ l q > dq (ql~) = 1.

This enables a probabilistic interpretation in the sense that I(q I~k)l 2 dq represents the probability of an event in the interval between q and q + dq during the time-interval q0 and q0 + dq0" A notion of a "density of events" is now ascribed to I(qlq~)l 2 which assuming finite time durations of processes, i.e. normalization between two space-like surfaces [ 1,7], introduces also the necessity for finite time intervals for relativistic quantum measurements. Space-time four-integrations are explicitly introduced in order to account for the explicit existence of particle/antiparticle transition processes in the proper time dependent relativistically invariant Schr6dinger theory.

In what follows we wish now to apply the Hilbert space and operator structure defined above to a generalization of the Wigner function examination estab-

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lished by Royer [9] for the non-relativistic case. By this approach we think that we confirm the main result of a previous publication of our group [10] con- cerning a relativistic generalization and interpretation of the Wigner function.

As a first a t tempt one could define a scalar relativistic phase-space distribution function of the Wigner- Moyal type in the following form (assuming h = 1)

F(X, P) = (270 -4

0 0

× f d4z exp(-ZiPuzU)t~*(x- z)~(X + z) - - o o

o o

= (2rr)-4 f d4K e x p ( - 2 i X u K g )

X $ * ( P + K ) $ ( P - K ) . (8)

Then along the line of Royer [9] we can represent it as follows:

F(X, P) = (270 -4 (4 Illxp[~O) • (9)

The operator IIxp has the following expression in our representation

nxe = f d4z exp(-2iPuzU)lX - z)(X + zl

: f d 4 r exp(-2iXuKU)lP + K)(P - KI, (10)

where IX) is an eigenstate of the operator 2 and IP) an eigenstate of/~. In order to find out the meaning of IIxp, consider a special case X = 0, P = 0 and having IIx=0,p= 0 = II we can write

1I = f d4z I - z > < z l = f d4K [ K ) ( - K I . (11)

The action of II on qJ(q) can be immediately investi- gated:

II~O(q) = f d 4 z I-z)(zl(ql~) = ~b(-q). (12)

II thus turns out to be the PT operator about the ori- gin, a fact that can also be easily seen by the following operator relations:

HXI I = - 2 , II/SH = - / 5 . (13)

IIXp now results out of 17 by means of a unitary trans formation

1]Xp = D(X, P) HD -1 (X, P ) , (14)

where DO(, P) is a phase-space displacement operator [ 11 ], of the following type

D(X, P) = exp [i(PX - XP)] , (15)

which produces the following actions on the 2 and/5 operators

D-I(x,P)2D(X,P)= X + X,

D-I(x,P)/SD(X,P) = /5 + P , (16)

where the same type of relations hold for an arbi- trary function of X and/5. Using relations (13), (14) and (16) we can show

n x e ( 2 - x ) n x e : - ( 2 - x ) ,

nx+,(/5 - P) n x e : - ( /5 - e ) , (17)

which reveals the operator IIxp to perform a reflec- tion about the phase-space point (X, P) and thus to be the PT operator about the same point. Therefore the Wigner function in its relativistic generalized form represents the expectation value of the PT operator about the phase-space point (X, P). Alternatively, tak- ing over the picture advanced by Royer [9], "the func tion F(X, P) is proportional to the overlap of ~ with its mirror image about (X, P), which represents a mea- sure of how much 4+ is "centered" about (X, P)", this result holding with the necessary generalizations, advanced above, also in the relativistic case.

Now, by noting that ( I Ixp)2 = 1 one can readily write the eigenfunctions f f~p corresponding to eigenvalues +1

nxpl+~e) = +-I+~ce), (18)

which are symmetric (+) or antisymmetric ( - ) with respect to (X, P). These eigenfunctions can also be ob- tained by means of displacement of the phase-space functions symmetric or antisymmetric about the ori- gin

I ~ X p ) = DO(, P)[~ + ) , (19)

which are eigenfunctions of H, i.e. HI~ +) = + l~O). Now every function ff can be separated into symmetric or antisymmetric components with respect to X, P by

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Volume II4A, number 8,9 PHYSICS LETTERS 17 March 1986

• + 1 means of a projector Pj-/, = 7(1 + IIxp), i.e. + +

I~0~p) = P~plq:), (20)

and using eq. (9) as well as the relation lIxp = P~p - P~.p we can finally represent the relativistic scalar Wigner function in the following form

F(X, P) = (2rr)-4(ll¢~cell 2 - [[ffxpl[2). (21)

Therefore F(X, P) reveals to be the difference of the norms of the PT-symmetric and PT-antisymmetric parts of ~ with respect to (X, P), exactly as it is in the non-relativistic case with respect to the space-parity operation. Therefore one sees that the relativistic analog to the non-relativistic parity comes out to be the combined PT-symmetry operation. Since this symmetry is tied to the particle/antiparticle symmetries in relativistic quantum mechanics one will be tempted to identify the non-positive definiteness o fF(X, P) with particle/antiparticle symmetries. As will be shown in what follows, although such an assertion proves not to hold it nevertheless leads to an interest- ing extension of F(X, P) that definitely meets the PT- symmetry requirements.

To this purpose take the PT-reversed solution of the Klein-Gordon equation. ~PT(x) = ~k(--x). Then we can write for the corresponding scalar relativistic Wigner function

FPT(x, ? ) = (27r)-4

× ~ a

I _ e , o

d4z exp(--2iPuzU)~*PT(x - z)t~PT(x + z)

= (2rr)-4 f d4z exp[-2i(-Pu)zU ]

X ~* ( -X - z )d / ( -X+z )=F ( -X , -P ) . (22)

Therefore according to our preceding considera- tions the PT-inverse solution, which represents an antiparticle solution, corresponds to a difference of the norms of the PT-symmetric and antisymmetric parts of with respect to -X, -P, i.e.

FPT(x,p) = (2r0-a(llqJ+__x,_ell 2 -- IIq~-x,_ell2).

(23)

Since our main concern is to interpret the occurrence of negative values in this phase-space distribu-

tion function as corresponding to antiparticle contributions, we readily recognize that the form of the chosen function F(X, P)is not the appropriate one: If the PT-symmetry operation reversed the sign of F(X, P) then one could map the negative particle contributions to F(X, P) to a set of positive antiparticle contributions. This is evidently not the case since F P T(X, P) = F(-X, -P).

In this context it is quite evident that one could try instead a generalized vector form for a possible Wigner distribution in the relativistic case. A possible candidate for this purpose was advanced in ref. [10] and reads as follows

Fu(X , P, e) = m-I [pu _ eAU(x)] F(X, P). (24)

If we write now the PT-transformed Wigner vector distribution function then it takes the following form:

FPu T(x,P, e) = [Pu - eAu(-x)]F(-X' -P)

= - [ ( - P u ) + eAu(x)] F(-X, -P) = -Fu(-X, -P, -e) (25)

This clearly shows that the negative values of Fu(X, P, e) can be mapped onto positive values of a distribution function that corresponds to an antiparticle at - X , with momentum - P and a charge - e , namely to FPT(-x, -P, -el.

This immediately shows the advantage of identify- ing the scalar F(X, P) function as a difference of PT- symmetric and antisymmetric parts of ~ with respect to (X,P). Since we could recognize that F(X, P) = (~0 [I lxe[~) does not change sign by means of the PT- symmetry operation we were forced to seek a different distribution function that satisfies this property, in view of the fact that PT-transformation is related to particle/antiparticle transitions. Therefore, the solution of this problem by multiplying F(X, P) with a PT-antisymmetric factor is straightforward, physically and intuitively well founded. Furthermore the relativistic four-vector Wigner distribution function Fu(X, P, e) can now be interpreted as a current distribution function where the negative values of the probability density FO(x, P, e) for particles may be interpreted as a positive probability density for antiparticle mo- tions at - X and with opposite four-momenta - P and charge - e .

Finally we can argue that the non-relativistic Schr6dinger-Wigner quasi distribution function

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Volume l I4A, number 8,9 PHYSICS LETTERS 17 March 1986

which is the non-relativistic limit o f F ( X , P), i.e.

Fs(x,p,t)= f F(X,P) dP 0 (26) _ c o

can only be brought to a form of difference of pari ty symmetric and antisymmetric parts o f ~ about x and p, as shown by Royer [9], and therefore does not en- able the interpretat ion of antiparticle contributions that we advanced for FU. Since the antiparticle states vanish in going to the Schr6dinger limit, we are simply left with the limit of the scalar F(X, P) with no possibility of sign reversal by any non-relativistic symmetry operation and therefore no chance of interpret- ing the occurrence of negative values in FS(X, p, t). Nevertheless the whole approach shows, as we believe, the formal and intuitive advantages of the defined Hilbert space structure which permits to transpose in the relativistic frame the scheme of the Royer interpretation of the Wigner function, and enables us to support by a somehow different route the already ob- tained results in our a t tempt to generalize the Wigner distribution function for relativistic quantum mechanics.

Three of us (C.D., P.R.H., A.K.) would like to thank the Royal Society, the SERC and the French

Government respectively for financial support which made this research possible. The authors are specially grateful to Professor Z. Marig for an important sugges. tion and many helpful discussions that enabled the completion of this work.

References

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[3] E. Wigner, Phys. Rev. 40 (1932) 749; J.E. Moyal, Proc. Camb. Phil. Soc. 45 (1949) 99.

[4] N. Cufaro-Petroni et al., Phys. Rev. D32 (1985) 1375; F. Guerra and R. Maria, Phys. Rev. D28 (1983) 1916.

[5 ] F. Guerra and D. Ruggiero, Lett. Nuovo Cimento 23 (1978) 529.

[6] R.P. Feynman, Phys. Rev. 80 (1950) 440, appendix A. [7] O. Costa de Beauregard, Pr6cis de la m6canique quan-

tique relativiste (Paris, 1961 ). [8] L.P. Horwitz and F. Rohrlich, Phys. Rev. D24 (1981)

1528. [9] A. Royer, Phys. Rev. AI5 (1977) 449.

[10] P.R. Holland, A. Kyprianidis, Z. Maric and J.P. Vigier, Relativistic generalization of the Wigner function and its interpretation in the causal stochastic formulation ot quantum mechanics, IHP preprint (Dec. 1985).

[11] R.J. Glauber, Phys. Rev. 131 (1963) 2766.

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Relativistic Wigner function as the expectation value of the PT operator