Probability representation in quantum field theory

5 November 1998

Physics Letters B 439 Ž1998. 328–336

Probability representation in quantum field theory V.I. Man’ko 1, L. Rosa, P. Vitale

2

Dipartimento di Fisica dell’UniÕersita, ` and INFN Sez. di Napoli, Mostra d’Oltremare, Pad.19, I-80125, Napoli, Italy Received 25 June 1998 Editor: L. Alvarez-Gaume´

Abstract The recently proposed probability representation of quantum mechanics is generalized to quantum field theory. We introduce a probability distribution functional for field configurations and find an evolution equation for such a distribution. The connection to the time-dependent generating functional of Green’s functions is elucidated and the classical limit is discussed. q 1998 Elsevier Science B.V. All rights reserved.

The use of statistical methods for describing quantum physics gives the opportunity to describe classical and quantum phenomena in a unified approach. Since the beginning of quantum mechanics there have been attempts to understand its nature in a classical-like context, namely to describe quantum states in terms of a classical distribution of probability. It is this philosophy which inspired the so-called quasi-probability distribution functions of Wigner, Husimi, Glauber and Sudarshan w1x. The original goal was not completely achieved Žthe above distribution functions are not always positive defined or they do not describe measurable variables. until Cahill and Glauber in Ref. w2x introduced a class of distribution functions, known as marginal distribution functions ŽMDF., which enjoyed all the properties of a density of probability. Nevertheless, it was realized only recently w3x that quantum mechanics could be described entirely in terms of a distribution of such a family, suitably defined for a random variable, which we will specify below. In Ref. w3x a consistent scheme has been proposed, the so-called probability representation, which has been shown to be completely equivalent to the ordinary formulation. Quantum states are described by a distribution of probability, the MDF, and the time evolution by an integro-differential equation for the MDF. Invertible relations have been established between the MDF and the density matrix w3,4x and between the Green’s functions of the related evolution equations w5x. In this framework classical and quantum phenomena, both statistically described, only differ by the evolution equations of the distributions of probabilities for the relevant observables. The quantum evolution equation, of Fokker-Plank type, is seen to reduce to the Boltzmann equation for the classical distribution of probability when the classical limit is considered.

1 2

On leave from Lebedev Physical Institute, Moscow, Russia. E-mail: manko,rosa,[email protected]

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 0 3 3 - 8

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In this letter we generalize the probability representation first to the case of N interacting particles, then to non relativistic quantum field theory. We consider a system of interacting oscillators which describe, when the limit to the continuum is performed, a self-interacting scalar field theory with generic self-interaction potential. We introduce the notion of MDF for the quantum state of such system and derive the evolution equation both for the discrete and continuous cases. The MDF is seen to be a distribution of probability with the same arguments used for 1-d quantum mechanics. Interestingly similar ideas have been developed by Wetterich in Ref. w6x in connection with the approach to equilibrium in non-equilibrium quantum field theories. There an evolution equation is found for a suitably defined time-dependent generating functional w7x. We establish the connection between our MDF and Wetterich’s generating functional in terms of a Žfunctional-. Fourier transform. In Section 1 we briefly review the probability representation of quantum mechanics for the simple case of a one-dimensional quadratic Hamiltonian. The derivation of the evolution equation for the MDF is performed in detail in a slightly different manner from its original derivation w3x, but more suitable for generalizations. In Section 2 we consider a quadratic Hamiltonian describing N interacting particles. We define the MDF and show that this is a well defined probability distribution. We then find the evolution equation. In Section 3 we consider a scalar field theory with self-interacting potential, which may be seen as the limit to the continuum of the previous model. We define the MDF which is now a probability distribution functional, and derive its exact evolution equation. In Section 4 we derive the evolution equations for the above mentioned systems, directly in terms of the Fourier transform of the MDF, the quantum characteristic function. We show then how our results may be connected to those found in Ref. w6x.

1. The probability representation of quantum mechanics The MDF of a random variable X was introduced in Ref. w2x as the Fourier transform of the quantum ˆ characteristic function x Ž k . s ² e i k X :, to be w Ž X ,t . s

1 2p

yi k X ² i k Xˆ :

Hdk e

e

,

Ž 1.1 .

where Xˆ is the operator associated to X, ² Aˆ: s TrŽ rˆ Aˆ., and rˆ is the time-dependent density operator. It is shown in Ref. w2x that w Ž X,t . is positive and normalized to unity, provided Xˆ is an observable. This theorem may be easily proven taking for simplicity rˆ to be the density operator for a pure state. Then, evaluating the ˆ it can be verified that Ž1.1. yields w Ž X,t . s r Ž X, X,t . which is trace in Ž1.1. on eigenstates of the operator X, positive and normalized to unity. We recall that the quantum characteristic function is, up to factors of i, the generating function of the ˆ Hence it plays in quantum statistical momenta of any order, for the probability distribution of the operator X. mechanics the same role as the generating functionals for the Green’s functions in quantum field theory. In Ref. ˆ w4x X is taken to be a variable of the form Xsmqqn p ,

Ž 1.2 .

where m , n are real parameters labelling different reference frames in the phase space. m is dimensionless, while w n x s w my1 xw t x. Thus, X represents the position coordinate taking values in an ensemble of reference frames. For such a choice of X it was shown that there exists an invertible relation among the MDF and the density matrix, respectively in Ref. w4x for the 1-d case, and in Ref. w8x for the 2-d case. This relation was originally understood through the Wigner function: the MDF was expressed in terms of the Wigner function which is in turn related to the density matrix and viceversa. The evolution equation of the MDF was then found starting from an evolution equation for the Wigner function established by Moyal in Ref. w9x. This intermediate

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step in terms of the Wigner function is not necessary. We can directly invert Ž1.1., when the variable X and the associated operator are given by Ž1.2.. The evolution equation is then obtained Žin the Schrodinger representa¨ tion. by means of the Liouville equation for the density operator, in coordinate representation. In view of the subsequent generalization to N degrees of freedom and to field theory, let us derive these results in some detail for a one dimensional system. Eq. Ž1.1. is explicitly written as w Ž X , m , n ,t . s

1

1

ˆ

yi k X ²

Z < rˆ e i k X < Z : s

Hdk HdZ e 2p

yi kw Xy m Ž Zyk n " r2.x

2p

Hdk HdZ r Ž Z,Z y kn " . e

.

Ž 1.3 . The MDF so defined is normalized with respect to the X variable: HdXw Ž X, m , n ,t . s 1. Performing the change of variables ZX s Z, ZXX s Z y k n " we may reexpress the MDF in the more convenient form: w Ž X , m , n ,t . s

1

X

Hr Ž Z ,Z 2p < n < "

XX

,t . exp yi

ZX y ZXX

n"

ž

Xym

ZX q ZXX 2

/

dZX dZXX

Ž 1.4 .

which can be inverted to

r Ž X , X X ,t . s


Hw 2p

ž

Y ,m,

XyXX "a

/

ž

exp i a Y y m

XqXX 2

/

d m dY ;

Ž 1.5 .

a is a parameter with dimension of an inverse length. The density matrix is independent of a . In facts, using the homogeneity of the MDF, w Ž a X, am , an . s < a
r Ž X , X ,t . s

1 2p

i

X

Hw Ž Y , m , X y X . exp

"

ž

Yym

XqXX 2

/

d m dY

Ž 1.6 .

where the variables Y, m have been rescaled by a . It is important to note that, for Ž1.4. to be invertible, it is necessary that X be a coordinate variable taking values in an ensemble of phase spaces; in other words, the specific choice m s 1, n s 0 or any other fixing of the parameters m and n would not allow to reconstruct the density matrix. Hence, the MDF contains the same amount of information on a quantum state as the density matrix, only if Eq. Ž1.2. is assumed. We now address the problem of finding the evolution equation for the MDF, for Hamiltonians of the form Hˆ s

pˆ 2 2m

q V Ž qˆ . .

Ž 1.7 .

Using the Liouville equation

Erˆ Et

i q "

Hˆ , rˆ s 0

Ž 1.8 .

and substituting into Eq. Ž1.4., we have w˙ Ž X , m , n ,t . s y

"2

i

H y 2m 2p < n < "

=exp yi

Z y ZX

n"

ž

ž

E2

E2

EZ2

Xym

y

E ZX 2

Z q ZX 2

/

/

Ž V Ž Z . y V Ž ZX . . r Ž Z,ZX ,t .

dZ dZX .

Ž 1.9 .

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331

Integrating by parts and assuming the density matrix to be zero at infinity, we finally have w˙ Ž X , m , n ,t . s

½

1 m

E m

En

žž /

V y

"

y1

E

žž /

yV y

y1

E

i q

EX

EX

E Em

E Em

in " E q

2 EX

in " E y

/5

2 EX

/

w Ž X , m , n ,t . ,

Ž 1.10 .

where the operator Ž ErE X .y1 is so defined y1

E

ž / EX

gŽZ. X

Hf Ž Z . e

dZ s

f Ž Z.

H gŽ Z. e

gŽZ. X

dZ.

Ž 1.11 .

This equation, which plays the role equation in the alternative scheme just outlined, has been ˆ of the Schrodinger ¨ studied and solved for some quantum mechanical systems w10,11x. The classical limit of Ž1.10. is easily seen to be w˙ Ž X , m , n ,t . s

½

m E m En

E

y1

E

E

žž / / 5

qn V X y

EX

Em E X

w Ž X , m , n ,t . ,

Ž 1.12 .

where V X is the derivative of the potential with respect to the argument. Eq. Ž1.12. may be checked to be equivalent to Boltzmann equation for a classical distribution of probability f Ž q, p,t . , Ef p Ef EV Ef q y s 0, Ž 1.13 . Et m Eq Eq Ep after performing the change of variables 1 w Ž X , m , n ,t . s f Ž q, p,t . e i kŽ Xy m qy n p. dk dq dp ; Ž 1.14 . 2p Hence, the classical and quantum evolution equations only differ by terms of higher order in ". Moreover, for potentials quadratic in q, ˆ higher order terms cancel out and the quantum evolution equation coincides with the classical one. This leads to the remarkable result that there is no difference between the evolution of the distributions of probability for quantum and classical observables, when the system is described by a Hamiltonian quadratic in positions and momenta. For this kind of systems, the propagator is the same w5x. Of course, what makes the difference is the initial condition.

H

2. Generalization to N degrees of freedom We consider now a system of N interacting particles sitting on the sites of a lattice Žwe choose it to be one-dimensional for simplicity.. The Hamiltonian of the system is N

Hˆ s

Ý is1

pˆi2 2 mi

q V Ž qˆ . .

Ž 2.1 .

We assume the masses to be equal to unity. We take the potential to be of the form: N

V Ž qˆ . s

Ý is1

ž

1 2 a2

Ž qˆiq1 y qˆi .

2

q U Ž qˆi .

/

Ž 2.2 .

where a is the lattice spacing and UŽ qˆi . is the part of the potential which depends only on the position of the i-th particle. This specification is not essential for the purposes of this section, but it will become necessary for

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understanding the limit to the continuum, which will be considered in next section. The quantum characteristic function for the N dimensional system may be defined as

x Ž k 1 ,... k N . s² e i Ý k i Xˆ i :, Ž 2.3 . i ˆ ˆ ² : Ž . Ž where A s Tr rˆ A , and rˆ is the density operator of the system. In case there is no interaction between different sites of the lattice the density operator may be factorized and the characteristic function is just the N product x Ž k 1 ,... k N . s P is1 x Ž k i . .. Performing the Fourier transform of Ž2.3. the MDF is then given by 1 ˆ w Ž Xs , ms , ns ,t . s dk 1 ... dk N eyi Ý k i X i² e i Ý k i X i : , Ž 2.4 . N i i Ž 2p .

H

where s is a collective index. It may be shown that this is a probability distribution, namely that it is positive definite and normalized, provided Xˆi are observables. The proof goes along with the one-dimensional case. We first suppose that there is no interaction between different sites at some initial time t 0 and we assume for simplicity that the system be in a pure state < c : s < c 1 : m ...m < c N :. Using the factorization property of the quantum characteristic function, Eq. Ž2.4. may be seen to reduce to the product w Ž Xs , ms , ns ,t 0 . s Ł i r i Ž X i , X i ,t 0 . s r Ž X, X,t 0 ., which is positive and normalized. Then the Liouville equation guarantees that this result stays valid when the interaction is switched on. In analogy with the one-dimensional case we now introduce the variables X i s m i q i q n i pi Ž 2.5 . with Xˆi accordingly defined. Introducing the notation < Zs : ' < Z1 : m ...m < ZN : we rewrite Ž2.4. as w Ž Xs , ms , ns ,t . s

N

N

1

Ž 2p .

N

N

ˆ

Ł dk i dZi eyi Ý k j X j ² Zs < rˆ e i Ý k l X l < Zs :

H is1

js 1

ls1

N

1

N P is1 dk i

yi

dZi r Ž Zs ,Zs y ks ns " . e N H Ž 2p . where r Ž Zs ,ZsX . s ² Zs < rˆ < ZsX :. Performing the change of variables s

ZiX s Zi ,

Ý k w X y m Ž Z yk n "r2.x , j

j

j

j

j

Ž 2.6 .

j

jq 1

ZiXX s Zi y k i n i " ,

Ž 2.7 .

we have w Ž Xs , ms , ns ,t . s

N

1

Ž 2p " .

N

H is1 Ł

ž

dZiX dZiXX
ZXj y ZXXj

N

/

r Ž ZsX ,ZsXX ,t . exp yi

Ý

nj "

js1

ž

Xj y mj

ZXj q ZXXj 2

/ Ž 2.8 .

which can be inverted to

rŽ

Xs , XsX

,t . s

N

1

Ž 2p .

N

H is1 Ł dm

i

dYi w Ž

Ys , ms , Xs y XsX

. exp

N

i "

Ý js1

ž

Yj y m j

X j q X jX 2

/

.

Ž 2.9 .

Once again, we recall that the inversion of Ž2.8. is made possible by choosing the variables X i as in Ž2.5.. We now use the Liouville equationŽ1.8. to get N

yi w˙ Ž Xs , ms , ns ,t . s

Ž 2p " .

N

H is1 Ł

ž

1
N

=r Ž

Zs ,ZsX

"2

N

dZi dZiX

,t . exp yi Ý ls1

/

Ý

y 2m

js1

Zl y ZlX

nl"

ž

ž

Xl y ml

E2 E Z j2

E2 y

Zl q ZlX 2

E Z jX 2

/

.

/

q Ž V Ž Zs . y V Ž ZsX . .

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333

Integrating by parts and assuming the density matrix to be zero at infinity, we finally have w˙ Ž Xs , ms , ns ,t . s

N

½

E mi

Ý

En i

is1

E

žž /

V y

" y1

E

žž /

yV y

y1

E

i q

E Xs

E Xs

E Ems

q

y

Ems

i ns "

E

2

E Xs

/5

i ns "

E

2

E Xs

/

w Ž Xs , ms , ns ,t . ,

Ž 2.10 .

where the inverse derivative is defined as in Ž1.11.. We report for future convenience the term containing the potential when explicitating the interaction between neighbours: y1

E

E

žž / žž /

V y

E Xs

Ems

E

s U y i" y a

2

y

y1

E Xs

N

E Xi

is1

E

2

E Xs

E y

Ems E

Ý ni

i ns "

i ns " 2

ž

E X iq1

E

/ žž / / žž / yV y

E Xs

Ems

E

E

yU y

E Xs

y1

E

y1

E

E

/

Em iq1

E

Em i

y1

E Xs

E y2

q

q

E Xi

i ns "

E

2

E Xs

E Ems

ž

q

i ns "

E

2

E Xs

y1

E E X iy1

/

/ /

E .

Em iy1

Ž 2.11 .

When considering the classical limit we have w˙ Ž Xs , ms , ns ,t . s

½

N

E

Ý

mi

is1

En i

q ni

E

E

1 a

2

y1

E Xi

ž

E X iq1

E

E

Ems

E Xs

žž / / 5

qVi y

E Xs

y1

E

/

E Em iq1

E y2

Em i

E q

w Ž Xs , ms , ns ,t . ,

E Xi

ž

E E X iy1

y1

/

E Em iy1

Ž 2.12 .

where Ui is the derivative of the self-interaction potential with respect to the i y th variable. Eq. Ž2.13. may be seen to be equivalent to the Boltzmann equation as in the one-dimensional case. Moreover, Hamiltonians which are quadratic in positions and momenta yield the same evolution equations for classical and quantum probability distributions.

3. Generalization to field theory We now consider a scalar quantum field theory described by the Hamiltonian d

Hˆ s d d x

H

1 2

pˆ 2 Ž x . q 12

Ý Ž E b fˆ Ž x . .

2

q U Ž fˆ Ž x . . .

Ž 3.1 .

bs1

d is the spatial dimension, while UŽ f Ž x .. is the self-interacting potential, polynomial in the field fˆ . The Hamiltonian Ž3.1. is easily seen to be obtained by the discrete Hamiltonian Ž2.1. by taking the limit to the continuum Ž a ™ 0. with the following rules: ayd r2 qˆi ™ fˆ Ž x . , ad Ý ™ d d x ,

H

i

ayd r2 pˆi ™ pˆ Ž x . ,

ayŽ d r2q1. Ž qˆiq1 y qˆi . ™

Efˆ Ž x . E xb

.

Ž 3.2 .

V.I. Man’ko et al.r Physics Letters B 439 (1998) 328–336

334

In analogy with the discrete case we introduce the field

Fˆ Ž x . s m Ž x . fˆ Ž x . q n Ž x . pˆ Ž x .

Ž 3.3 .

yd r2

yd r2

where m Ž x . s lim a™ 0 a m i , and n Ž x . s lim a™ 0 a ni . The quantum characteristic functional, which now will play the role ˆ of generating functional for correlation functions of the fields, may be defined as

x Ž k Ž x . . s e iHd

¦

d

d i d x k Ž x . Fˆ Ž x .

x k Ž x .Fˆ Ž x .

;s Tr ž rˆ Ž t . e H

/.

Ž 3.4 .

F ., The functional Fourier transform of x Ž k Ž x .., what we will call the marginal distribution functional ŽMDF still defines a probability distribution. This can be understood by recognizing that it is the limit of the MDF for the discrete N-dimensional system considered in the previous section: w Ž F Ž x . , m Ž x . , n Ž x . ,t . s D k eyiHk Ž x .F Ž x . d x x Ž k . s lim w Ž Xs , ms , ns ,t . ,

H

where Ł i

dk i Ž2 p . N

Ž 3.5 .

a™0

™ H D k. Also, the density matrix functional may be defined as the limit of Eq. Ž2.9. to be

X

X

r Ž F ,F ,t . s Dm DC w Ž C , m ,F y F . exp

H

½

i

Hdy C Ž y . y m Ž y .

"

F Ž y. yFX Ž y.

ž

2

/5

.

Ž 3.6 .

Then, the evolution equation for the probability distribution functional is easily obtained by taking the limit of Ž2.11.: w˙ Ž F Ž x . , m Ž x . , n Ž x . ,t . s

½

Hd i

q "

yU

d

d x mŽ x .

U

ž

ž

dF Ž x .

dF Ž x .

q 2n Ž x .

dn Ž x .

yd

yd

d

y1

/

/

d dm Ž x .

y1

dF Ž x .

d q

y

The inverse functional derivative Ž drdF Ž x . .

ž

d dF Ž x .

y1

/

yi k Ž y .F Ž y . d y

H Dk e

H

in Ž x . "

d

2

dF Ž x . d

2

dF Ž x .

dm Ž x .

y1

/

d dm Ž x .

5 Ž 3.7 .

is so defined: i

s

dF Ž x .

in Ž x . "

=w Ž F Ž x . , m Ž x . , n Ž x . ,t . y1

ž

D

d

yi k Ž y .F Ž y . d y

H Dk k Ž x . e

H

,

Ž 3.8 .

while the notation Dw f Ž x .x stands for f Ž x q D x . y f Ž x .. Performing an expansion in powers of " the classical limit may be obtained as in the previous sections.

4. The quantum characteristic function as a generating function In this section we discuss the connection between the probability representation described above both for quantum mechanics and quantum field theory and a slightly different point of view developed in Ref. w6x, where evolution equations are found for a suitably defined euclidean partition function. There are two main ingredients in our approach: one is is the probabilistic interpretation for the distribution describing the observables, the other

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335

is the equivalence between the description based on the MDF and the conventional description based on the density matrix. The first aspect is guaranteed by the Glauber theorem which states that the Fourier transform of the quantum characteristic function associated to observables is a probability distribution. The second aspect, namely the invertibility of the MDF in terms of the density matrix, is achieved by introducing configuration space variables which take value in an ensemble of reference frames in phase space, each labelled by the two parameters, m , n . Thus, the evolution equations which we have found ŽŽ1.10., Ž2.11., Ž3.7.., together with suitable initial conditions, completely characterize the state of the given quantum system. These equations assume a simpler form when their Fourier transform is performed. We have

x Ž k , m , n ,t . s dXe i k X w Ž X , m , n ,t . ,

Ž 4.1 .

H

with obvious generalizations to the N dimensional case and to field theory. For the one-dimensional quantum systems considered in Section 1, the Fourier transform of Eq. Ž1.10. yields an evolution equation for the quantum characteristic function itself:

x˙ Ž k , m , n ,t . s

½

E

1 m

m

i

En

V

q "

ž

1 E

kn "

ik Em

y 2

/ ž yV

1 E

kn "

ik Em

q 2

/5

x Ž k , m , n ,t . .

Ž 4.2 .

For the N-dimensional quantum systems considered in Section 2, the Fourier transform of Eq. Ž2.11. yields

x˙ Ž ks , ms , ns ,t . s

½

N

1 m

yV

E

Ý mi En

is1

ž

i "

i

E

1

iks Ems

V

q

q

ž

ns ks " 2

1

E

iks Ems

/5

y

ns ks " 2

/

x Ž ks , ms , ns ,t . ,

Ž 4.3 .

while Fourier transforming Eq. Ž3.7. we have, for quantum field theory,

x˙ Ž k Ž x . , m Ž x . , n Ž x . ,t . s

½

Hd

d

x

qi U

mŽ x .

m

dn Ž x . d

1

y

y 2 i"k Ž x . n Ž x . D

d

ik Ž x . dm Ž x .

q

1

d

ik Ž x . dm Ž x .

kŽ x.n Ž x. "

ik Ž x . dm Ž x . 1

yU

d

1

2 kŽ x.n Ž x. "

5

2

x Ž k Ž x . , m Ž x . , n Ž x . ,t . .

Ž 4.4 .

Now the comparison with the results of w6x may be easily understood. Let us stick to quantum field theory for definiteness. The quantum characteristic functional which is a generating functional for correlation functions of the F field coincides with the generating functional considered in Ref. w6x Z Ž mX , n X ,t . s Tr rˆ Ž t . exp

ž

½H

d d x mX Ž x . fˆ Ž x . q n X Ž x . pˆ Ž x .

5/

,

Ž 4.5 .

after rescaling the parameters m and n to mX s ik m and n X s ik n Žof course, the same holds for quantum mechanics.. Consequently, the evolution equations for the characteristic functional may be seen to be equal to those found in Ref. w6x for the generating functional Ž4.5. provided the parameters mX and n X are rescaled as specified. Going back to the initial remark of this section, we may conclude that, the quantum characteristic function and its evolution equation Žor the generating functional in Ž4.5.. are more interesting from an operative point of view as they determine the correlation functions and their time evolution. On the other hand the introduction of

336

V.I. Man’ko et al.r Physics Letters B 439 (1998) 328–336

the MDF is both relevant and necessary from a theoretical point of view. In facts it allows a unified description of classical and quantum phenomena in terms of probability distributions obeying different evolution equations. Also it justifies the introduction of the X variable, as a variable taking values in an ensemble of reference frames Ž1.2., in view of the invertibility of the MDF in terms of the density matrix. This seems to us the profound motivation for introducing such a combination of phase space variables in the quantum characteristic functional and in the generating functional Ž4.5.. We stress once again that X and its field analogue F are, for each couple Ž m , n ., configuration space variables in the transformed reference frame labelled by Ž m , n ..

5. Conclusions In this letter we have presented an extension of the probabilistic representation of quantum mechanics to quantum field theory. In this framework classical and quantum phenomena, both statistically described, only differ by the evolution equations of the distributions of probabilities for the relevant observables. Quantum observables are described by a distribution of probability, the MDF, and the time evolution by an integro-differential equation for the MDF. We recently addressed the problem of finding the Green’s function for the time-evolution equation of the MDF w10x. The problem was solved for quadratic Hamiltonians, and a characterization of such a propagator in terms of the time-dependent invariants of the system was found. This propagator represents the transition probability of the system from a quantum state to another. Thus, a generalization to quantum field theory would be interesting in our opinion, and is presently under consideration. Another promising application of the probabilistic point of view is suggested in Ref. w6x where it is used to study the approach to equilibrium of non equilibrium quantum field theories. An extension to relativistic quantum field theory would be also interesting, although it poses problems of interpretation which are not understood at the moment.

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