Stochastic quantization in Minkowski space

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

STOCHASTIC QUANTIZATION IN MINKOWSKI SPACE H. H O F F E L and H. RUMPF lnstitut fi~r Theoretische Physik, Universitat Wienl A-1090 Vienna, Austria Received 15 June 1984

We propose a generalization of the euclidean stochastic quantization scheme of Parisi and Wu that is applicable to fields in Minkowski space. A perturbative proof of the equivalence of the new method to ordinary quantization is given for the self-interacting scalar field. It is argued furthermore non-perturbatively that the method generally implies the Schwinger-Dyson equations.

Parisi and Wu [1] introduced a stochastic quantization scheme which is based on the Langevin equation of non-equilibrium statistical mechanics. The method can be applied to the quantization of scalar, Dirac, and gauge fields in euclidean space-time [2]. In this paper we propose a modification of the original Parisi-Wu approach allowing to quantize scalar fields in Minkowsld space right away: We introduce a Langevin equation with complex drift term; the Green functions in Minkowski space are obtained as a "weak equilibrium limit" in the distributional sense of the correlation functions of the complex process defined by the generalized Langevin equation. We give a perturbative proof of the equivalence of the quantization method for the case of a self-interacting scalar field by using a diagrammatical technique. In addition we supply a non-perturbative argument that the Schwinger-Dyson equations will hold in general for the Green functions obtained by the weak limiting procedure. We propose to replace the Langevin equation of Parisi and Wu [1] for the euclidean scalar field ~/i

O~( x, t ) /Ot = --SSE[ ~b]/SdP( x, t) + 71(x, t)

(1)

by the following generalized Langevin equation for the field # in Minkowski space:

d ~ ( x , t)/Ot = i ( S S [ * ] / 8 ~ ( x , t)) + *l(x, t).

(2)

Here S E and S denote the action in euclidean and Minkowski space, respectively; the fictitious time parameter t should not be confused with the physical time x 0. The random source 7/is a gaussian white noise with correlation function

( Ti( x, t )Tl( x', t))n = 2(2*r)48(t - t')g(4)( x - x').

(3)

Note that the process ~(x, t) defined by eq. (2) is complex and that the existence of an equilibrium for t --* oo constitutes a more subtle problem than in the euclidean case. Actually what we are going to show is that the equilibrium limit exists in the distributional sense, namely that the correlation functions converge to the quantum Green functions for t --* oo only when interpreted as tempered distributions. 104

0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

As an example we discuss the stochastic perturbation theory of a self-interacting scalar field with lagrangian L = ½(0¢)

2 --

(g/3!)

¢3

(4)

(we have put m 2 = 0 only for convenience; moreover all our results can be generalized for any polynomial self-interaction). The corresponding generalized Langevin equation is (5)

= - i ( r q ~ + ½g¢2) +*lWe consider first the free case g = 0. In this case the unique solution of (5) with initial condition ,/i(t = 0) = 0 is simply

(6)

¢(°)(k, t) = fo/drexp [ik2(t - r)] ~l(r). Owing to the white noise correlation function (3) of ~1 the correlation function of the free field ¢(0) is (~(O)(k,t) ¢(0)(k,, t') )n = i(2~r)'8(4)(k

=

f0min(t' '') dCexp [ik2(, + t ' - 2~')] 2(2~r)48(4)(k + k')

+ k')k- z{ exp (ik2lt -/'1)

- exp [ i k 2 ( / + / ' ) ] }.

(7)

(8)

Now although the limit t = t' --* o0 of (8) does not exist in the ordinary sense, the expression converges if interpreted as a tempered distribution (see e.g. ref. [3]) of k, k', which is the proper interpretation in quantum field theory (all states being defined in terms of smooth wave packets). Starting from the definitions [3] it is easy to prove the following relations that hold in the theory of tempered distributions: lim exp(ixs) -- 0, s~o~

lim P ( 1 / x ) exp (ixs) = i~rS(x)

(9)

$ - - ~ oo

where P denotes the principal value. Therefore

(¢I'(°)(k,t)¢(°)(k',t'))=

lim t~t'--,

i(21r)48(4)(k +

k')[P(l/k 2) -irrS(k2)]

oo

1

(10)

= i(2~r)48(4)(k + k') k2 + i0"

We have thus obtained the Feynman propagator from the generalized stochastic quantization. An alternative, and less sophisticated, way of arriving at (10) is to add a negative imaginary mass term - l i c ¢ 2 to the lagrangian (4) and let c tend to zero after all calculations have been performed. Then the second term in (8) is exponentially damped, since k 2 is replaced by k 2 + ic. Let us now turn to first order perturbation theory. The first order correction to (6) is ~(1)(k,/)

=

_ i f o t d ~ . e x p [ i k ~ ( t _ r)] g

( d4P ~(O,(p, r)~(O,(k3_p,~.). "

J

(11)

In order to calculate the three-point Green function we consider the three-point correlation function which 105

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

is given to first order by

(1~(0) (kl ' / ) ~b(o)(k2 ' t) ~O)(k3 ' t))7 + 2 similar terms.

(12)

A typical contribution to (12) is -ifotd~'ex p [ i k 2 ( t - ~ - ) l g

f

d4~P4( ~ ( ° ) ( k l , t ) ~ ( ° ) ( p , ~ ) > n ( ~ ( ° ) ( k 2 , t ) ~ ( ° ) ( k 3 - p , t ) )

(2¢r)

n

= ½ig(2¢r)43(i)(kl + k 2 + k3) k21k2 f j d r e x p [i(k~ + k 2 + k 2)t ]{ e x p [ - i ( k 2 + k22 + k 2 )'r] - e x p [ - i ( k 2 - k 2+ k 2)~] - e x p [ - i ( - k

2 + k 2+ k 2)~-] + e x p [ - i ( - k

t - k 2+ k 2)~-]}

(13)

which tends (upon addition of the small imaginary mass term to the lagrangian) for t ~ oo to 1

1

1

½g(2~)48(4)(kl + k 2 4- k3) - - k 2 + i 0 k22+i0 k2 + k2 + k2 + iO "

(14)

Adding finally the terms arising from all the possible permutations in k 1, k2, k 3 one obtains

( ~ ( k l , t)dP(k2, t)~b(k 3, t))~nl)t~-~°* _ g(2~r)4/j(')(kl + k 2 + k3) - -1

1

1

_ _ _ _ k 2+i0k 2+i0 k~+i0'

(15)

which complies exactly with the result of ordinary perturbation theory. We proceed to show the equivalence of our proposed quantization scheme to the usual Minkowski space quantization for all orders of perturbation theory. We shall use a diagrammatic method that was employed in a similar proof [4] for stochastic quantization in euclidean space. For more details on the common features of that proof and the present one, the reader is referred to ref. [4]. Transforming the Langevin equation (2) into an integral equation and solving the equation by iteration we arrive at a power series expansion of the field ~ in the coupling constant(s), expressing ~(x, t) as a functional of the white noise 7/. We consider an N-point correlation function (~(Xx, t ) . . . ~ ( x N, t)>n and substitute for • its perturbative expression. When the random average over 71 is taken diagrams are obtained which we call stochastic diagrams. They have the form of an ordinary Feynman diagram apart from crosses on the lines where two ,/'s have been joined together (these crossed lines represent the two-point correlation function). To every Feynman diagram there exists a number of stochastic diagrams of the same topology. We are going to show by induction that the sum of all stochastic diagrams corresponding to a given Feym'nan diagram yields just this Feyuman diagram in the weak limit t ~ oo. Assume, then, that we have proved this equivalence for all numbers of vertices smaller than N with any number of external lines L (the case N = 1 was already demonstrated above). Consider a stochastic diagram S F belonging to some Feynman diagram F with N vertices and L external lines, each vertex carrying a (fictitious) time ~'i which has to be integrated over. As in ref. [4] we introduce time orderings of the vertices and perform the time integrations starting with the lowest vertex time and going successively to higher ones. Leaving out the momentum denominators of the two-point correlation functions, and dropping the second term in the bracket in (8) (which does not contribute as t ~ oo), we get due to the ordering of vertex times for the k th time integration

( - ' g•) f 106

"0

d~.kexp ( _ , ~ . k ~ p / = ( _ i g ) i ( ~__,( p ~ wk ! wk

+i0/)

--1

"'',

wk

(16)

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

where the elements of the index set W k label all those momenta which attach to the vertices corresponding to the time z 1, ~'2. . . . . r k but which do not connect any two of them. The other momenta do not show up as their contribution has been cancelled by the preceding ~'i integrations. Strictly speaking this is true only if the contributions of the lower integration boundaries can be neglected. But this is indeed the case for t ~ o0 (weak limit). The same remark applies to the lower integration boundary of the ~'k integration itself, whose asymptotically vanishing contribution is indicated by the dots on the right-hand side of (16). The asymptotic suppression of these contributions occurs through multiplication with the Langevin Green function (appearing in the integrand on the fight-hand side of (6)) or the two-point correlation function corresponding to every uncrossed or crossed external line of the diagram, respectively. Another consequence of these multiplications is that the whole exponential

exp(i

p2) external momenta

(that remains after the TN integration) is reduced to unity. Now given a fixed time ordering of vertices with, say, ~'N maximal (hence the corresponding vertex being external), we define a stochastic diagram SF,, which is obtained from S F by dropping the "ru vertex with its attached external line(s); similarly we define the Feynman diagram F ' . The essential ingredient of the proof is now the fact (following from (16) and the accompanying remarks) that the N-fold z integration of S r equals (up to a simple combinatorial factor) the ( N - 1)-fold ~- integration of SF, times a factor K which is given by

K=(-ig)i

(

E

(P 2+i0)

)1

(17)

external momenta

As S F, contains N - 1 vertices, the induction assumption may be used so that upon summing and integrating over all time-ordered stochastic diagrams S r (keeping ~N as the largest time) we obtain essentially KF" in the limit t ~ oo. To complete the proof one has to show that the sum over all possible places for the largest vertex time ~'N just gives the Feynman diagram F. For the case of single external lines attached to the external vertices this follows immediately from the identity

1

--

--)-~-PN

2 ~1o~F ' - K = 2 1o~

=]Ep~ loe

E

(P 2+i0)

(-ig)

F"

external momenta

E (P2+i0)

F=F,

(18)

k external momenta

where ~'loc is the sum over all positions of the N t h vertex and PN is the external momentum attached to it. The factor 1 / 2 on the left-hand side of (18) is the exact result of the sum over stochastic diagrams. The exact result for this sum in the case of double external lines is

1 1 )KF'= P2,1 +P~,z 2 K F ' = ( p21 + pN,2)KF. 2 ~ ~ 2 PN,1 + i 0 PN,2 + i0 (PN,1 + i0)(p2,2 + i 0 )

(19)

Summation over all locations again yields F. This completes the diagrammatic proof. 107

Volume 148B,number 1,2,3

PHYSICSLETTERS

22 November1984

In the euclidean case the validity of the Parisi-Wu method has been shown also non-perturbatively for non-gauge theories [2]; for gauge theories see ref. [5]. All these non-perturbative proofs are based on the Fokker-Planck equation associated with the stochastic process under consideration. In minkowskian field theory the Fokker-Planck equation for the positive definite probability distribution describing the complex process defined by (2) apparently ceases to be a useful tool: Due to the complicated structure of the equation no obvious candidate for an equilibrium distribution could be found. However it was observed recently [6] that one can define a complex valued probability distribution on the real axis such that { F [ ~ , t])~ = / D [ ~ ]

t'[~R, *] F [ ~ R ] ,

(20)

where P[~R, t] satisfies the Fokker-Planck equation (of the usual form) for the real field q~p, subject to the complex drift force i S S [ ~ a ] / 8 ~ R. We expect from the perturbative treatment of the Langevin equation that (20) makes sense and that P[ ~/iR, t] approaches indeed its formal equilibrium limit (i.e. the stationary solution of the Fokker-Planck equation) peq[ ~ , ] = e x p (iS[ #R l) /

fD[ a,. ]exp (iS[*R]).

(21)

We do not attempt in this paper to perform a rigorous treatment of the existence and the convergence properties of P[~R, t]. Rather we formulate the relaxation hypothesis by postulating that lira { ~ ( x , t ) F [ ~ ] ) , = 0

(22)

t---~ O0

for any functional F[~]. In the euclidean case (22) can be derived from the Fokker-Planck equation and the existence of an equilibrium limit [2]. In the following we prove that (22) implies the Schwinger-Dyson equations for minkowskian field theories. The proof generalizes elements of the equivalence proofs given by Nakano [7] and Alfaro and Sakita [8] for the euclidean case. Consider the functional Zt[J ] defined by

We have

O=fD[nl n(x,t ) exp( -1 -4fo dtfd4x'~2+ifd4x'J(x')~(x"t))

(24)

Upon using the Langevin equation and [81

8e(x',

t) -- ½8(')(x - x'),

(26)

(25) becomes

( - ½i~(x, t) + ½i[SS/Se(x, t)] + ½i J(x, t))s, c 108

(27)

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

Now (22) implies

( (~( x, t ) ) j, tt---~ O.

(28)

Hence

( i( SS[xl/SX)lx=(1/i)8/s J + iJ )Zt[ J ] ---) 0,

(29)

i.e. Zt[J ] obeys in the limit t ---, o0 the same functional differential equation as the conventional generating functional

Z[J]= f D [

~] exp ( i S [ ~ ] + i f

d4XJ(x)~(X))

(30)

of Minkowski space quantum field theory. As is well known [9], this functional differential equation implies the Schwinger-Dyson equations for the Green functions of the theory. Note that for linear field theories (29) implies formally lim

t---~ O¢

Zt[J ] = Z[J],

(31)

owing to the normalization condition Zt[0] = 1. The general validity of (31) requires e.g. the existence of the functional Fourier transform of lim ZAJ ] which is not obvious to us. The above proof may be extended even to gauge theories, if "stochastic gauge fixing" [10] is introduced. As an example we consider scalar electrodynamics. If we transform Aa(x, t), q~(x, t) and ~*(x, t) to

Ba(x,t)=A,(x,t)-O~X(X,t),

~(x,t)=exp[-iex(x,t)]q~(x,t),

O*(x, t) = exp [iex(x, t)] ,~*(x, t),

(32)

then the Langevin equation for Ba picks up an additional drift term due to the dependence of X on the fictitious time t:

Ba(X,l)

= i(BS[ B, ~, ~* ]/SBa(x,t)) - Oa~((X,t ) -b il/2B~(X, t).

(33)

S[B, ~, ~*] is the action of the Maxwell field (without gauge fixing term), coupled minimally to the charged scalar field; ~la denotes a gaussian "quantum variable" with formal path integral measure D[~] exp (¼i7/2). It is important to note that one may disregard the additional drift terms - i e x ~ , ie~(~* in the Langevin equations for ~ and ~*, so that we have ~(x,t)=i(SS[B, ~, ¢*]/S¢(x,t)) +~(x,t).

(34)

This fact can most easily be understood by considering a generic element

F= f d x d y d z [q~(x)~*(y)]"A(z)"

(35)

of a gauge invariant functional of the fields. Under infinitesimal gauge transformations 8~, 8~* one has

( SF/Sq~ )aqJ + ( 3F/aqJ* )~q~* =

0.

(36) 109

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

Therefore the time evolution of gauge invariant quantities is not affected by the additional terms under discussion, so that they m a y be disregarded. The new drift term for the gauge field B~ renders possible the existence o f a stationary solution of the F o k k e r - P l a n c k equation. Indeed if one chooses

= ia-lOaB~

(37)

then eq. (29) implies for t ~ oo just the S c h w i n g e r - D y s o n equations of ordinary q u a n t u m field theory with the covariant gauge fixing term ( 2 a ) - t ( a B ) 2 added to the action S[B, ~, ~*]. W e conclude with the following remark. It m a y look superficial to insist on a Minkowski space f o r m u l a t i o n of q u a n t u m field theory with all its mathematical complications, as the existence of a well-defined euclidean counterpart is usually taken for granted. However this is not true in the important case of the gravitational field. The main motive of this paper was to provide the prerequisites for the stochastic quantization of the gravitational field. The latter will be the subject of a forthcoming paper. W e acknowledge helpful discussions with B. Baumgartner, G. Ecker, H. Grosse, and H. Rupertsberger. H . H . is also very grateful to M. Mintchev and G. Parisi for valuable discussions.

References [1] G. Parisi and Wu Yong-Shi, Sci. Sin. 1 (1981) 483. [2] J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B233 (1984) 61, and references cited therein. [3] M.J. Lighthill, Introduction to Fourier analysis and generalized functions (Cambridge U.P., London, 1958). [4] W. Grimus and H. Hi~ffel, Z. Phys. C18 (1983) 129. [5] L. Baulieu and D. Zwanziger, Nucl. Phys. B193 (1981) 163. [6] G. Parisi, Phys. Lett. 131B (1983) 393. [7] Y. Nakano, Prog. Theor. Phys. 69 (1983) 361. [8] J. Alfaro and B. Sakita, in: Gauge theory and gravitation, Proc. Intern. Symp. (Nara, Japan, 1982) (Springer, Berlin, 1983). [9] C. Itzykson and J.-B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). [10] D. Zwanziger, Nucl. Phys. B192 (1981) 259.

110