Stochastic quantization, non-markovian regularization and renormalization

Nuclear Physics B300 [FS22] (1988) 128-142 North-Holland, Amsterdam

STOCHASTIC QUANTIZATION, NON-MARKOVIAN REGULARIZATION AND RENORMALIZATION

Roberto IENGO International School for Advanced Studies, (SISSA), Trieste, Italy International Centre for Theoretical Physics, Trieste, Italy and Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Trieste, Italy

Sergio PUGNETTI International School for Advanced Studies, (SISSA), Trieste, Italy and Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Trieste, Italy

Received 9 February 1987 (Revised 26 October 1987)

We consider the stochastic quantization of field theories, generalized to suitable nonmarkovian processes which can be used as an analytic ultraviolet regularization, and also to provide a new approximation scheme for computing critical exponents. We discuss the renormalization procedure for the general case, making use also of the recently introduced methods based on BRS transformations, and we present a general formulation of a background field method for the Langevin equation.

1. Introduction and summary W e c o n s i d e r here the stochastic q u a n t i z a t i o n m e t h o d [1], b y which the G r e e n f u n c t i o n s o f a euclidean field theory are defined b y i n t r o d u c i n g a fictitious time a n d w r i t i n g the following Langevin e q u a t i o n 0

6S

o~-:-:'P(x; t) + ~,~(x;)t ~ - ,l(x; t).

(1.1)

Here, S is the classical action, ¢p = ep(x; t) is a scalar field on a d - d i m e n s i o n a l e u c l i d e a n s p a c e evolving in the time t a n d 71 is a white G a u s s i a n noise, i.e. a s t o c h a s t i c p r o c e s s with the correlation

(,l(x, t),l(x', c)) = 2~(t - t') ~"(x - x'). 0169-6823/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(1.2)

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×

+2

×

+0

(Z')

Fig. 1. E x p a n s i o n in tree diagrams.

The Green functions are then obtained by considering the T-~ oo or equilibrium limit of the average with respect to */ (cp(xl)...cp(xn)) = lim T-~oO

r)),.

(1.3)

The perturbative solution of this equation can be obtained by defining a stochastic propagator G ( x - y ; t - t') as the inverse of the operator acting linearly on cp in the Langevin equation and expressing cp as a sum of tree diagrams [2]. The branches of the tree end on the noise ~ (which will be graphically denoted by a cross), each line is a stochastic propagator G and each vertex, generated by the nonlinear part of ~ S / 8 ~ , carries a coupling constant (for an example see fig. 1). The right-hand side of eq. (1.3) is then represented by a sum of stochastic graphs obtained by contracting together the noise ~ according to eq. (1.2), which will give rise to loops with crossed and uncrossed lines (for an example see fig. 2). We remark that the symmetries of the system and the related Ward identities also hold for the stochastic graphs [3]. It has been pointed out [4] that it is convenient to generalize the previous picture to the case where the correlation of the Gaussian process ~ is different from 8 ( t - t'), implying a non-markovian process; namely, one can consider

(~/(x; t)r/(x'; t')) = 2ao(t- t')8~(x -x').

(1.4)

We have introduced a parameter o such that l i m a~( t - t') = 8 ( t - t ' ) . o "-'~0

(1.5)

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-4-

(e3) Fig. 2. Graphical representation of the average.

The main motivations for this generalization rely on the renormalization of the theory, and in fact we can consider two related perspectives. First, one can introduce a o ( t - t') as a device for regularizing the theory. By some suitable choice of ao one obtains that the divergences appear as poles in o, and one can devise a minimal-subtraction scheme in which the poles are reabsorbed into the renormalization constants, while keeping the dimensionality d fixed, for instance d = 4 in the gep4 theory. The advantage of keeping the dimensionality fixed should be relevant, for instance, in the case of supersymmetric theories [5]. Second, one can consider a theory defined through the Langevin equation for o ~ 0. This should give a non-markovian theory where the average (~o... ~p)n is not equal to the corresponding physical Green functions of a euclidean field theory. One can nevertheless imagine a way to renormalize that theory, which would be renormalizable for values of the dimensionality d related to o, and also introduce the corresponding renormalization group equations, and study the critical phenomena. This appears to be convenient because of the observation that for d fixed, say d = 3, for the typical critical phenomena one is interested in, there exists a particular value o = o* for which the non-markovian theory is renormalizable and asymptotically free in the infrared. This provides a systematic method for a subsequent extrapolation to the physical point o = 0, in the same spirit of the e-expansion algorithm for computing critical exponents, here (o - o*) playing the role of the expansion parameter [6]. Of course, since our aim is that of reaching the physical theory by continuing o to o = 0, we take the analytic continuation in o as our tool for computing Green functions. Then, also, our discussion of the renormalization will be based on this continuation. The u.v. divergences appear then, as said above and similarly to what happens for the analytic continuation in the dimensions, in form of poles in o and

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131

we will verify that they indeed correspond to the possible logarithmic singularities one can foresee by power counting. As is well known, the renormalization schemes based on a regularization obtained by an analytic continuation differ from the schemes based on a cut-off in the treatment of divergences higher than logarithmic. In particular, the analytic regularization avoids the introduction of additional dimensionful terms in the problem (i.e. relevant operators), as is suitable for studying the critical behaviour, and this will be an ingredient of our algorithm. Numerous questions arise when considering the renormalization pattern of the non-markovian theory. In this paper we will discuss the features of the theory defined in general by eqs. (1.1) and (1.4), including the markovian and nonmarkovian cases. First of all, in sect. 2, we will discuss a proper definition of the non-markovian case and we will introduce a particularly suitable a o satisfying eq. (1.5), i.e. the one used to provide an analytic regularization [7]. We also discuss, by means of perturbation theory, the convergence of the T---> ~ limit defined in eq. (1.3) for the general non-markovian case, for the Fokker-Planck equation cannot be derived. Second, in sect. 3, we will discuss the renormalizability of the theory. The problem of the renormalization for stochastic processes described by a Langevin equation has been extensively considered in a different framework, i.e. in the study of the Dynamical Critical Phenomena ([8], [9] and references therein). Here, we will see (by applying suitable, recently introduced, techniques based on BRS invariance [10]) that it is possible to renormalize in general both the markovian and the non-markovian theory by introducing renormalization constants in the Langevin equation

ZtO/~= -[(-Z~,O,O~+ma+rm2)ep+ Z,,SV/dd?l + z,l*l,

(1.6)

where we consider, as examples, the two cases: V = gRq54/4! and V = ~Rff3/3!. In fact, if we study the renormalization properties of the stochastic quantization in the path-integral formulation [10], based on the functional integration over ghost and auxiliary fields, it turns out that, to renormalize the effective action A, appearing in this formulation, it is sufficient to introduce the counterterms directly in the Langevin equation as in eq. (1.6). In the particular markovian case there is an additional supersymmetry of the effective action A, which implies Ward identities among some renormalization constants. In sect. 4 we will complete the discussion about the renormalization, introducing also a general formulation for the background field method based on the Langevin equation. As an example, we will consider the interaction 2,4~3 and discuss also the differences between the markovian and the non-markovian cases. 2. Definition of the non-markovian ease

Let us now discuss the possibility of introducing a consistent non-markovian theory, i.e. a suitable noise measure and a convergent (for T ~ oo) perturbative

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expansion. In order to define the measure [dpO1)] for the Gaussian process, we look for the " m o m e n t u m representation", i.e. we describe ~(t) by means of its Fourier transform

(2.1)

1 fdte_i,,rl(t )

In this representation the kernel a -1 of the bilinear form, which defines the Gaussian distribution, is diagonal and we get [dp(7/)] = N e x p ( - ¼

(2.2)

The requirement is then that the kernel t~-l(v) is positive definite. As we want to use it for regularizing the theory, we consider a family of kernels ao, labelled by the real variable o, such that, for a --* 0, we get back 8(t - t') and that the ultraviolet divergences are expressed as poles in a. A useful form is the following

so(.) = cos(o~12)r(o +

1)iv I -~,

which is positive for o ~ I., where I . is the interval I , = ( - 1

(2.3) +4n;1+4n)

for

n>~O. By Fourier transforming [11] to the variable t we get

ao(t-t')=½olt-t'l

a-l,

(2.4)

which was verified [7] to provide the desired analyticity properties. We see that the value a* = - 1 considered in ref. [6] is within I0, i.e. an allowed interval. Notice that an analytic continuation in a, if needed, is implicit in the above formula for the Fourier transformation. Analytic continuation is also required (see ref. [4]) to provide the desired regularization of ultraviolet divergences, because one must reach the region where a~(t - t') has zeros of sufficiently large order for t ~ t', and so we must take a ~ I , with n sufficiently large. After the divergences have been cancelled, an analytic continuation from I, to the interval of interest (usually I0) through unphysical regions will be needed*. We still want to prove the convergence for infinite fictitious time of the regularized graphs of the perturbative expansion. The proof of the convergence of the perturbative expansion of the usual (markovian) stochastic quantization [12] is based on the Fokker-Planck equation, but in the non-markovian case we cannot use it. The main reason is that [13] the Fokker-Planck equation is a consequence of the * The content of this analysis has been discussed with N. Parga, who also pointed out the relation between eqs. (2.3) and (2.4) (in a private communication).

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i33

characterizing property of the markovian systems for the conditional probabilities

p ( x , tlz, t'; y z ) = p ( x t l z t ' ) ,

for t ' > ~"

(2.5)

In order to discuss the non-markovian case, let us consider the graphs of the perturbation theory. According to the discussion of sect. 1, each graph appearing in the perturbative evaluation of eq. (1.3) is obtained by contracting together some tree diagrams according to eq. (1.4). Hence, any graph can be decomposed into the tree diagrams which have generated it. Each vertex and each cross in the tree diagrams involves an integration over the corresponding time variables. Therefore, a tree diagram with V vertices gives, for a ~,~0n theory, V time variables t~ and (n - 2 ) V + 1 variables ~'~ from the vertices and the crosses, respectively. The variables ~'~,~-~' appear in ao(~"~ - z,~), and also in the propagators. All these time variables have T, the external time at which the Green functions are computed, as an upper bound. A line starting at t and ending at t' (our convention is t > t') in the tree diagram is a stochastic propagator

G( p; t - t') = O( t

-

t')e

-(p2+m2)(t-t')

.

(2.6)

The presence of the 0-functions induces an orientation and chronological ordering in the sets of the ti's and r,'s. Moreover, every oriented line G ends either on a vertex or on a cross, and not more than one G can end at the same point, because of the topological properties of the tree diagrams. We have a one-to-one correspondence between the integration variables and the set of G's by associating to each time variable the stochastic propagator having that time variable as smaller time. In this way, we have associated to each time variable a difference of times (the argument of the associated G, see eq. (2.6)). Let us perform the change of integration variables from the ti's and T~'s to the associated differences z~'s and y,'s. These are suitable integration variables since they are positive definite because the 0-functions and the exponentials in eq. (2.6) are always damping factors. So the integration is performed over damping exponentials times the functions ao which have, as arguments, sums and differences of z's and y 's. The integration interval for each variable runs from 0 to a time value tsup bounded by T, i.e. tsup = [ T - (a sum of zt's) ] and this is the only place where T appears. When we take the T ~ oo limit the upper integration value goes to infinity and the integration is convergent due to the exponential damping, at least for m q= 0 (we consider the case when, like in eq. (2.4), ao(Z - y ) behaves as a power of (z - y)). In the case m = 0, which is interesting for the theory of critical phenomena, the problem of the limit T ~ oo is the same as the problem of the infrared divergences, indeed, T provides an infrared regulator and we can always put m = 0 before T--* oo (this could be useful for discussing the renormalization of non-linear o-models for which a symmetric infrared regularization is desirable).

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It is difficult to obtain an explicit proof of the existence of this limit in the general case. However, assuming its existence, we can study the scaling properties of the corresponding Green functions by using standard arguments based on the renormalization group. Actually, in the interesting case where one uses the non-markovian theory as a starting point of an extrapolation in o towards the physical markovian point o = 0, as discussed in sect. 1, one can first perform the computation of the renormalization-group structure function and critical exponents, extrapolate them in a and finally go to the T ~ oo limit. In fact, anticipating the discussion of sect. 3, where we show it is possible to renormalize both the markovian and non-markovian theories, we observe that the renormalization constants, and therefore the critical exponents yo, are independent of T, by general dimensional reasons (this has also been verified in ref. [6]). Schematically, a Green function ~ which scales according to the critical exponent 7o will behave as "to

Po ]

2

\

Po

]

Considering values of T such that T >> p-2, the Green function is very close to the equilibrium situation, and since 7, is independent of T, eq. (2.7) gives information about the behaviour in p2 << pg, also for the equilibrium limit.

3. The renormalization of the theory In order to cover both the markovian and the non-markovian case, we will make use of the general discussion of the renormalization, based on BRS invariance, due to Zinn-Justin [10], which will be summarized for the relevant points. The crucial observation [14] is that the stochastic quantization can be set in a path-integral formalism. The averages of eq. (1.3) are generated by

z(J)

= f[d~o][dc][d?][dto] e -A +J~,

(3.1)

where an auxiliary field to and two anticommuting ghost fields c and ? have been introduced. The action is 8L A = - W(to) + L ( ~ ) ~ - ~7--c,

(3.2)

where L(tp) is equal to the 1.h.s. of eq. (1.1) and W(to) is the generating functional

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135

of the noise-noise correlations, defined using the measure of eq. (2.2) as

eW(~)=f [dp(n)l e~".

(3.3)

We will consider the case where the function %, defining the noise correlation as in eq. (1.4), has the form of eqs. (2.3) or (2.4). It has been shown in ref. [10] that A is, in general, invariant under the following BRS transformation 8q~ = T(c,

8c = O,

8(o = 0,

87 = y(to,

(3.4)

where ~ is an anticommuting parameter. Since our regularization, based on eqs. (2.3) and (2.4), defines a particular measure, it does not break this symmetry, and then the renormalized action A R satisfies the Ward identity induced by eq. (3.4) &4 R

--c+ 8q0

&4 R to-~-

=

0

(3.5)

"

If one is able to prove, by power commuting and fermion-number conservation, that A R is at most quadratic in the ghost fields, i.e. A R has the form A R = -?M(q0; to)c + ~(¢p; to),

(3.6)

then the Ward identity (3.5) implies

M(tp; to)

to) = toLR(

8q0

; to) - W

(to).

(3.7)

Comparing eqs. (3.6) and (3.7) with eq. (3.2), we see that the theory is renormalizable if the dimensional analysis, applied to LR(~; to), does not allow a dependence on to, i.e. L R is a function of ~0 only, and moreover it is a polynomial in ~0 and its derivatives of the same form as eq. (1.1). The renormalization scheme can be based on the renormalized Langevin equation (1.6), when the dimensional analysis fixes WR(to ) to be at most quadratic in to. We will consider mainly the theory g~4 for d > 2, since it is relevant for the application to the critical phenomena, both in the markovian and non-markovian case. As an additional example, we also discuss the theory hrg3.

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From eq. (2.4) we get the canonical dimensions (in length) =

[c] =

= x -d/2,

[~k] = X - a + d / 2 - 3 .

[ g ] ~- X - 2 a - 4 + d ,

(3.8)

We will be interested in the case where the coupling constant is dimensionless, and therefore we require 2 o + 4 = d,

(3.9)

2o + 6 = d ,

(3.10)

for the case g~4, and

for the case X~o3. Following the discussion of sect. 2, we also require o ~ I 0 and therefore restrict 2 < d < 6,

(3.11)

4
(3.12)

for the case gcp4, and

for the case Xcp3. Since a four-ghost term has dimensions

[f ~c~cdtdax] = x -a+2

(3.13)

it is not ultraviolet divergent for d > 2, and therefore eq. (3.6) holds. Let us discuss the form of L R: it has canonical dimensions x "-d/2-1. The non-interacting part of it, i.e. the part containing only ~ and only linearly, will have the form

Zt atcp+ z~,(-at, a~')cp + (m 2 + 8m2)cp.

(3.14)

For the case of the gcp4 theory, the interaction part of L R will be of the form: ZvgR993/3!. Notice that terms of the form q~2 or w~ are forbidden by the symmetry cp ~ - ~ 0 , w ~ - w . If there is an additional X~03 interaction, besides g~4; still keeping eq. (3.9), one sees that a term wtp would appear to be multiplied by one power of X at least, making it non-divergent. For the case of the X~3 theory, the interaction part of L R will be of the form: ZVXRCp2/2, except for the particular case d = 4, corresponding to a non-markovian situation with o = - 1 , due to eq. (3.10), where a term wcp can appear. Let us repeat the same dimensional analysis on WR(W). In a minimal-subtraction scheme, a potentially divergent term of the form fwrdtddx appearing in WR(w)

R. Iengo, S. Pugnetti / Stochastic quantization

137

can give a logarithmic divergence (i.e. the relevant one) only if its dimensionality is a positive even integer. It turns out that, in the case of the gq04 and 7~qo3 theories, a logarithmically divergent term, quadratic in ~, occurs for d = 4 and d = 6, respectively, i.e. in the markovian case ( o = 0). The function W(o~) generating the noise-noise correlations remains quadratic under renormalization, and the renormalization constant Z,, appearing in W(o~l=Z~fdtdt'ddx~(x;t)a,(t-t'l~(x;t

'1

(3.15)

is actually required only in the markovian case, being equal to one in the nonmarkovian cases (in eq. (1.6) the noise 7/has been rescaled so that eq. (3.15) holds). We conclude that grp4 is renormalizable in general when eqs. (3.9) and (3.11) hold, and Xep3 is renormalizable when eqs. (3.10) and (3.12) hold. In both the cases L a ( ~ ) can be set in the form dSa La(W) = Z, OtW+ 8W ,

(3.16)

where S a is the renormalized version of the action S (eq. (1.1)). Notice that at o = -1 (the lower extreme value of I0), corresponding to d = 2 for gq04 (eq (3.9)) and d = 4 for ~03 (eq. (3.10)), the non-markovian case is not renormalizable. In fact, for gq04, four-ghost terms and terms containing higher powers of ~ and q0 can appear, while, for ~o 3, non-quadratic and ~0-dependent noise-noise correlations would be required. Let us remind ourselves of an important feature of the renormalization of the markovian case. It has been pointed out [10,14] that the action A is not only BRS invariant but also supersymmetric. In fact, if we introduce the superfield q~(x, t; 0, 0) = ¢p + 0c + ?0 + 00~0,

(3.17)

the action A is now manifestly supersymmetric under the transformation t ---, t - 0X, 0 ~ 0 + X, 0 ~ 0 + X- This supersymmetry, which is related to the time-reversal properties of the system (ref. [15]), implies an additional Ward identity [10] Zt=Z ~ .

(3.18)

the contrary, if the process is non-markovian, it is not possible to write A in the supersymmetric form because now W(~o) is no more local in time, while only local terms can appear in the supersymmetric action. In this case, eq. (3.18) does not work any more; a non-trivial Z t will be, in general, required, while, as previously seen, Zn = 1. A further discussion on Z t and Z~ will be presented in sect. 4. Finally, let us mention that in the physical application presented in refs. [6,16], the validity of the above renormalization scheme has been verified up to the non-trivial order of three loops. On

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R. lengo, S. Pugnetti / Stochastic quantization

4. The background field method Let us rewrite eq. (1.1) in the presence of an external deterministic source ~b(x; t) = -Fo[qO(x; t)] + (e-7/(x;

t) -J(x; t),

J(x; t) (4.1)

where

F0[q~(x; t)]

~s[,] 6q~(x;t)'

(4.2)

and e is a small parameter playing the role of h. As an example we will consider the action S of a kq03 theory where, for generality, a linear term in rp has been added

S=fdax

½q~(-0~0~'+m2)rp+~.q0 +Kcp .

(4.3)

We want to compute directly the "effective force" for the stochastic process in analogy with the definition of effective action in the usual quantum field theory. In the path-integral approach, one defines the effective action F[q~] by means of a Legendre transform of the functional generating the connected Green functions. The argument of r , O(x), is the expectation value (in the sense of the path integral) of the quantum field q~(x) in the presence of an external source J(x). The effective action generalizes the classical action, in the sense that the equation of motion for q~ is equal to the "quantum equation of motion"

( asia] ) _ 8r[~] = -S(x).

(4.4)

We define the effective force for the stochastic system, extending the validity of eq. (4.4) to finite time, as

( Vo[~(x; t)l). = F~ [$(x; ,)1,

~(x; t) =(op(x; t)) n,

(4.5)

where now the averages must be understood over the noise measure, eq. (2.2). Taking the average of eq. (4.1) we get an equation for F elf 0,4; = - Fef~[0] - J .

(4.6)

During the calculation of Fell(0) we will meet divergences and hence, as we have seen in sect. 3, counterterms must be introduced

Z,O,O=

-F~fe [0] - J ,

(4.7)

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139

where F~ff means the renormalized effective force, i.e. in F~ff the renormalization constants Z,, ~m 2, Z v , 8K appear. The computation of eq. (4.7) can be done using a background field method [17]. We define the background field ~(x; t) as the solution of eq. (4.1) when e has been set equal to zero

o,,(x; t) = - r 0 [ , ( x ;

t)] - J .

(4.8)

This deterministic equation (a heat equation) corresponds to the tree-level approximation of eq. (4.7). Let us shift ~(x; t) = ~(x; t) + ~(x; t),

(4.9)

d~(x; t) = ~b(x; t) + ((x; t),

(4.10)

where ( i s the mean value of the stochastic field (. The crucial observation in the background field method is that [17] the effective action £, expressed as a functional of q~, can be computed by summing only vacuum graphs (i.e. graphs with no external ~ leg) in the background q~. Here, we will see that the same feature occurs for the effective force F~ff(q~). Of course, since the force is the derivative of the action, now we will have vacuum diagrams with one external amputated leg, at the point (x, t) corresponding to 8/SeO(x;t), represented by a dot in fig. 3. Let us compute directly F~ff(4~). From eqs. (4.1) and (4.9) we get (4.11)

,e= _ { Fo,(,#),~ + ½Fo,,(,~)~= + ... } +,fin.

This equation can be formally solved by defining a stochastic propagator in the background q,(x; t) by

[O,+F~[eo(x;t)l]G(x-y;t-.c;eo)=Sd(x-y)8(t-.r).

(4.12)

Eq. (4.11) is solved by $(x; t ) =

fddyd,rG(x-y; t-,; eo)(-½Fo"(eo)li2(y; ,r) 1 Fd"~3+ ... +Vt{t/(y; ~.)}. 3!

(4.13)

By taking the average we get (= <~> =

n=l

~",~, = c .

-{£o"(~2>

- --F"<,~3>

3! 0

+

--"

"

(4.14)

Remembering the usual perturbative expansion in the stochastic quantization scheme, we note that the expansion in e is nothing but a loop expansion. Defining

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R. lengo, S. Pugnetti / Stochastic quantization

b) Fig. 3. Diagrams for the effectiveforce,

~o -- q, we can solve, order by order, the equation for

F elf

oo

O, E eNI;N= -- ~ eNFu[q~] - J N=O

(4.15)

N=O

We produce in this way a set of algebraic equations defining F N in terms of the preceding F/s; the general form is N

JG;o+ E K=O

~.mi<~N

E

1 -K-!F

N(K) --(ml + .

.+mK)~m . . . 1 .

~m K = -- Ot~gN,

(4.16)

1 tn 1 . . . trl K

where F/(K) is the Kth derivative of F/ with respect to ~ computed at ~ = q~ and convolutions in (xi; q) are understood. For the reader's convenience, we show here

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141

the first equations

v=o,

J+Fo[,] = -o,,,

N = 1,

F 1 q- F(1)~1 = - 0 , ~ 1 ,

N = 2,

F 2 -[-- FI(1)~I --[- Fo(1)~2 -4- 1 F ( 2 ) ( ~ 1 ) 2 = - Ot~ 2 .

(4.17)

We consider the example of eq. (4.3). In this case, F (2) -~" ~ while higher derivatives vanish, and we can, as usual, represent eq. (4.13) as an expansion in tree diagrams (fig. 1) (the lines now represent the propagator eq. (4.12) and the crosses the noise */). Eq. (4.14) is represented by the sum of stochastic graphs of fig. 2 (the cross here represents the correlation as in eq. (1.4)). The terms in brackets of fig. 2 are ix, and 42, respectively. Eq. (4.17) for N = 1 gives the first correction to the force (see fig. 3a)

F~[q,(x; t)] =

1)~fddydrxdz2G(x- y ; t -

~1;q~)G(x - y ; t -

~'2; q))2a,,(~'l - ~'2)(4.18)

Notice that the external stochastic propagator of ~1 has been removed because of eq. (4.12). We then expand Fl(,~(x)) in the background field and compute the divergent part of the graphs with zero, one and two background fields as external legs. These divergences will be absorbed by the renormalization constants 8K for zero external legs, Z~,,Zt, 8m 2 for one and Z z for two. The second-order correction can be obtained from eq. (4.17) for N --- 2. It is easy to see that the various terms, appearing in the equation for F2, arrange themselves in such a way that F 2 is expressed only by means of 1PI graphs with no external leg, as shown in fig. 3b. A final remark following the discussion of sect. 3, one should also care about the renormalization of the noise correlations (eq. (3.15)). But, it must be noticed that the only place where the renormalization constant Zn appears is the crossed line (for zero background) P

t

Z2 ( ( Z4,p2+rn2+Sm2[t_t,[ × t'= Zt[Z~p2+m2+Sm2 ] exp Zt

(p'+m2+m2 )1

-exp-

~

]t+t']

. (4.19)

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R. lengo, S. Pugnetti / Stochastic quantization

In the markovian case, the Ward identity discussed in sect. 3 (eq. (3.18)) holds and we have not to care about Zn. On the other hand, for the non-markovian case, we have seen in sect. 3 that Z~ = 1. We then conclude that the renormalization scheme based on the effective force, as previously discussed, is sufficient, and the explicit introduction of a renormalization constant Zn is not required. Further, by using eq. (4.19) and applying the topological arguments about the structure of the stochastic graphs (sect. 2) to the graphs defining the correction F n to the effective force, it is possible to show that Fn ( t;

Zt) = ( Z t- 1Z2 )nFn( t/Zt; 1),

(4.20)

where we have indicated only the dependence of F n from the external time t and from Z t. In the markovian case, o = 0, eq. (4.20) shows that the equilibrium (t ~ oo) does not depend on Z t because of eq. (3.18), so that the only wave-function renormalization is given by Z , (the usual renormalization constant Z of quantum field theory). On the contrary, in the non-markovian case Z, = 1 and the equilibrium wave-function renormalization needs both Z t and Zo. We would like to thank N. Parga for suggestions and discussions about sect. 2, and J. Helayel-Neto and F. Legovini for suggestions and discussions about sect. 4. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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