Z. Phys. C - Particles and Fields 18, 129-134 (1983) Zeitschrift P a r t i c l e s for Physik C

and F lds (~) Springer-Verlag 1983

Perturbation Theory from Stochastic Quantization of Scalar Fields

W. Grimus

CERN, CH-1211 Genf 23, Switzerland

H. Hiiffel*

Institut fiir Theoretische Physik, Universit~it Wien, A-1090 Wien, Austria

Received 4 November 1982

Abstract. By using a diagrammatical technique it is shown that in scalar theories the stochastic quanti- zation method of Parisi and Wu gives the usual per- turbation series in Feynman diagrams.

1. Introduction

Parisi and Wu [1] introduced a stochastic quanti- zation method which is based on the Langevin equa- tion [2] of non-equilibrium statistical mechanics. They applied their method to the quantization of scalar and gauge field theories with the main aim of constructing a new perturbation theory for gauge theories without introducing gauge fixing. Therefore for non-Abelian gauge theories their method is free of the Gribov ambiguity [3].

In this paper we deal with scalar theories and show diagrammatically that the perturbation theory of Parisi and Wu gives the usual Feynman diagrams when calculating Green functions. Our paper is or- ganized as follows: In Sect. II we recall the basic fea- tures of the stochastic quantization method and how to obtain "stochastic diagrams" from an iterative so- lution of the Langevin equation. In Sect. III we show how to perform the time integrations of stochastic diagrams by introducing time ordering of the ver- tices and in Sect. IV we prove the equivalence of sums of stochastic diagrams and Feynman diagrams in the case of self-interacting scalar fields. According to a remark in [1] this has been shown in [-4] but we find it hard to see what has been proved in this connection.

* Supported in part by a fellowship of the French Government

2. Langevin Equation and Stochastic Diagrams

The starting point of the discussion of stochastic di- agrams is the introduction of a fictitious time for the fields whose time evolution is given by the Langevin equation

~ ( x , t) ~S - - ~-r/(x, t) (2.1 a)

,~t ~ ( x , t)

D (1 2 1 2 2 . S=~d x ~ ( ~ ) +~m q)-I-3~g~3-q-4~q~4 }. (2.1b)

We have written down (2.1) for a real self-interacting scalar field. S is the Euclidean action in D space- time dimensions. The fictitious time t should not be confused with the physical time contained in x. r/ is a random source with Gaussian distribution

( r/(x, t) r/(x', c) ).= 2 6 (x - x') 6 ( t - c) (2.2 a)

(r/(X1, t l ) --" r/(X2n+ 1, t2n+ 1))r/= 0 (2.2b)

(r/(x,, tO... r/(x2~ t2,)b = ~ [I Ql(xi, tl) r/(xj, t j)),. (2.2c)

possible pairs pair comb,

Explicitly we can write the random average over r/

S [dr/] (...) exp( - �88 S dDxdtr/2(x, t)) ( ' " ) " = ~ [dr/] exp(-�88 t)) (2.3)

Now the crucial point is that in the limit t ~ oo equilibrium is reached and the random average of correlation functions of ~b(x,t) tends to the corre- sponding Green functions of the quantum field theory with action (2.1b)

lim (~(x~, t)... (1)(XL, t )~ = ((~(X1)... (1)(XL)). (2.4) t~OO

130 W. Grimus and H. Htiffel: Stochastic Quantization of Scalar Fields

This has been shown in [1] for scalar theories by using the Fokker-Planck equation [2]. For gauge theories the situation is more complicated [5]. In contrast to [1] we prove (2.4) by a diagrammatical technique in the case of scalar fields.

The Langevin equation (2.1a) can be transformed into an integral equation in momentum space

�9 (k, t )=~ dzG(k, t - z ) tl(k, r) (2.5a) 0

1 . dDp 2 ! g j (~)Dn)D Cb(p, z) cI,(k-p, z)

1 " d~ dDq t 3 ! 2 J (~ )D S (~)D q)(p, z) cl)(q, r) cI)(k-p-q, z) d

G(k, t - r) = e x p ( - (t - z) (k 2 + rn2)) (2.5 b)

where the boundary condition ~b(x, 0 )=0 has been used. However, any trace of a specific boundary con- dition should die out with t--* oo due to the equilib- rium property of the system. In momentum space (2.2a) reads

(tl(k,t)~(k',t'))n=2(2~)D~(k+k')(3(t-t'). (2.6)

Solving (2.5) by iteration one arrives at a power se- ries expansion of ~ in the coupling constants, which can be written diagrammatically

~= ~:+ g~+_l_ 1 X~R + - - ~ + 2! -3!

~.. +-.. (2.7)

where we denote G by a line and t/ by a cross; in- tegration over the momenta and the fictitious times at all the vertices and crosses is included.

Let us now consider the L point function (42(x 1,t)...c~(xL,t)) . and substitute for q~ its diag- rammatical expansion (2.7). When the random av- erage over the t/'s is taken all crosses are joined to- gether in all possible ways due to the Wick-decom- position property (2.2c). In this way diagrams are obtained which we call stochastic diagrams. Each of them has the form of an ordinary Feynman diagram of the theory described by the action S apart from crosses on the lines where two r/'s have been joined together (Fig. 1). Conversely, to every Feynman dia- gram there exists a number of stochastic diagrams with the same topology [6]. Actually we will show in Sect. IV that the sum of all stochastic diagrams to a given Feynman diagram yields just this Feynman diagram. Obviously cutting a stochastic diagram of

§

Fig. 1. All stochastic diagrams belonging to a given Feynman dia- gram. This is an example of what is proved in Sect. IV

O ( T 1 , T l , ~ G ( ' r 3 - ' r 4 ) G(t-'rl) ( ~ 0 ( ~ " ~ t

G(TI-T2~"---]---"~ D( T2,T 3 ) T2

Fig, 2. A stochastic diagram and the times and propagators as- sociated to it. In each of the G's the first time must be greater than the second one because of the time integration in the expan- sion (2.7). Momenta have been omitted

an L point function at all crosses gives L connected trees contained in (2.7).

The momentum c5 function in (2.6) guarantees momentum conservation at a cross and eliminates one momentum integration, so momentum inte- gration is reduced just to the usual one of Feynman diagrams. Also the time integrations on both sides of a cross can be performed and lead to the pro- pagator

min(r, "if) D(k, z, z ' )=2 ~ d z " G ( k , z - ~ " ) G ( k , # - # ' )

0 1

-- k2 + m2 (exp( - I z - z'] (k 2 + m 2)

- e x p ( - (z + z') (k 2 + m2))). (2.8)

z, z' are the times of the neighbour vertices. This means that in a stochastic diagram each line repre- sents the Green function G and each crossed line the propagator D respectively (Fig. 2).

3. Integrations Over the Fictitious Times

In this section we will show how to perform the time integration of stochastic diagrams by introduc- ing time ordering of the vertices. Let us consider a

w. Grimus and H. Hiiffel: Stochastic Quantization of Scalar Fields 131

stochastic d iagram belonging to some F e y n m a n dia- g r am with L external lines and N vertices. Each ver- tex carries a t ime 27~ which has to be integrated over between 0 and the t ime of some next vertex accord- ing to (2.7). Due to the absolute values of t ime dif- ferences in the D propaga to rs it is however con- venient to introduce fixed t ime orderings of the ver- tices. Given the F e y n m a n graph we will al low all t ime orderings of vertices which are compat ib le with at least one of the corresponding stochastic dia- g rams (e.g. in Fig. 2 27 2 or 27 4 cannot be greater than all the other 27's, because one could not go then f rom vertex 2(4) to an external leg via an increasing se- quence of times at the vertices for any distr ibution of the crosses). In general, a given al lowed t ime or- dering belongs to several stochastic diagrams. In per forming the t ime integrat ions we will only keep the largest terms thus leaving out the second te rm in D and always dropping terms, which come f rom the lower boundar ies of the integrations. We will see that the integrat ion over the largest terms compen- sates exactly the exponent ia l t dependence of the G's on the external lines and all smaller terms will go to zero exponent ia l ly in t.

Let us concentra te now on a fixed t ime order ing of the N vertices which we can choose to be

0 < 2 7 1 < 2 7 2 < . . . < 2 7 N < t . (3.1)

We denote the set of m o m e n t a of the i-th vertex by Vii and leave out the m o m e n t u m denomina to r s of the D's for the moment . Fo r convenience of no ta t ion we substi tute p2 +m2 __~p2 for all momenta . Thus masses are trivially contained in our discussion. N o w we in- tegrate over 271. Because it is the smallest t ime all 271 exponents in G's and D's have positive signs (except those we have already d ropped f rom D, which do not concern us anymore) and we obtain

~2 exp(% E p 2) -- 1

S d271 exp(271 ~ p 2 ) = v, (3.2) o v~ 2 p 2

v1

We drop the 1 in (3.2) and go to the 272 integration. N o w there are two possibilities: a) vertex 2 is not a ne ighbour vertex of the first one, which means that all ne ighbour vertices of vertex 2 have times larger than 272. In this case again all z 2 exponents have positive signs and

~3 exp(273 ~'~ p2 )_ 1 Sdz2 e x p ( % ( Z p 2 + Z p2))_ v~v~ 0 V1 V2 2 p 2

v,,~v~ (3.3)

The 1 will be d ropped as before.

b) vertex 2 is a ne ighbour of vertex 1. In this case special care has to be taken for the lines which con- nect vertices 1 and 2 as the corresponding G's and C's depend on bo th 271 and 272; they are of the form e x p ( - ( z 2 -271)p2) before 271 integration. F r o m this one sees immediate ly that all 272 exponents coming f rom momen ta , which connect vertices 1 and 2, are can- celled by 271 integration. On the other hand all o ther ne ighbour vertices of 2 will give posit ive 272 ex- ponents. Explicitly we have

~3 ~ d z 2 e x p ( % ( Z P 2 - Z p 2 + Z p2)) 0 V1 V1 (3 V 2 V 2 -- V 1 (3 V 2

exp(% ~ p 2 ) _ 1 = w2 (3.4a)

~p2 Wz

W 2 = V 1 w V 2 - V 1 c~ V 2 (3.4b)

(3.4) also includes (3.3), so it is the general result of the second integration. Using the same a rguments as in (3.4) one obtains for the k-th in tegrat ion

"Ok+ 1

dzkexp(zk( Z p2 Z p2 0 Wk i U(ViC~Vk)

l < i < k + 2 p 2 ) )

V k - U(Vi n Vk) 1 6 i < k

exp(27k+ 1 E p 2 ) - 1 = wk (3.5 a)

~p2 wk

~ = ( % 1- 0 (v, aE))w(v~- 0 (v, nvk)) l <i<k l <i<k

k

U i=1 l<--i<j<=k

W I ~ V , , 27N+1=-t ( 3 . 5 b )

(3.5b) can easily be shown by induction. N o w we would like to m a k e some remarks to

(3.5). a) The procedure is independent of the n u m b e r of m o m e n t a leading to a vertex. b) The cont r ibut ion f rom the N- th t ime in tegra t ion is given by

exp(t ~ p 2 ) - i external

lines

p2 external

lines

(3.6)

so that the t exponents of the G's and D's corre- sponding to external lines are exactly cancelled. We see at this point that the neglect of non- leading con-

132 W. Grimus and H. Hiiffel: Stochastic Quantization of Scalar Fields

P2 k2

"c 1

I h P3 kl Ik 1

T 1 <"t" 2 .~T 3

Fig. 3. This figure illustrates remark d) in Sect. III. The second diagram has been obtained from the first one by removing the vertex with the largest time z 3 and the external line k 3

tributions is justified due to their exponential fall off in t.

c) From all time integrations we finally get for

N 1 I~ p2 (3.7)

k = l E Wk

d) Note that leaving out the last time integration in (3.7) we get exactly the expression of the time in- tegration of a diagram obtained from the original one by dropping the zN vertex with its external line(s). In the example of Fig. 3 for instance we get

1 1 1 k2+p2+p2 3 k2+k2+p2+p 2 k2+k2+k 2

1 1 2 k l 2 2 2 2 2 2" q-pl q-p3 kl q-k2 q-P2 q-P3

(3.8)

e) Given a time ordered stochastic diagram we find therefore

k= 1 E p2 crosses p2 Wk

(3.9)

where the second product of momenta arises simply from the so far neglected denominators of the D's.

f ) To obtain the full contribution of a stochastic di- agram one has to sum (3.9) over all allowed time orderings.

4. Equivalence Proof

In this section we prove by induction on the number of vertices N of a given Feynman diagram that the sum over all stochastic diagrams with the same to- pology gives exactly the Feynman diagram in the case of the real scalar field theory (2.1b).

For N = I we only have the 3 point and the 4 point vertex function in lowest order. According to the simplest application of the time integration rule (3.7) and (3.9) we get

-~g

= - g

k3 k3 k3 \

kj kl kl ] (4.1 a)

1 [ 1 1 1 \ - g ~ + ~ + ~ -

k 2 + k ~ + k ~ tk , k 2 k 2k 3 k 3k 1) k~k2 2k 2

1 X + + . . . . 3! 3!

ka k 2 kl k~

1 ( 1 1 ) =-2k2+k2+k23+k~ ~ t +... kl k2 k3 k~ k~ k~

k 21.21,21~2" (4.1 b)

2! (3 !) is the number of possibilities of joining the t/'s together to obtain the desired Feynman diagram and for the crosses we have inserted the propagators stemming from the D's according to (3.9). We have exactly obtained the corresponding Feynman dia- grams. Because of the expansion (2.7) all Green func- tions we consider are non-truncated.

Now we assume that we have proved the equiva- lence to Feynman diagrams for all numbers of ver- tices smaller than N with any number of external lines.

Let us consider a Feynman diagram F with N vertices and L external lines and a stochastic dia- gram S F with the same topology. In F and S v mo- mentum integration is not contained. It is easy to see that S v has the number of crosses

Z = L + N3 + zN4 (4.2) 2

where N3(4} is the number of 3(4) vertices of F and N=N3+N4.

Now we introduce time ordering of the vertices compatible with S e. The largest time has to be at a vertex with at least one external line (see (2.7) and Sect. III). For an external line there are five to- pological possibilities shown in Fig. 4. All lines from vertex N leading into the blobs of Fig. 4 are inner lines. We call the largest time zN. With k's we denote external momenta and with p's inner momenta. In- tegration over zN yields

1 L ----- K. (4.3)

i=1

We discuss now the different cases of Fig. 4:

Case (a). Because rN is the largest time there cannot be a cross o n k 1. Removing the momentum k 1 and

W. Grimus and H. Hiiffel: Stochastic Quantization of Scalar Fields 133

k z k L T N L

k k~, k

(a) k3- k2 kl (c) k2 .kl

k3 k L

k k L k~

(d) (el Fig.4a-e. The topological possibilities for an external line in a self-interacting scalar field theory with action (2.1b)

TN p p'

k2 kL ,~ k2 kL

k3 k4 a F F'

kzN/kl p

k) ~ k k ,. k 3 ~ kk

Fig. Sa and h. Removing the vertex with the largest time z N and its external lines from F one obtains F'

kz .k 1 k 2. k~

k 3 kL k 3 . kt

[k,'- (a) (b)

Fig. 6a and b. The two possibilities for crosses on the external lines of the vertex with z N of Fig. 4b

- 1 / 2 ! g & - 1 / 2 ! g ~ . ' ~ ~ k z +k2 .

I~ ' k~/'

Fig. 7. This figure shows how the 1/2! at the vertex with time zs in Fig. 6a is cancelled by two identical terms in the expansion of ~(k9

Case (b). There mus t be now either a cross on k 1 or k 2, so we consider the two stochastic d iagrams of Fig. 6. Because the iterative expansion (2.7) of qS(k2) in Fig. 6a for example contains two identical pieces we have only - g at the vertex zN (see Fig. 7) in ac- cordance with the F e y n m a n rules.

N o w we remove k!, k 2 and - g f rom F and ob- tain F ' (Fig. 5b) and S F. The change in the n u m b e r of crosses going f rom F to F ' is A Z = - 1 . Again SF is a correct s tochast ic d i ag ram with topo logy F'. Doing the same s u m m a t i o n as in case (a) we get

- 1 / 2 ! g f rom F we obtain a F e y n m a n d iag ram F ' (Fig. 5a). Doing the same with S F we get SF. The factor 1/2! comes f rom the first i terat ion of ~b(kl) and is exactly needed for a loop with two identical bosons. The change in the number of crosses going f rom topo logy F to F ' is AZ=O (4.2). Because there was no cross on k 1 this means that S F has the right number of crosses to be a stochastic d i ag ram with topology F' and because of r emark d) in Sect. I I I the t ime integrat ion of S v with the given t ime ordering is just (time integrat ion of SF)x K. N o w summing over all t ime ordered stochastic d iagrams Sp keeping "c N as largest t ime one obta ins (sum over all s tochas- tic d iagrams of F')x K. According to our induct ion assumpt ion the sum over all s tochastic d iagrams of F' gives F', because F ' has one vertex less than F. So we obta in for the sum over all t ime ordered stochas- tic d iagrams of F with z N as largest t ime

1 ( 1 ) 1 F ' = k~-F.L (4.4) 2!gKF'=k~K - ~ . g k2 Z k2

i=1

( 1 + 1 ~ k21+k 2 - g \k~ k~] KF' L F (4.5)

i - 1

1/k~ and 1/k 2 come f rom the D p ropaga to r s at the places of the crosses in Fig. 6.

Analogous ly case (c) yields the same as (a), case (d) the same as (b) and case (e) gives

k21+ k2~ + k~ F. L

~, k~ (4.6) i--1

Thus summing over all possible places for the larg- est t ime z N we get a cont r ibut ion k~KF f rom each external leg with m o m e n t u m k/ and all the contri- but ions sum up to give F.

We want to stress that the correct m o m e n t u m integrat ion of the F e y n m a n d iag ram has a l ready been obta ined by the iterative expansion of 4) (2.7) and the r a n d o m average over the t/'s (2.6).

We have shown now the equivalence in the case of one real scalar field. It is s t ra ight forward to gen-

134 w. Grimus and H. Hiiffel: Stochastic Quantization of Scalar Fields

e ra l i ze o u r c o n s i d e r a t i o n s to m a n y sca la r fields a n d

c o m p l e x sca la r fields. T h e n every field has its o w n

r a n d o m source , a lso the cha rge c o n j u g a t e of a c o m -

p lex field [1]. T h e t i m e in t eg ra t i ons are the s a m e as

in Sect. I I I and the on ly poin t , whe re o n e has to be carefu l in Sect. IV, is to get the r igh t c o m b i n a t o r i a l

fac tors o f the F e y n m a n d i ag rams , bu t this can easi ly be seen f r o m the i t e ra t ive e x p a n s i o n o f the fields.

Acknowledgements. H.H. is very grateful for the hospitality at the Laboratoire de Physique Th4orique de l'Ecole Normale Sup- &ieure in Paris and thanks especially J. Iliopoulos for stimulating discussions. Both authors want to thank E. Etim for valuable dis- cussions and J.S. Bell for a critical reading of the manuscript.

References

1. G. Parisi, Wu Yongshi: Sci. Sin. 24, 483 (1981) 2. See e.g.A.H. Jazwinski: Stochastic processes and filtering

theory. In: Mathematics in science and engineering. Bellmann, R. (ed.), Vol. 64. New York: Academic Press 1970

3. V.N. Gribov: Nucl. Phys. B 139, 1 (1978) 4. C. DeDominicis: Lett. Nuovo Cimento 12, 567 (1975) 5. D, Zwanziger: Nucl. Phys. B192, 259 (1981); D. Zwanziger:

Phys. Lett. l14B, 337 (1982); L. Baulieu, D. Zwanziger: Nucl. Phys. B 193, 163 (1981)

6. A. Niemi, L.C.R. Wijewardhana: Ann. Phys. 140, 247 (1982). Though their method of stochastic quantization is quite dif- ferent from that of Parisi and Wu, the diagrammatic represen- tation is the same in both methods, but the interpretation is different