IL NUOV0 CIMENT0 VoL. 91A, N. 4 21Fobbraio 1986

Finite-Temperature Green's Functions by Stochastic Quantization.

W. Gm3~us

I~sti t~t fiir Theoretische Physilr Uqviversit~t Wien - Wien

G. I~ARX)U-~I

Dipartimento di ,Fisica deU' Universith - Bar i Istit~to _Vazionale di E~sioa 1Vuvleare - Sezione eli Bari, I talia

(rioevuto il 20 Se~tembre 1985)

Summary. - - T h e stochastic quantization method of Parisi and Wu is extended to include thermal effects. The proof of equivalence between stoohastic and usual quantization at T :/: 0 and some examples of one- loop oalculations for gauge theories are discussed.

PACS. 11.I0. - Field theory.

1 . - I n t r o d u c t i o n .

The stochastic quantization method of Parisi and W u (1) has mainly been introduced for gauge theories to avoid gauge fixing, and therefore the Gribov ambiguity (2). Though there are no l~addeev-Popov ghosts in this formulation 9

it has been shown in second order tha t their contributions show up in the cal-

culation of Green's functions coming from longitudinal terms of the stochastic version of the gauge field propagator (~). There exist proofs of equivalence with

(1) G. P~RISX and WU YONGSHI: •Ci. ~q/Tb., 24, 483 (1981). (2) V. 1~. GRIBOV: _ArUVZ. Phys. B, 139, 1 (1978). (3) IVY. NAMIKI, J. 0HBA, K. AKANO and Y. YAMAI~AKA'- Prog. Theor. Phys . , 69, 1580 (1983).

384

FI~IT]~-TEMPERAT~R:E GR:EEN'S F ~ C T I O ~ S BY STOCHASTIC QUA~TIZATIO~ 385

the usual me thod for gauge theories (~,5), bu t t hey rely on a modificat ion of the original scheme. For scalar theories the s i tuat ion is much simpler and there are different methods to establish equivalence (5.7).

The a im of the present pape r is to introduce the rma l effects in stochastic quantizat ion. I n order to obta in fmi te - tempera tm 'e T Green's functions, we modi fy the Langevin equat ion b y considering Gaussian r andom variables which are periodic (or antiperiodic for fermions) in the Eucl idean t ime, with per iod length fl ~ l i T . We prove t h a t the free-field Green's fuact ions obta ined in this way are jus t the f in i te - tempera ture Green's functions in the so-called imag inary- t ime formal i sm (8). For in teract ing fields, we show t h a t the proofs of equivalence between stochastic and canonical quant iza t ion are val id also a t T r 0. We discuss some explicit examples of one-loop calculations for the electric mass of gauge bosons based on the Langevin equat ion with periodic r andom variables.

The plan of the pape r is as follows: in sect. 2 the basic formal i sm and the s ease are discussed. I n sect. 3 we consider a self- interacting scalar field theory and sect. 4 contains ~ discussion of interaetSng gauge field theories.

2. - S tochast ic q u a n t i z a t i o n at T ~ 0 : f ree - f i e ld theories .

To begin wi th we recall the basic features of stochastic quant izat ion. Let S[~b] be the action for a Eucl idean field theory in D dimensions (*) (we assume

to be a scalar field for simplicity). The stochast ic quant iza t ion of Parisi and W u consists in introducing a fictitious ~ t ime ~) t; the evolution of the fields in the ex t ra t ime is assumed to be governed b y the Langevin equat ion

(2.1) 8~5(x, t) 8%[~] 8t - - 8q)(x, t) + ~/(x, t ) ,

where ~(x,t) is a Gaussian r~ndom var iable and x : (Xo, xz, ... x~_x). Let ~b(x, t) be a solution of the Langevin equat ion; then one can construct correlation

(a) D. ZWANZlGX~: _,Vuct. Phys. B, 192, 259 (1981); Phys. 3~ett. B, l l 4 , 337 (1982); L. BAULI]~U and D. ZWANZIGER: AYq~ct. Phys. B, 193, 163 (1981). (s) E. FLORATOS and J. ILIOrOVLOS: 2Yucl. Phys. B, 214, 392 (1983). (a) W. GRI~US and H. H0~FEL: Z. Phys. C, 18, 129 (1983). (7) H. NAKAZA%O, IV[. I~A~IKI, J. 0HBA and K. 0KAlVO: Prog. Theor. Phys., 7@, 298 (1983); E. SEt. EGORIAN and S. KALITSIN: Erevan Physics Institute preprint EFI-638 (28)-83 (1983). (a) S. W~INBXI~G: Phys. ~ev. D, 9, 3357 (1974); C.W. BXl~ARD: Phys. ~ev. D, 9, 3312 (1974); L. DOLA~ and R. JACKIW: Phys. Bey. D, 9, 3320 (1974). (*) In the examples of sect. 4 we will put D -- 4.

386 w. GRI~S and o. lgARDULLI

functions of the fields

(2.2) <~5(xl, t l ) . . , q~(xL, tL)>n =

~[d~/] q)(x~, tx) ... r tL) exp [ - �88 dt ~*(x, t)] f[dz/] exp [ - �88 dt~2(x, t)]

and the equivalence between stochastic and canonical quantizat ion is proved if (2.2) converges to the usual Green's funct ion in Eucl idean space in the l imit

Le t us now suppose tha t the Gaussian noise is periodic (*) in the xo-variable with period fl > 0:

(2.3) ~(zo, x, t) = ~(xo + fl, x , t ) ,

where x = (x~, . . . , x~_~) . Then it is clear t ha t also the solution of the Langevin equat ion is periodic in x~ with period fl, provided t h a t the initial condition of (2.1) is periodic too : ~(x0, x, 0) = q~(Xo § fl, x, 0) (**). We define new corre- la t ion functions by modifying the ~-averages as follows:

ftd ].A[ l e, p [ - t I dxjdO- xdtr t)] (2.4) <A>n = o

0

where the subscript fl means t ha t we integrate over all the ~'s which are periodic in xo with period ft. Then the propert ies

(2.5)

<~(x, O~(x', t')>, = 2 ~ ( x - x ' ) , ~ ( t - t') ,

<~}(Xl , t l ) " '" ~(X2n-{-1, t2n-{-1)>' = 0 ,

<~(xl, tl) ... ~(x2~, t~)>, = ~ ~ <v(x,, ~,)v(x~, t~)>, , l~oSstble l)airm

~atr comb.

which hold for the fl = ~ ease are obviously still valid. To see the effect of our modifications we consider first a free scalar field

theory described by

~gb 2

(*) For Bose fields; for Fermi fields antiperiodieity must be imposed. (**) However, the large time limits are independent of the choice of initial value.

FII~IT:E-T]~MP:EI%ATI71~ GI~:EE:N'S IelTI~CTIONS B Y $TOCItASTIC QISA:NTIZATIO:N

The Langevin equat ion is given by

8q~(x, t) _ (0~_ m2) q)(x, t) ~- V(x, t), (2.7) 8t

where V(x~ t) obeys (2.3). We in~roduce the re tarded Green's function

G(x, t) = O(t) ~ j ~ exp [ik..x] exp [-- t(k~ ~- m2)] , (2.8)

where

(2.9a)

(2.9b)

387

(2.1~) D ( x - - x'; t, t') = ( r t) r t ' ) ) ,

(2.12)

where

(2.13)

and one obtains af ter some algebra

xz f d'- 'k D(y; t, t') = ~ ~ D(w +, k; t, t') exp [ik~.y],

D(o~ +, ]r t, $') :

I { e x p [ _ ( k ~ + m ~ ) ] t _ t , i ] _ e x p [ _ ( k ~ _ m 2 ) l t _ ~ t , ] ] } .

Now it is clear t ha t for t - - - - t ' -~ oo

(2.14) lira D(y; t, t') -~ 1 f d ' - lk 1

k. = (~+., k ) ,

+ 2~n C O s - - ~ "

The solution of (2.7) together with the initial condition ~(x, 0) --~ 0 is given by

~(x, t) = I d~ldyo f dO-ly e (x - - y, t - - ~),(y, ~) (2.10) o o

as can be immedia te ly checked. One can now evaluate the correlation function

$88 W . GRIMUS ~Iis G. NARD~3LLI

Equat ion (2.14) shows that ~he equal-time correlation function (2.11) tends to the finite-temperature Green's function for the free scalar field in the imaginary t ime formalism (s).

Then we turn to a free gauge field theory described b y the action

(2.15) S = ~ ,

t t a where F~--~ 0 ~ A : - - ~ , A ~ (a--~ 1, ..., N). The fields Au sar ~he Langevin equation

(2.16) ~A~(x, t) at - (~ ~ - 0,, 0,)A~(x, 1) + ~ , ( z , t ) ,

where ~/~(x, t) ~;s a Gaussian noise

a , ~ a t ~ bs0gr (2.17) <%,2~ 0 , <%,(x, )~/~( , t')}~ 2 ~ u ~ = ~ ( x - - x ' ) ( ~ ( t - - t ')

and periodic in the xo-variable with period ft. Following the same reasoning as before, one obtains

~ f d f f ~ A u ( x , t ) = ~ dy0 d ' - X y G ~ , ( x - - y , t - ) ~ L ( y , v ) , 0 0

(2.18)

where

(2.19)

and

(2.20)

1 f dD-lk G;~(y, ~) = ~ Z ~ 1 e~p [~k..U]~(k., 1)

(2.21)

one obtains

(2.22)

(2.23) ,,b . D ~ ( k , t, t ') =

__ ~,b {(~u ~ k~k~'l,k~_ l -~ . . . . . / ~exp [ -

ab G~v(k , t) = O(t)(~ab{((~tt~- ]~tt]c~/k 2) e x p [ - - k2 t ] + kuk~/k ~} .

Constructing the correlation function of two gauge fields

ab a D ; , ( x - ~' ; t, t') = <Au(x, t )A~(x' , t ' )>, ,

ab . 1 /" d D - l k : ~ [~k. y ]DL(k . , t, t')

2 ku k~ } k~lt - - t'l] - - exp [-- k2(t -[- t')]) @ k 2 min(t, t') .

t ab o In the limit t = t ' -~ c~, D~,,(k, t~ t') tends to the fmite-temperature vector

FINITF~-TEMPERATUR]~ GRE:EN'S FUNCTIONS BY STOCHASTIC QUANTIZATION 389

boson Green's funct ion in the imag inary- t ime formal i sm in the Landau gauge, plus a longitudinal t e r m propor t ional to t

Formulae (2.22)-(2.24) are the T :/: 0 version of the analogue results obta ined b y P ~ I S I and W u at zero t e m p e r a t u r e (*).

F ina l ly we invest igate the free fe rmion case. The Eucl idean action is

(2.25)

where the ~, matr ices sat isfy

(2.26) {r~, r ,} = - 2 ~ . , .

The s t ra ight forward approach to stochast ic quant izat ion fails for fermion fields because the opera tor ( i ~ - m) has bo th posit ive and negat ive eigenvalues. This leads one to consider the following modified Langevin equat ion (9):

(2.27) ~ ( x , t) 8S

We can consider the equat ion for ~ too:

(2.28) ~v~(x, t) ~t - - ( ~ - m~) ~7(x' t) 4- ~ ( x , t)

and the an t i eommnt ing r andom variables ~, ~ are assumed to sat isfy

< ~ ( x , t)~a(~', t ')>, = 2[- - i ~ + m].a(~(x - - x') ~(t - - t') ,

(2.29) ( ~ } , = ( ~ } , = O,

(~75, = ( ~ 5 , = o .

Moreover, we assume t h a t ~v, ~v are ant iperiodic in xo, in order to introduce

(*) Of course eq. (2.24) is formally divergent with t --> c~; this is due to the fact that (2.21) is not a gauge invariant quantity. (9) J .D . BREIT, S. GUPTA and A. ZAKS: Stochastic quantizabion and regularization, preprint, Princeton (March 1983) ; P. H. DAMGAX~D and K. Tsoxos : Maryland preprint 83/218 (1983).

3 9 0 w . ~R~M~JS and O. ~LRDULLI

f i n i t e - t e m p e r a t u r e effects :

(2.30) ~Tto(xo, x , t) = - - ~7~(Xo ~- fl, x, t)

and a s imi lar e q u a t i o n for ~ .

One can eas i ly check t h a t the so lu t ion of (2.27) is

p

(2.31) ~f(x, t )=jd~jayo jd , -~G(x- -y , t - -~ )W, (Y , ~) , 0 0

where G(x, t) is t h e r e t a r d e d Green:s f unc t i on def ined in eq. (2.8) w i th

(2.32) k . -~ (o)~-, k ) , o): = (2n § 1)g/ f t .

I f we n o w c o n s t r u c t t he cor re la t ion func t i on of t he two f e r m i o n fields ~, v~:

~ ( x - - x ' ; t, t') = < ~ ( x , t)(fs(x', t')>,, (2.33)

we o b t a i n

wi th k'. = (o~-, k) and

(2.35)

f dD-lk (2~)D_ 1 S~(k~'; t, t ' ) exp [ik'~.y]

S ~ ( k ; t, t') =

(~ § m)~a - - k 2 + m s {exp [-- (k 2 ~- m~)[t -- t'l] -- oxp [ - - (k 2 -~- m2)(t -~ tt)]}.

I t is n o w clear t h a t for t ---- t'---> c~, we ge t

(2.36) l im S ~ ( y ; t, t) = 1 I" dD-lk ( ~ -~ m)a~ . o exp E k; yl + . 3 ,

which is t h e f i n i t e - t e m p e r a t u r e f ree f e r m i o n p r o p a g a t o r in t he i m a g i n a r y t i m e f o r m a l i s m (s).

3 . - S e l f - i n t e r a c t i n g s c a l a r f i e l d t h e o r y .

As a f irst e x a m p l e of i n t e r a c t i n g fields, we n o w cons ider t he ac t ion of a se l f -coupled scalar field

PI~IT:E-TEMP:EI~AT~7:R:E GIt:E:EN~S FUNCTION8 BY STOCHASTIC QUANTIZATION 391

and obtain the Langevin equat ion

(3.2) 8r t) 1 6 8t - - ( ~ ' - m~) q)(x, t) - ~ gO~(x, t) - 2r t) -4- r/(m, t ) ,

where V is periodic in xo with period length ft. Defming the Fourier t ransform of a funct ion ] by

(3.3) k)-_f o [- oXo]J'dO-, [- 0

we can write down eq. (3.2) in momen tum space

(3.4) ~O(~o+~, k, t) _ (k~ § ,r~) ~ ( o ~ +, k, t) - - ~t

~g~f dD--l~ 2 (2~)- -~ r176 p ' t) ~ (~+_~ , k - - p , t) - -

1 1 f dD-lp ( dD-lq 6 '~ ~'2 ~ i - ~ .J (2:T~) D-1 (~((L}~, p , t) r q-, q, t) r162 k -- p -- q, t) q-

§ n(o~+., k, t).

The correlation funct ion between the ~'s is now given by

(3.5) (U((o,, p , t)~(op~ q, t')>, : 2fl(2z~)'-~6~+~.o6(p + q)6( t - - t ' )

and r, s are integer numbers. Now we see tha t we get the f ini te- temperature case from the T ~ 0 case by the substi tutions

(3.6) ~ ~ , p --+ p,,=- \ p ]

for all integrations and momenta in the Langevin equation, and

(3.7) (2~) ~(po + qo) -> fi~,+s,o

in the correlation funct ion of the ~'s. Therefore, the momen tum and t s t ructure of the theory are quite the same as in the case T : 0, and we can over take the diagrammatic proof of equivalence between stochastic and canonical quan- t ization from ref. (e). In conclusion, we obtain that , for scalar theories, the imaginary t ime formalism is obtained by stochastic quantizat ion with random sources periodic in Eucl idean t ime with period length fl ~ 1/T.

392 w. ~.RIMVS and G. NARDULL][

4. - Gauge theories .

We shah now discuss thermal effects in the stochastic quantizat ion of inter- acting gauge fields. Le t us consider the Langevin equat ion for the gauge

field A t

~A~(x, t) ~ [ A , ...] (4.1) ~ - ~A~(x,t) + n~(x,t),

where S is the Euclidean action containing both free and interact ion parts (dots represent other fields). :Now equivalence between stochastic and usual quantizat ion is only expected if gauge-invariant quantit ies are considered. This conjecture has been proved for self-interacting non-Abelian gauge fields obeying eq. (4.1) at the one-loop level in ref. (3). A general proof has been given in ref. (4); it relies on the following modification of eq. (4.1):

(4.2a)

where

(42b)

~A~(x , t) ~ [ A , ...]

~t ~A~(x, t) D ~ b V ~ - ~ ] ~ ( x , t ) ,

D ~b : 5 ~ a~, - - g / 'b 'A~(x , t)

and V b z Vb[A, x] is an a rb i t ra ry (gauge noninvariant) functional of the fields A~. The insertion of the new piece D~av b in the Langevin equat ion does not change random averages of gauge-invariant functionals E [ A ] , but it does allow us to calculate the limit t --> cc of ( F [ A ] } , , where E is not gauge invariant , provided V a breaks gauge invari~nce. For special Vb~s one obtains the usual description with gauge fixing and Faddeev-Popov ghosts (4) for t --> oo.

a Let us now choose a random variable ~,(x, t) which is periodic in Euclidean t ime xo, a periodic functional V b

(4.a) V~[A; Xo, x] = W[A; Xo + fl, x]

and periodic initial conditions for eq. (4.2); then it is clear t ha t its solution will be periodic too. F rom this, it follows t ha t the proof of equivalence between stochastic and usual quantizat ion of ref. (4,~) can be over taken with those modi- fications and A t na tura l ly describes stochastic fields in a heat ba th at tem- perature T ~-- 1/fl (s).

Firs t it might be instructive, however, to see what happens if the original eq. (4.1) is considered without modification. Even though the formal proof of equivalence seems to be difficult in this case, eq. (4.1) could be useful for diagrammatic calculation because the corresponding F ey n m an rules look simpler. As an illustration, we shall consider a one-loop calculation in spinor

FI~IT:E-T:EMP:E:RAT~TI~:E GRE]]:N~S FTJNCTIO~ BY STOCttASTIC Q~ANTIZATION 3 ~

QED. The action is given by

(4.4)

The Langevin equations in m o m e n t u m space are (9)

(4.5)

with

8An(k, t) ~t -- ( k~G-- ~, ,k~)A,(k, t) q- Ir~(k, t ) ,

~ ( k , t) - (k~ + m~) ~(k, t) + A(k, t ) ,

~t

~(k , t) ~t

- (k~ § m~) ~7(k, t) + .~(k, t)

(4.6) { Y.(k, t) = ~.(k, t) + efdq(p(q, t)y.y~(k - - q, t ) ,

A(k, t) = ~v(k, t) + ef q (~ - m) y.F(q, t)A~(k - - q, t ) ,

where k = (co~, k) (w + for bosons, o)~ for fermions) and the correlation functions of the U~s are modified according to (3.5). The integral over q contains a discrete s u m

(2~)D--1 "

We can now apply the set of eqs. (4.5) to the calculation of the electric mass of the photon

(4.8) m 2 = :Zog(k 0 = O, k - ~ - O ) e l

which is different f rom zero at T J: 0. Such a calculation is feasible by stochastic methods, because the vacuum polarization tensor z ~ is a gauge-invariant quant i ty in QED. We can solve eqs. (4.5) by i terat ion as applied in ref. (1.3.e.9). In the one-loop approximation, we have the set of diagrams a), b) of fig. 1 which describe the correlation function

(4.9) < A , ( x , t ) A . ( x ' , t ')>, .

In the stochastic diagrams of fig. 1, a crossed wavy (solid) line means a factor D,~(p; T, r (S~(T; z, z')), and a wavy (solid) line without crosses represents a factor G,~(p; r, ~') (G(p; z, z')) (see sect. 2 for their definitions). Loop integrations are performed according to (4.7) and at each ver tex we have a (( t ime >> t integration and conservation of momen tum and discrete energy.

~ 9 4 W. GlCIMUS a n ~ G. ~ARDULLI

k t t r p v

a)

qP=k +q ql qt

+ ( q ~ q O 4-(q ~ q O

b)

c)

Fig. 1. - Stochastic 4iagrams contributing to the photon propagator up to O(e~). Diagrams c) appear only in scalar QED.

Finally, F e y n m a n rules for the vert ices can easily be read off f rom the Langevin equations. Taking the l imit t ~ - t ' ~ co in (4.9) we can ex t rac t the gauge- invar iant v a c u u m polarizat ion tensor and obtain the result (g)

(4n0)

w i t h

(4.11)

zt~,v(k) ~ e 2 V~,,(q, q') q* + q,: + M ~- 2 m 2"

�9 ,2+m *q2+ms~-q2~-~ +q,~+m~l-

f ~q 1 = ~ v..(q, q') -(q~+ ~,~)(q,~ + m~)

V..(q, q') = Tr{r~(q' § m)r.(q § m)),

which is jus t the analyt ic expression for the usual F e y n m a n diagram. T h e

sum over the discrete energies in (4.10) can be per formed b y using the formula (lo)

(4.12)

o~ co+~e

= i(.) Jr + Ir - - co - - c o + i s

exp [-- iflz] T 1 '

where upper signs refer to bosons, lower signs to fermions; this formula expli- ci t ly separates T = 0 and T # 0 contributions. Thus we obtain the electric

(lo) p .D . N[OR:L~Y and M.B. KISSLZ~O~R: Phys. Re/). C, 51, 63 (1979).

FINITE-T:E]~IPEttAT~J1KE GR]~]~N'S F~NCTIO:N8 BY STOCHASTIC QXYANTIZATIO~ 3 ~

mass of the photon

2e ~ T~ (4.13) m,~ -~

r

0

wi th ~ ~ m!T. In the l imi t 2 -* c~ (*) one gets

(4.14) m~, - - - - m~ T~ exp [-- m / T ] ,

(~r = e~/4~) which agrees wi th the resul t of ref. (1~), and for 2 - . 0

e2 T ~ (4.15) m. ~, =

3

Stochastic quant iza t ion a t finite t empe ra tu r e can be applied to the scalar QED case in the same way. ~ o w also the diagrams e) of fig. 1 contr ibute and af ter a s t ra ight forward calculation, one again obtains for a massless part icle (~) the result (4.15).

On the other hand, in non-Abel ian gauge theories~ the electric mass of the gauge boson m , cannot be calculated in this way. The reason is t h a t for non- Abel ian gauge theories ne i ther the gange boson propaga tor D~b~(k) nor the self-

_ab Zk~ energy ~%~ ; are gauge- invar iant quanti t ies and the l imit t : t '-->cr of ~ b does not exist. One could t r y to calculate the one-loop contr ibut ion to m~, as

(4.16) !

m~,2 ~_ lim~_.~ lim~_.o Ikl~(-- Do0 (ko = 0, k; t, t)) ,

where D ' is the one-loop propagator , because lk l ' ampu ta t e s the external legs and ~0o (ko ~ 0, k - * 0 ) is gauge invar ian t (18,14). Even though (4.16) looks reasonable, it can easily be shown t h a t i% leads to the wrong result, me~ = 0. As a m a t t e r of fact , t ak ing the l imit k -+ 0 for fixed t implies tha t the (~ damping force ~ (-- k~c~ ~- k~k~)A:(k, t) in the Four ier t r ans fo rm of eq. (2.16) is absent even for noniuteraet ing fields. On the other hand, the v~lidi ty of the stochast ic quant iza t ion approach is based upon the presence of a damping force which ensures the existence of a l imit ing value of the fields irrespective of the initial values (**).

(*) T could be assumed to be of the order of 3 K. (11) j . F . Ni~vv, s, P .B . PAr. and D. G. U~Q~I~: Phys. t~ev. D, 28, 908 (1983). (12) O.K. KALASE~IXOV and V.V. KLI~ov: Phys. Zett. B, 95, 234 (1980). (la) D.g. GROSS, R.D. P~SAYr and L. G. Yi:er~: ]~ev. Mod. :Phys., 53, 43 (1981). (14) O.K. KALASHNIKOV and V.V. KLIMOV: Soy. J. s Phys., 31, 699 (1980). (**) For discussions see ref. (1,3,5).

3 9 6 w . Gt r a n d G. ~ A R D U L L I

The above difficulties can be avoided by using the modification (4.2a) of the Langevin equation, because in this case the limit t--> co does always exist, even for non-gauge-invari~nt quantities. In t roducing

( 4 . 1 7 ) V , ( x ) = - 1 ~ , , A ~ ( x ) O~

in (4.2a) (a ---- 1 would correspond to the Fey n m an gauge), one obtains for the gauge boson self-energy ~ in a pure gauge theory, ~he usual result plus the following te rm (15):

~ ( k ) : k~-- 2k ~ 5~) fdq q~(k - - q)~" (4.18) ~~b g2/~d/btd(k" 1

I t is clear, however, t ha t

(4.19) lira ~~b k-,o ~~176176 I-- 0, k) = 0

so tha t the usual result (~8.1~) in SU~

g3 (4.20) m~, = ~ T2s

for the one-loop gauge boson electric mass at T # 0 is obtained. Let us finish this discussion by stat ing tha t even though the original Parisi-

Wu approach cannot be used to calculate thermal masses in non-Abelian gauge theories through two-point correlation functions, it can be used to calculate gauge-invariant Green's functions and whenever the method is applicable a t T--~ 0, the extension to T r 0 is allowed and straightforward, as explained in the previous sections. This can be seen explicitly in the example

(4.21) ~ x t ) F L ( x , l im <F~( , t )>, , f--r

which has been considered in detail ill ref. (3) at T ~ 0. In conclusion we want to say tha t f ini te-temperature effects can be incor-

porated quite natura l ly in the stochastic quantizat ion scheme by choosing random variables which are (anti) periodic in Euclidean t ime x. with period fi = 1/T ~n4 the theory obtained in this way is equivalent to the imaginary t ime formalism of a f ini te- temperature field theory.

* * *

We thank D. A ~ I for discussions and the Theoretical Physics Division at C E I ~ , where this work was par t ly done, for its hospitality.

(1~) It. NAC~AOOSm, M. NAMIKI, J. ORBA and K. AxA~o: Prog. Theor. Phys., 70, 326 (1983).

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