Spinodal decomposition in quantum field theory

ANNALS

OF PHYSICS

190,

Spinodal

32-58 (1989)

Decomposition

in Quantum

Field Theory

ESTEBAN CALZETTA*

University

Institute of Alberta,

for Theoretical Physics, Edmonton, Alberta, Canada,

T6G 2JI

Received March 9, 1988 We investigate the dynamics of spinodai decomposition in quantum field theory. We consider a A44 scalar field with tachyonic mass nZ < 0 which is suddenly brought into contact with a heat bath at zero temperature. By using the two-particle irreducible closed-time-path effective action we give a detailed description of how fluctuations in the infrared end of the spectrum grow to give rise to a Bose-Einstein condensate. The later time behavior of the phase decomposition is described by mean field theory. 0 1989 Academic Press, Inc.

I. INTRODUCTION The process of phase transition through spinodal decomposition (SD) has been described by Gibbs [l] and observed experimentally in studies of phase change in alloys [2]. Together with nucleation they are the two main mechanisms through which phase transitions proceed. More recently, spinodal decomposition has become the center of attention for physicists studying phase transitions in the Early Universe. It has been proposed that the observed homogeneity of the universe rules out nucleation processes between the Planck and Grand Unified eras, leaving SD the only candidate for a dynamical theory of phase change in those early epochs [3]. We have thus two different traditions in the literature concerning SD, one mainly oriented towards condensed matter problems [4], and the other studying SD in the context of inflationary models [3,5]. Although very different in method and scope, these two traditions share the problems associated with the most difficult aspects of spinodal decomposition. Naturally not all the theoretical approaches to critical dynamics attempted in the condensed matter context (such as numerical simulations, studies based on spin-flip dynamics for Ising models, or dynamical renormalization group methods) [4] are readily generalizable to phase transitions in field theory. Moreover, there is not at present a full model of SD derivable from first principles. There is, however, a substantial body of work, culminating with the work of Langer and co-workers [6] and others [7], which has withstood experimental tests and provides a yardstick against which new contributions should be measured. * Supported by the Natural Sciences and Engineering Research Council of Canada.

32 OOO3-4916/89 $7.50 Copyright Q 1989 by Academic Press. Inc. All rights of reproducfion in any form reserved.

SPINODAL

DECOMPOSITION

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FIELD

33

THEORY

There are three basic approaches to a dynamical theory of SD. They are mean field (Landau-Guinzburg) theory (developed in this context by Cahn and Hilliard [2,8]), the Langevin equation method (developed by Cook [9]), and the FokkerPlanck equation, developed by Langer et al. [6,7]. The problem of interest is a system (in our case, a quantum field 4) which at time t = 0 is suddenly “quenched,” that is, is brought into contact with a heat bath at a temperature much below the spinodal point. As a result of the quench, the temperature of the field drops to that of the bath. The quench is assumed to be so fast, moreover, that no appreciable phase separation occurs during the temperature drop. In the Cahn-Hiiliard approach, the system is described by a single, c-number field 4(x), the mean field. The mean field obeys an equation of the form (in a nonrelativistic theory)

(1.1) where F( 4) is the Landau-Guinzburg free energy. Phenomenological terms can also be added to (1.1). Temperature is introduced explicitly F(4) to be temperature-dependent. Typically

dissipation by allowing

F($) = 1 d3x [i (@)* + 4 (T* - T,Z) $* + higher powers of 41.

(1.2)

The spinodal point T, is the temperature at which 4 =0 becomes an unstable solution for Eq. ( 1.1). The fact that below T, the symmetric point is not simply metastable but unstable is of course the central feature of SD. Cahn-Hilliard theory completely ignores fluctuations around the mean field. To re-introduce them, Cook added to the RHS of Eq. (1.1) a random source J(x) [9]. The expectation value of J(x) is zero, and the correlation of J at different points is so chosen that (4(x) 4(x’)) takes, for large t, t’, the value corresponding to thermal equilibrium with the bath. The c-number field $(x) cannot be interpreted as a mean field. Rather, it is now a stochastic field, which derives its random character from that of the source. It can be interpreted as an “instantaneous” field. Cook’s theory is of course purely classical. The field 4(x) may take different values with well defined probabilities, but these different potential values do not interfere with each other. Thus Cook’s theory may be at best a phenomenological approximation to quantum behavior. Because 4(x) is now stochastic, we may define a probability density p in the space of c-number field configurations. Under fairly general circumstances, this probability density obeys a Fokker-Planck equation [lo],

34

The probability current equilibrium. For example,

ESTEBAN

CALZETTA

J is chosen ad hoc to describe

the approach

to

(1.4)

where y is the same constant as in Eq. (1.1 ), F is the Landau-Guinzburg free energy equation (1.2) and p = (KT)-', where T is the temperature of the bath. The system (1.3) and (1.4) is generally quite formidable. It is simpler to use it to generate equations of motion for the momenta of the probability density p. For example, if we multiply Eq. (1.3) by 4(x) and integrate over the space of field configurations, we obtain

; (4(x)>= -Y ($)3

(1.5)

which reduces to Eq. (1.1) in the mean field approximation. We can also obtain equations for higher momenta; if the correlations of more than a certain number of fields are neglected, we obtain a finite system of equations. Those equations are the base of Langer and co-workers’ approach [6,7]. An aspect in which Langer’s theory is a definite improvement over earlier formulations is that it correctly accounts for “coarse graining,” that is, the shift towards the infrared of the fluctuation spectrum during the phase separation. The methods of analyzing SD we have described have several shortcomings. First of all, they are nonrelativistic (although it is easy to derive a relativistic LandauGuinzburg equation from an effective action (see Ref. [ 1l] and Section V)). Besides, the Langevin or Fokker-Planck equations are only phenomenologicaf approaches to SD; neither can be derived from first principles for a realistic system. Although the Fokker-Planck approach is more comprehensive than that of the Langevin equation, the derivation of a closed system of equations for the correlation functions from Eqs. (1.3) and (1.4) is not straightforward; in their derivation, Langer and co-workers [6] made use of repeated ansatz on the form of p and other reduced distribution functions (see also Ref. [7]). The relationship between system and bath is much oversimplified in these models. The temperature dependence of the free energy is assumed, with no discussion of its dynamical origin. Also the possibility of a change in the system-bath interaction during SD because of this dependence is disregarded. Finally, the very early development of the phase transition is trivial because of the “sudden” quenching approximation, but this is not always a justifiable assumption. In fact, in many experiments substantial phase separation is observed before the quench is completed [ 123. A highly asymmetric treatment of system and bath is undesirable in cosmological theories, which at least in principle are expected to account for “everything.” The study of inflationary models [3, 51 has spurred a large literature on phase transitions in cosmological settings. However, the points we raised in the previous

SPINODAL

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IN QUANTUM

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35

paragraph have not been satisfactorily answered yet. The simplest approach to inflationary cosmology is a time-dependent Landau-Guinzburg equation as derived from an effective action

-

8,g””Ji

ad(x)+ V(4,T)= 0,

(1.6)

where g”” is the metric tensor describing the evolving spacetime, g = det gpv, and V is the effective potential (which may or may not include quantum or other corrections), for example,

It is remarkable that in general Eq. (1.6) contains terms linear in first derivatives of the field which can be interpreted as dissipation. The use of mean field theory as in Eq. (1.6) has been strongly criticized [ 131. Careful analysis has established the validity of mean field theory when the selfinteractions of the scalar field are feeble [14] or else when the initial fluctuations around the mean field are small [ 151. Also we must mention that in general the equations derived from the conventional effective action [16] are neither real nor causal [ 171; therefore, they do not admit an initial value problem, and 4 cannot be interpreted as the expected value of a field. An interesting attempt at going beyond mean field theory makes use of a Langevin equation [18]. The idea is to interpret Eq. (1.6) as the Heisenberg equation of motion for a quantum field 4. Splitting 4 into a background 4 and a fluctuation contribution & (4 is a c-number, while & is a q-number) we obtain 0 fj + V’( fj) = - 0 fjr + corrections.

(1.8)

To obtain a Langevin equation, the RHS of Eq. (1.8) is replaced by a c-number random source g(x), whose self-correlation is computed from that of & under the approximation that & is a massless free field. However, because the #r are massless the instability of the fluctuations is suppressed around the symmetric point. Equation (1.8) turns out to be a good description of the later time development of the phase transition (cf. Section V) but it fails to describe the beginning of it. This brief account of the existing literature on SD points to the need for a full quantum field theoretical treatment of SD, which must be at least able to furnish a causal description of the process, go beyond mean field theory, take into account the instability of the fluctuation field, and describe coarse graining. We believe such a formalism is provided by Schwinger’s “Closed-Time Path” functional techniques [19]. The extension beyond mean field theory is obtained by going to a two-particle irreducible formalism [20]. We will sketch the basic ideas of the method [21]. Consider the theory of a scalar field QH(x); suppose that at time t = 0 the state of

36

ESTEBAN

CALZETTA

the field is described by a density matrix p. Schwinger’s closed time path (or “In-In”) generating functional is defined as [ 17, 19,213 z[Jl,

~21 = Tr(p~(~-iSd4-~JZ(X)~Ho) T(eiJd4-~J’(x)~~(-r))),

(1.9)

Consider a complete set of states I$, 0) and I#, t+ ) which correspond to states of definite field at time t = 0 and t = t+ (t’ = + co), respectively. Completeness means

Jo~~~,o)(~,o~=J~~/~,r+)(~,t+~=l, where the integral ranges over all three-dimensional can be written as

z[J,, J*]=Jo+v01)’

D$((I)~,

ZCJ’, J~I =Jwl

field configurations.

Then Z

01 F(e-iJd4XJ2(x’Q~(X)) I$, t+ >

x (+, t+ 1 qeiJd4dXP&)

The source brackets have conventional us to write

(1.10)

1 I~‘~ O>(lcI’T 0 IPI ti2Y 0)).

path integral representations,

W’W’L

(1.11)

which allow

0~01 P I$~(., o),o>

xexp(i(S[$‘]+J’$‘-S*[ti2]

-J2ti2)),

(1.12)

where S is the classical action and J$ = 1 d4x J(x) $(x). The integral ranges over all four-dimensional field configurations such that $‘(x, t’) = $‘(x, t’). At this point it is convenient to consider the labels 1,2 as indexes in some internal space and introduce a metric tensor cab = diag( 1, - 1). Introducing the action

SCIc/“l= fw’l

- s*cti’1,

(1.13)

we rewrite (1.12) as

z[J,, = J D$“($‘l p I$‘> e’(s[ti’I+-‘@)

(1.14)

and define W = - i In Z. To generate mean field theory we take the Legendre transform of W with respect to J, q-p]

= W[.fJ

- .I,(&

(1.15)

where j0 is the solution of 8 WI aJ, = &. The solution 6 of (1.16)

SPINODAL

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37

THEORY

corresponds to the expectation value of QH with respect to the state p. Equation (1.16) may be considered as the relativistic Landau-Guinzburg equation. It is real and causal [17]. To go beyond mean field theory, we observe that if p can be written as p=Cexp

(1

-

d3kflI,a:

(1.17)

ak , 1

where the a;, aL are the interaction picture creation and destruction operators (we define the interaction and Heisenberg pictures to coincide at 1= 0), for some positive numbers /II, (for a thermal state, /Ik = j(k* + m2)‘/*), then ($‘I

p II)‘)

= Cexp f [ d4x d4x’ K$)(x, x’) +“(x)

I,/I~(x’)

(1.18)

for some nonlocal source KIp). This suggests defining

We perform the full Legendre transform f [&, Gab] = w[.&, k,b] -&I++~~~(G~~+&$~), where kab is the solution correlation functions G”b(X9 x’) = (

to a WI a&b = i(Gab + p$“). (T(@H(X) @Ax’))) ( GH(X) @ff(X’))

(1.20)

The Gab is a matrix

(@H(X’) @H(X)) (T&(x) @“(XI))) ) ’

of

(1.21)

where ( ) stands for expectation value with respect to p, and T, T are temporal and anti-temporal ordering, respectively. p and Gab are the solutions of the system

fj

[t’ = i’ = ,$,Gob] = -Kbg@

-j$

[t’ = q$‘= 4, Gab] = -5 KS).

(1.22)

Because actually K$)(x, x’) = Khp)(x, x’) s(t) @t’), the second equation may be considered as a homogeneous equation with a nontrivial initial condition. Equations (1.22) are real and causal, and we have gone beyond Landau-Guinzburg theory, since we now have the correlation functions of the field as new independent variables. More complex initial conditions may be handled by adding further nonlocal sources. The central point is of course that r[& G] is easy to compute, at least in perturbation theory. It is the sum of the two-particle irreducible vacuum graphs of a

38

ESTEBAN

CALZETTA

theory with action S[&], with G”b as free propagator [20]. The details are given in Ref. [21] and references therein. If only l-loop order terms are retained in Eqs. (1.22) they reduce to the large N approximation to the O(N) symmetric model [22]. The objective of this paper is to demonstate the application of the closed time path technique to the description of a quantum field undergoing a phase transition. Concretely, we will consider a quantum field 4 which at time t = 0 is brought into contact with a heat bath below the spinodal point. We will follow the subsequent evolution of the field by studying Eqs. (1.22). To avoid the need for a “sudden quench” approximation, and to be able to follow in detail the dissipation processes at work, we will consider a concrete system-bath interaction. The bath will be given by a second quantum field $ which interacts with the first field, 4; because $ is a heat bath, its correlation functions are always thermal, irrespective of the “back reaction” of the 4 field. Because of the 4 -+ -4 symmetry, Eqs. (1.22) admit a solution with I= 0 (the phase transition proceeds through Bose-Einstein condensation and is reflected by an infrared peak in the autocorrelation functions Gab), so that only the second equation of (1.22) should be considered. However, as time proceeds and the characteristic wavelength of the condensate becomes of the size of the observed volume, it is convenient to formally consider this condensate as a background field, retaining in Gab only shorter wavelengths. Thus to study the long time behavior of the field we shall use the full system (1.22). This later time solution should be matched to the early time solution (6 = 0, Gab from (1.22)) at some intermediate time, according to Gab (early time) = PEb (late time) + fast varying terms.

(1.23)

The evolution of the 4 field is dictated by its interaction with the $ field and with itself. For simplicity, we shall assume that the $ field is at zero temperature; its function is mainly to provide dissipation mechanisms to the 4 field, trying to cool it down to absolute zero. To lowest order, the effect of the self-interaction is to introduce a state dependence in the effective mass of the 4 field. Rather than solving Eqs. (1.22) for the self-interacting field, we will consider the equivalent problem of a noninteracting field with a time-dependent mass. By adjusting the evolution of the mass to that of the state of the field, we may achieve a self-consistent solution. Moreover, we shall assume that the mass is slowly varying. In this approximation, we may study the evolution of the field by dividing time in intervals, so that inside each interval the mass may be considered to be constant. The constant mass problem may be solved exactly, as shown by Schwinger [19]. The global evolution is then found by patching the different intervals together. The emerging picture resembles that from Langer’s theory [6]. In the first stage (quenching) the fluctuations of the 4 field drop, until eventually the effective mass becomes tachyonic. At this point the now unstable infrared modes begin growing in amplitude, producing the infrared condensate and pushing the effective mass again

SPINODAL

DECOMPOSITION

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39

towards zero; when the point of zero effective mass is reached, the spectrum of fluctuations will have shifted substantially towards the infrared. In general at this point the amplitude of the infrared peak will be too large to rely on a linearized equation. On the other hand, the wavelength of the condensate is large enough that there is no substantial loss of accuracy in substituting the infrared peak by a scalar background field as in Eq. (1.23). The rest of the evolution may be studied by using the full nonlinear equation for the background field, and a linearized equation for the fluctuations around the background. These equations describe the “rolling down the hill” of the background field towards the true vacuum state [3], and the coming to equilibrium of the fluctuations at the temperature of the bath (absolute zero). Before going on, we point out an important limitation of our method, and of all other theories of spinodal decomposition mentioned so far. By concentrating on SD, the theories exclude the possibility of nucleation of bubbles of true vacuum either during the quenching, or between the spinodal point and completion of the phase transition. In other words, these theories assign to a metastable state above the spinodal point an infinite lifetime, which is not correct [23]. A satisfactory theory of phase transition should be able to account for the occurrence of both SD and bubble nucleation simultaneously. In fact, it is not always possible to distinguish these two mechanisms experimentally [24]. The need for such inclusive formulation has been noted in both the condensed matter [25] and field theory contexts [26]. Because our field theoretical model does not correspond to a realistic theory, we shall not pursue its solution in enough detail to address such questions as scaling. The discussion is detailed enough, however, to be relevant to the usual inflationary models. This paper is organized as follows. Section II considers the evolution of a free field 4 in contact with a heat bath. Building on Schwinger’s work [ 191, we give estimates for the Green functions of the field 4. We also consider the case in which $ has a tachyonic mass. Section III is devoted to the early development of the phase transition in the selfinteracting case. In SectionIV we consider the late time evolution of SD, in which the dominant process is the “rolling down the hill” of the background field towards the true vacuum [3]. Section V contains a few final comments.

II.

FREE

SCALAR FIELD

IIa. Statement of the Model In this section we begin the study of the quantum field theoretical problem. We will set up the model, under one important simplifying assumption, namely, we will ignore the self-interactions of the 4 field. We will go back to the general problem in Section III.

40

ESTEBAN

CALZETTA

We are interested in a system of two quantum action

fields, 4 and @, with classical

We have chosen this model for simplicity, as will be clear later on. Nevertheless, we must warn the reader that it is somewhat pathological from the point of view of quantum field theory. For one thing, as it stands it is not renormalizable; to obtain a renormalizable theory we should add terms linear, cubic, and quartic in 4. Also the presence of the dimensionful coupling g is an unusual feature. We will assume throughout that unwanted side effects, such as the explicit breaking of the 4 + -4 symmetry, are compensated by external factors and can be ignored. The only role the $ field plays for us is that of a heat bath for the 4 field; as such, the cubic interaction will lead to the same overall behavior as in more realistic (and also more complex) renormalizable interactions. The 2PI In-In effective action is a functional of two background fields 4” and a 2 x 2 matrix propagator Gub [ 19,201 G%,

x’) = (T(d(x)

4(x’))>

G12(~,x’) = (4(x’) d(x) > (2.2)

G2k ~‘1= (0) 4(x’) > Gz2(x,~‘1= (fV@, 4(x’)) > plus their counterparts t,V and the $-propagators Gzb. In Eq. (2.2), ( ) stands for expectation value, and T(T) means (anti) temporal ordering. The computation of the 2PI In-In effective action r is no different from that of a single scalar field. In this and the next sections we will assume that all four background fields I#“, +” are identically zero. Also we are only interested in the equations of motion for the Gab’s, since by assumption the Gzb’s are equilibrium vacuum propagators (that is, the $ field acts as a heat reservoir for the Q field). The relevant terms in r are [ZO] r[G”“]

=f ln Det((Gab)-‘) -- 1 2

d4x c,b( 0 + ,u;) Gab(x, x)

d4x d4x’ G$?(x, x’) G$!(x, x’) Gcg(x, x’),

(2.3)

SPINODAL

DECOMPOSITION

IN QUANTUM

FIELD

41

THEORY

where cab, h,,. are 1 (- 1) if all entries are 1 (2), and zero otherwise. We have overlooked possible wave function and coupling constant renormalization, since they will not be necessary to this order. From variation of (2.3) with respect to Gab, we find the equations of motion. On general grounds [ 17,19,21] we know that the counterterms in the In-In formalism are the same as in the conventional effective action. Thus we write 112d=+

-&[&l)]),

(2.4)

where p2 (which can be either positive or negative) is the renormalized arbitrary scale fixing parameter, and E= d - 4. To avoid the proliferation of mass scales we will choose

mass, p,, an

po = (471)- u2 6,

(2.5)

where d2 is the mass of the $ field. From variation of Eq. (2.3) with respect to Gab, we find the equations of motion (O,+~~)G”(x,x’)-~gZ

5f”>O

d4x”

c,Jac(x,

x”)

Gdb(xil, x’)

= -icab 6(x-x’)

(2.6)

and ( 0 .x’+ p’,) GUb(x, x’) -;

g2

I f”>O

d4x”

cc,,GuC(x, xl’) Zdb(xl’,

= - icab 6(x - x’),

x’)

(2.7)

where Cab = ( Gzb)‘. Because we want our analysis to be relevant to almost any kind of field-bath interaction, we will try not to rely on the explicit form of the Z kernels. Nevertheless these kernels enjoy a series of properties which follow directly from causality, Lorentz invariance, and the properties of equilibrium propagators. We can use these properties freely. The t; kernels admit a Fourier transform (C”

- C12)(x, x’) = (-i)

ddk j m t?k(x--X’) Z(k)

with X(k*)

= F((k”

- i~)~ - k2).

(2.9)

42

ESTEBAN

CALZETTA

The function F(z) is analytic on the whole complex plane (obeying F(z*) = (F(z))*) except for a branch cut along the positive real axis, where, to lowest order, Im F(x) =.0x)

= -!(4n)2 [l -462jx]“2e(x-4sz).

The full function F is given by (2.11) where we have performed one subtraction to obtain a convergent integral. That is, (2.11) represents F up to a polynomial in z. But these extra terms can always be absorbed in the local part of the theory. The other Z kernels can be written in terms of F. We find (.z22 - P)(x,

x’) = ( - l)(,P

- ,P)(x’,

x)

ddk dkcx ~ *‘) F[ (ko + iE)2 _ k2] =i s (2n)d (C”

+C22)(x,

x’)=

Rather than the propagators G,k

(-i)

ddk jme

+ iE)- F(k2 - ic)).

(2.13)

(2.2), it is convenient to use the combinations

x’) = (G12 + G”)(x,

x’) = ( {4(x,, 4(x’,} >

Gret(x, x’) = i(G” - G12)(x, x’)=

Gadv(x, x’) = ( -i)(G2’

rk(x-J’)(F(k2

iO(t-

t’)( [d(x),

4(x’)])

(2.14)

- G12)(x, x’) = ( -i) O(t’ - t)( [d(x), 4(x’)] >,

where 13is the usual Heavyside step function. It would not be realistic to assume that G, is even approximately translation invariant for finite point separation. However, if the initial conditions are spatially homogeneous, this condition will be maintained throughout, and we can perform a spatial Fourier transform G(x,x’)=Jf$e

-ik(x - x’) Gk(t, f),

(2.15)

where G is either G,, G,,,, or Gad,,. Taking the spatial Fourier transform of the Z kernels from Eqs. (2.8), (2.12), and (2.13), we find

SPINODAL

DECOMPOSITION

(i?;+w;)G,,(t,

=-

t)-;g2j;

g2

2

cc

QUANTUM

FIELD

+C22)k(f,

(8;. + w:) Gk,(t, t)+;

Gkadv(fll,

t”)

43

THEORY

tll)Gkl(tll,

dt11(Z11-Z12),(t,

dt”(C”

s0

IN

t’)

(2.16)

t’)

g2 SK dt” Gkl(t, +‘)(C22-C12)(t1’,

t’)

0

=-

(d;

+

w:)

:

g2

s0

Gkret(t, t”)(C”

m dt”

Gkret(cr I’)-;

+Z2*)#‘,

g2 1 dt”(.Z”

t’)

(2.17)

- C’2)k(t, t”) Gkre,(t’l,

t’)

= 6( t - 2’)

(2.18)

(3’8 + w’,) G,&t,

t’) + f g2 { dt” Gkadv(t, t11)(C22 - C12)k(t11, t’)

= 6( 1 - I’),

(2.19)

where wi = k2 + ,u*. Actually (2.20)

Gkadv(f, 0 = Gkret (t’, I),

since G,, (t’, t) is seen to satisfy Eq. (2.19) and vanishes if t - t’ > 0. The boundary conditions on G,, are that Gkl, arGkl, a,.G,,, and a$G,, must be continuous at t = t’ = 0. In the case in which the 4 field was at equilibrium for t, t’ < 0, these boundary conditions are Gkl(O, 0)= (2~~)~‘Cl+

(e80n’k- l))‘]

d,G,, (0,O) = a,.G,,(O, 0) = 0 a:,4

(2.21)

(0, 0) = w:Gk, (0, 01,

where PO corresponds to the initial temperature. We can write a formal solution of Eqs. (2.16) and (2.17) in terms of Gkret and Gkadv. First we observe that G,q(t, r’,=;j-om is a particular solution satisfying (2.21) is

dy[;= dy’G kret(c YP” with null boundary

Gk,(f, f’)=GkP,(h

+ ~22M~, conditions.

t’)+ (a,,G,,,,(t,O)

+ w:G,m(fr

v’) Gcad~‘~

t’) (2.22)

The complete

solution

~,G~adO,

f’)

0) G,cadv(O, 1’)) G,,(O, 0).

(2.23)

44 Thus our problem we can find Gkret .

ESTEBAN

(the evolution

CALZETTA

of Gk, (1, t) as a function of t) will be solved if

IIb. Evolution of the q5Field (Constant p2) In this section we continue the study of the free field case, considering the forms of the retarded propagator for different values of the mass p2, either “normal” (II’ > 0) or “tachyonic” (p2 < 0). In the case ~1’ > 0 we are dealing with the temperature equalization of the 4 and tj fields. This case has already been studied by Schwinger [19] and we include it here mainly for completeness. As we saw towards the end of last section, the main point of the analysis is to find a solution Gkret for Eq. (2.18). Because w: is time-independent, we may look for a solution of the form Gkret(t, t’)= I yrn ~e’“(“‘~G,,,,(w).

(2.24)

Because of the causal boundary conditions, Gkret(w) can be continued to an analytic function in the lower half complex x plane. The equation for GkrCt(w) is (cf. Eqs. (2.18), (2.19), and (2.10))

Let us consider the function (2.26) Schwinger [19 3 has shown that if the stability condition H(k2 +4a2)<0 is satisfied, H(z) has no (first sheet) complex zeros. Explicitly, the stability condition reads (2.27) If p2 > 0 and f is given by Eq. (2.10), this inequality will always hold for g and 6 small enough. Gkret(w) will be given by HP ‘((w - ic)‘), and Gkrel (t, t’) is then given by Eq. (2.24). An elementary contour deformation leads us to Gkrer (t, t’) = O(t - t’) f : $ ws

x $fb2-k2)]

(sin w( t - t’))

lGd~)l-~,

(2.28)

SPINODAL

DECOMPOSITION

IN

QUANTUM

FIELD

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45

where

(2.29)

w; = k* + 4h2.

The long time behavior of Gkrer (t - t’) will be determined by those frequency bands in which GLret( w ) is ’ 1east smooth. We identify two frequency intervals which dominate the asymptotic behavior of Gkrctr one near the two-particle threshold wg (the lower limit of the integral in Eq. (2.28)) and the other in the neighborhood of the point w0 where Re H(w’) vanishes [ 191. The contribution of the first band is approximately

W,k,(ht’)‘v$

sin “t:-“))

[w,(t-t’)]-3’20(t--‘)xconstant.

(2.30)

Let us look now for the contribution of the second band, near the zero of Re H(w2). Following Schwinger [19], we find (2.31) where

(2.32) and

y=$B(Zw,)-'f(w;-k'). In Eq. (2.32), the integral must be understood

P s'ma dxf(x) 462 (X

Equation

+ k - w;)’

= lim

WI- “‘0

P I uDdx

(2.31) leads to the time-domain

G&(t-

t')= B

462

(2.33) as

f(x) (x + k2 - w;)(x

+ k2 - w;)’

propagator

(2.34)

The approximate behavior of the propagator is given by the sum of Eqs. (2.30) and (2.34). Until times of order l/S, (2.34) dominates, showing a propagator which decays exponentially from the original amplitude; at larger times, the small amplitude power law tail (2.30) becomes the dominant contribution. This kind of behavior is similar to that observed in a classical oscillator in contact with a “bath” of other such oscillators. Because Gkret(t, t’)+O as It-P + co, at large times G, will be given by GP,

46

ESTEBAN

CALZETTA

Eq. (2.22). This propagator describes thermal equilibrium between the 4 and II/ fields. So far we have assumed that the two critical frequencies wg and w0 were widely separated. However, if we consider p2 as a variable parameter, w0 decreases with p (for p = 0, w0 = 0 is an exact zero of (Re Gk’)) and eventually we will reach a region in which w,,- wg. As w,, decreases, y + 0, so the decay of Gkre, is less pronounced, and the amplitude of the t- ‘I2 tail increases, until Eq. (2.30) becomes the dominant contribution to Gkre,. In the particular case in which w0 = wg, the same argument leading to Eq. (2.30) leads us to

Gkret (t,t’)- sinw;:-y[Ws(t-t’)]-‘/*e(t-t’). Since in this case there is not other independent contribution, this is the leading behavior of Gkre,. We now consider the case in which 4 has a mass p2 < 4a2. The difference with the previous case is that the stability condition Eq. (2.27) no longer holds. Thus in this case Get has a true pole at a (complex) value of w such that w2 < wi. If the pole is located at w& then we can eliminate p2 for wi and write G,;,(w)=(-1)(w2-w;) dxf(x)[(x+k'-(w-i.z)*)(x+k'-w;)]-'

1 .

(2.36)

We must distinguish two cases, depending on the sign of wi. If wi> 0, we find that Gkre,(t- t’), as given by Eq. (2.24), is the sum of the “cut” contribution (2.28) plus a “pole” contribution G~~~~)(t - t’) = B( w; ’ sin wO(t - t’)) 0( t - t’),

(2.37)

where B is given by Eq. (2.32), the “principal value” now being superfluous, since wi - k2 < 4~5~.The “cut” contribution still satisfies the estimate (2.30). It is clear that Gkre, (t) does not go to zero as t + co and that thermal equilibrium is therefore not established. But there is still some partial dissipation given by the amplitude B < 1. Observe that as w0 + wg from below, B-0. We get the picture then of a slow (power law) cooling of the 4 field, which does not reach full termal equilibrium, but in which the original power in the 4 fluctuations is reduced by a factor B. At the point where w0 reaches wg we no longer have a pole but a weaker singularity, and G, - t I/*, as in Eq. (2.35). If wi < 0, Gkret (t) does not admit a Fourier transform but being causal, it admits a Laplace transform

GM(S) =jom dte-‘“‘G,,,,(t),

s = w - ia.

(2.38)

SPINODAL DECOMPOSITION IN QUANTUMFIELD

THEORY

47

For (T>O large enough Gkret (s) is given by Eq. (2.36). The Laplace inversion theorem says that (2.39) provided 0 > JwOJ. Closing the contour in (2.39) from above we again find “cut” and pole contributions, the cut part given by Eq. (2.30) and the pole one given by G~~~~‘(t)=B(w,J-‘(sinhlw,l

1)0(t).

(2.40)

We have found the long sought explosion of those modes with k2 small enough that the zero of Eq. (2.26) corresponds to z
III.

SELF-INTERACTING

FIELD

In this section we will use the results of Section II to obtain a picture of the development of spinodal decomposition in the case in which the 4 field, besides its interaction with the thermal bath, has a A#4-type self-interaction. We will consider only the lowest order correction to the free field theory, in which a new term ; G”(x, x) Gclb(x, x’)

(3.1)

must be added to the right-hand side of Eq. (2.6) (a similar term goes into Eq. (2.7), with CL1 evaluated at x’). The formal manipulations of Section IIa go unchanged, only now the bare 4 mass is given by

(we are sticking to the choice of scale we made in Section II), Eqs. (2.16t(2.19) is now the time-dependent quantity w:(t)=k2+m2(t), 595/190/l-4

and w: in

(3.3)

48

ESTEBAN

CALZETTA

where m’(t) = I”’ +;i

1

G,(k

t)(x, t))(3.4)

which is independent of x under the assumption of spatial homogeneity. At time t < 0, the 4 field is assumed to be in thermal equilibrium at temperature T well above the critical point, so that w:(0)=tk2+;

T2%0

(3.5)

(that the initial spectrum of fluctuations be thermal is not really important, but the estimate (3.5) is). At t =0 the system is brought into contact with the bath and released. Our problem is to find the subsequent evolution. The simplest approach to the problem is the “adiabatic” one. We first identify the time scale on which the spectrum of fluctuations changes appreciably. Then we divide the time axis into intervals of about that duration. Inside each interval, the spectrum does not change, and so G1 (t) may be considered constant. This brings us back to the problem in Section II. Once the evolution inside a given interval is solved as in Section II, the result is used as Cauchy data for the next interval. The emerging picture is similar to Langer’s theory [6]. At the beginning we are in the situation of Section Ha; &field fluctuations are being strongly damped by radiation processes, and m2(t) will decrease. This in turn will cause a shift of the peak w0 of the 4 propagator towards wg. As w0 approaches ws radiation damping becomes less and less efficient, but the cooling continues and eventually w,, becomes less than ws. What happens then depends on the region of the spectrum that is being considered. For ultraviolet modes with k2 $lp2), the cooling continues, although at an extremely slow rate. For modes with k* 5 I,u21, however, the peak w0 eventually will approach zero and finally become imaginary. At this point the infrared modes with imaginary w0 will begin to grow, pushing m2(t) upwards. There results therefore a competition between the growth of the infrared modes and the cooling of the ultraviolet ones. As the short wavelength modes approach their zero temperature amplitudes, the infrared modes will become dominant. This event marks the end of the early stage of the phase transition. Let us be more concrete. Suppose we take Cauchy data for G, at some time t = t,. Then, for t > t, we will have G,, (t, t) = G,q”l’( t, t) + (a,,G,,,(c

td2 G,cl(t,, t,)

-2G~c,,,(t,

~~)~,,G,,,,(~,

f,)a,G,,(t,,

+(Gkret(h

t,)J2 WUf,r

t11,

?I)

(3.6)

SPINODAL

DECOMPOSITION

IN QUANTUM

FIELD

THEORY

49

where Gkq(‘l)(f, t) =$I R

dy dy’ Gkret(f, YW”

+ C2*Lh

Y’) G/cret(~ ~‘1.

(3.7)

TheregionR=([t,,t]x[t,,t])u([t,,t]x(O,t,))u((O,t,)x[t,,t]).Ifw2(t)>0, G,qC’)is dominated by the region y - y’ 5 t. Since y N t, we can assume that Gkret is approximately translation invariant. Gkrer(tr y) = 1 f$ ei”“-Y)Gkret(w,

t).

Using the fact that Gkret (w, t) is analytical if Im w < 0 and the identity ,mwdw Im Gkret(w, t) = I 7c

i “6

(3.9)

(which follows from Eq. (2.28) and the boundary conditions for Gkret), we find Gkq(‘l)(t, t) - w; l(t). As the other terms in Eq. (3.6) either go to zero or contain powers of B< 1 (Eq. (2.32)), in this situation we have the approach to equilibrium bf mode k, Let us introduce the spectrum at time t,

./i(f) = Kwdf) G,cl(h t) - 11.

(3.10)

For m*(t) close to zero, Eq. (3.4) reads (3.11) where (3.12) The logarithmic term in Eq. (3.11) comes from the sum of vacuum fluctuations (w;‘), that is, from the “tadpole” Feynman graphs. Because this graph is dominated by ultraviolet contributions, and the UV modes are stable, we can define the “tadpole” for m2 < 0 as the real part of the analytical continuation of the m2 > 0 tadpole [lo]. It follows that Eq. (3.11) is still valid for m2 5 0, provided we write ln(lm*\/6*) in the second term. We see that the solution of Eq. (3.11) for p2 + (A/2) M*(t) =0 is m2 = 0, so this equation describes the crossing of the zero mass threshold as the spectrum diminishes. If w$(t)O (it is convenient to choose t, so that am-w,), the dominant contribution to Gkq(‘l) comes from the region y - y’ 2 t, . Gkre, (t, y) is exponentially suppressed with respect to Gkret(t, f,) elsewhere.

50

ESTEBAN

CALZETTA

To find Gkret(t, y), we will look for WKB solutions of Eq. (2.18) with variable mass at both sides of the turning point t,, where w:(t2) = 0. (We assume this turning point to be linear.) For y > t, we neglect the nonlocal terms to get

Gkret(1, Y) ‘v

1 ’ Iw,Jt’)l dt’ Clw/cl(t) l%b4111~2exp s.v

(3.13)

plus exponentially small terms. If t, > y > y’, we analyze the contribution nonlocal term as in Ref. [ 191 to get

Gkret(Y,

Y’)

=

[Wt(y)

’

wk(y1)]1/2

(sin 1; wk(t’)

dt’) exp (-I:

y(f)

dt’).

of the

(3.14)

With y given by Eq. (2.33), we get Gkret (t, JJ) for t > t, > y by matching Eqs. (3.13) and (3.14) across the turning point. Observe that roughly we find

G,cret(t,v) = G/xet(ttt,)cosCwdtl)(y-t,)l i --

1 w!f(t1)

a,,G,,t(t,

xsinCwk(t,)(.vtI)l

tl)

I

exp(-y(t,)(y--t,)).

(3.15)

If we compare the G,,P(tl) from Eq. (3.15) with the other terms in Eq. (3.6) we see that the overall effect is to add, to whatever spectrum was there at time t,, a new contribution w; l, which is then amplified in the usual way by the exponential growth of Gkret. This new contribution corresponds physically to particles created from the vacuum around t N t,. Particle creation may occur here because, although particle number is an adiabatic invariant, the evolution is not adiabatic close to the turning point [27]. An important consequence of all this is that the growth of the infrared peak will occur even if these modes were not occupied in the original state, making SD inescapable. As the zero mass threshold is crossed, the ultraviolet fluctuations will continue to be damped away in a power-law fashion. On the other hand the infrared modes will grow explosively. At the beginning, the ultraviolet depletion will dominate, and m2 will become more negative. As Im2J increases, the width of the infrared band also rises, and the exponential growth becomes faster. Eventually, the balance tips towards the infrared growth, and (m2( begins to decrease, following an increase in M2, until the m2 = 0 limit is reached again, this time from below. The overall result

SPINODALDECOMPOSITION

IN QUANTUM FIELDTHEORY

51

has been a net transfer of power from the UV modes to the IR ones, since the farUV modes have been damped all the way, while modes with k* < jp]* have been increasing most of the time. In most cases, the amplitude of the infrared peak will now be too large to neglect nonlinear effects, and we will have to apply a different scheme to proceed with the investigation of the phase transition. We will do so in the next section.

IV. THE LATE DEVELOPMENT

OF THE PHASE TRANSITION

As we saw in the last section, the early phase of spinodal decomposition ends with the concentration of fluctuations in the IR end of the spectrum while in a state with effective mass mz - 0. If we were to continue this analysis as such, we would conclude that spinodal decomposition ends with a monochromatic (k = 0) BoseEinstein (BE) condensate, whose amplitude is fixed by the condition of vanishing effective mass, plus a thermal (T = 0) sea of fluctuations. This picture is flawed, however, because it does not take into account the nonlinear characteristics of spinodal decomposition. Near m2 - 0, the amplitude of the BE condensate is large enough to be in the attractive basin of the true vacuum. The final estate is dictated by the falling down towards this vacuum and not by the m2 - 0 condition. In the final moments of SD, as the C$field approaches the true vacuum state, we will have m2 well above zero and thus a finite correlation length. Fluctuation modes with wavelengths much larger than rn-' can be considered as a single, slowly varying background. This background obeys a dynamical equation derived from the 2PI In-In effective action: the initial conditions are given by the requirement of a smooth crossover to the initial regime described in Section III. The first step is then to find the 2PI In-In effective action, with a background field da included. The classical action is

The 2PI In-In

effective action [20] is r[p,Gab]

= SC@] +iln

Det(Gab)-’

1 a?s Gab + r2 [p, Gnb] + constant. '?&pa@ In the third term, the indices a, b must be extended to include derivations

(4.2)

w.r.t.

52

ESTEBAN

the (zero) $-field background. in previous sections, we find r[@,

Keeping only terms up to 0( g*) and G(A), as we did

Gob] = j dx (; cob a$@80” -; - $ h,b,d&$bpC$” -5 -;

s

d4X

(

CALZETTA

)

&,b(ihj

+ i ln Det(Gab)-’ .

c,b( 0 + &) + ; h,,,,&$f)

1 d4x(habr&G$?(x,

Gab(x,

X)

x))

- 2 j d4x hubrdGab(x, x) GCd(x, x) + f g*h,,,h, The equations of motion

1 d4x d4x’ GT(x, x’) Gff(x, x’) Gcg(x, x’) + . . . . for Gab and & are

-; (Gab)-'(x, xl)-; -2

(4.3)

( C,(m

+&++habcd&p(X)

hobcdGCd(x, x) 6(x - x’)

+; g*h,,,h,,G$x,

x’) Gf(x, x’) = 0

(4.4)

and (4.5) While in earlier sections we neglected the back reaction of the 4 field on the $ propagators, if we were to do so here we would miss an important physical process, namely, the dissipation of short wavelength components of the background field d through radiation damping. To find the correct +-field propagators Gzb on the presence of a nontrivial & background, we must extremize the In-In effective action. If &’ = 0, then

ar@q iYGzb(x,x’)

ar = q

[(=o]-$h,,,&(X)d(X-X’)+

... .

(4.6)

SPINODAL

DECOMPOSITION

IN QUANTUM

FIELD

THEORY

53

From (4.6) we deduce that Gzb(x, x’)Ipzo=

G;*(x, x’)lr=o - ighder d4x” G$x,

x1’) &x1’)

i

G$‘(xl’, x’) + O( g’). (4.7)

When Eq. (4.7) is substituted in the RHS of Eq. (4.5) we find two terms: a constant source of size 4J2g and a “fish”-type term. Assuming that the source-like term is compensated from outside, and introducing the notation Cab as in Eqs. (2.6) and (2.7), we find

--;g* -P2)(x, x’)6(x,) =0 sd4x’(P (where we are using the fact that for the physical solution Eq. (4.4) may be manipulated into

I

Ll~~+p:,+ZG’1(x,x)+2~(x)

>

6’ = d2= 4). In turn,

Gub(x,x’)

--fg2 s Gab(x, x’) (q...+~.+~G”(x’,x~)+~~(x’)) d4x”

and

1

(4.8)

--;g2I d4x”

ccd Zuc(x, xll)

Gdb(x”,

ccdGoc(x, xll) Zdb(x”,

x’) = icUb 6(x-x’)

x’) = -icab 6(x-x’).

(4.9)

(4.10)

Under the assumption that G”(x, x) and J’(x) are slowly varying functions of x we are led to the following equation for Gret:

d2x”(C” - Z”)(x, xl’) Gret(xll,

x') = 6(x-x').

(4.11)

If x is close to x’ we may assume that G,,, is almost translation invariant and perform a (four-dimensional) Fourier transform in the difference variable x - x’. As

54

ESTEBAN

CALZETTA

the effective mass is positive, we are in the situation of Section IIb, that is, we may approximate [ 191 -1 ,

(w-W,‘-k2-m2(~)-~m^‘cx,]

(4.12)

where m2 is given by the condition

(4.13) and (4.14) where wg = k2 + m2 + (A/2) p. Observe that we have changed our notation w.r.t. Sections II and III, now calling m2 + (L/2) 6’ the effective mass, which before was just m2. The Feynman propagator can be split into a “vacuum” and a statistical part G”=G” vat +G stat.

(4.15)

G,,,, is a solution of

q

(4.16)

,+m2(x)+2y(x)i

With boundary obeys

conditions

x)

GtA,(x 7

=

(m’

+

carried over from the Section III. WNW

(4n)*

2

lm2

+

where we have used dimensional regularization minimal subtraction of &, Eq. (4.13) reads

(WI

PI

62

;-W+ln [

and

1 m2=p2+2M2+(4n)2 J-(m2+~(2)h($~m2+~f+P~)

E

The vacuum part

(x)

1 ,

(4

1,)

= d- 4. After modified

SPINODAL

DECOMPOSITION

IN QUANTUM

FIELD

55

THEORY

where, as usual, M2(x) = Gstat(x, x). Now we can rewrite Eq. (4.8) as [28 J

q,cj+m2(X)fj+;d3

where (cf. Eqs. (2.8)-(2.11))

Given the nonlinearity of Eqs. (4.18) and (4.19), almost any initial condition will eventually give origin to a rather complicated background field configuration. On the other hand, we do not have to follow the evolution of the background field in full detail, since only the average background field over the observed volume L3 of space-time is seen as contributing to the BE condensate. Let us begin by performing a spatial Fourier transform on b;,

and write dk(l)

=

(2703&(t)

~3(W

(4.22)

+6:(f)>

where & is assumed to be small. 4, represents the sum of the infrared fluctuations. We find

iJ’(x,t)-di(l)+$g$

I&(t)\’

+ fluctuating

where we have used the fact that J(x) is real. Although second term in (4.23),

terms,

(4.23)

& is small for each p, the

(4.24) may be comparable to the first term due to constructive interference. The fluctuating terms. however, may be neglected, at least as far as the influence of @ on 6, is concerned. From Eq. (4.22) we find I’(x,

t) = f&t, + 302(t) q&)(t) (4.25)

56

ESTEBAN CALZETTA

From the point of view of the dynamics of &, they may be interpreted as a selfinteraction, a mass shift, and a noise source. Observe that the noise source is cubic in 6:. Terms linear in 6: vanish upon spatial average. The nonlocal term in Eq. (4.19) is analyzed in the same way as in Sections II and III. It produces a mass shift (which is cancelled by the last term in Eq. (4.19) and an ohmic dissipation term (due to radiation damping). Another dissipation mechanism is the radiation of 4 field quanta by the background. This mechanism appears to 0(1’). If we neglect the 4: components, the dynamical equation for 6, is then [28] (4.26) Equations (4.26), (4.18), and (4.14) are the closed system which describes the approach to the true vacuum, while Eq. (4.16) describes the thermalization of the 4 and Ic/ fields.

V. CONCLUSIONS We have presented here a detailed dynamical description of spinodal decomposition in a quantum field system. Our model is based on the use of the twoparticle irreducible In-In effective action. As compared with earlier work on critical dynamics in condensed systems [4], our model is qualitatively similar to that of Langer and others [6,7]. These models were not relativistic, and the use of the In-In effective action bypasses the delicate question of the correct generalization of the Fokker-Planck equation to quantum field theoretic systems. Our method is also free of the several phenomenological approximations made in the usual approach. The main advantage of the In-In effective action with respect to the usual effective action method [16] is that the In-In effective action allows a rigorous and systematic study of the evolution of a field theoretic system for given initial data. This is of course desirable for the problem at hand, although one can conceive problems in which mixed advanced and retarded asymptotic conditions may be of greater use than a purely causal formulation such as ours [17]. The In-In formulation is not technically more difficult than the usual one. Although formally there are more Feynman graphs to compute, usually the new graphs can be computed automatically by means of their relationship to the graphs in the conventional approach [29]. Moreover, in a causal formulation it is straightforward to identify the different dissipative mechanisms at work on the quantum field. Going from the one-particle irreducible to the two-particle irreducible effective action is essential in allowing us to go beyond mean held theory. Using the 2PI effective action, we can analyze in detail the very beginning of the phase transition. In particular, we do not have to make special choices of initial conditions (such as to

SPINODAL DECOMPOSITION IN QUANTUM FIELD THEORY

51

assume that the field is in an asymmetric state already at time t = 0) or to make a hybrid of spinodal decomposition and condensation theory by assuming that a “small” bubble forms, and only then spinodal decomposition begins. We therefore believe we have a strong case for the usefulness of the closed time path formulation of critical dynamics in quantum field theory. We also believe we have only begun to tap the power of this technique. It could be said that so far we have used the In-In effective action only as a clever device to write down causal Dyson equations. It would have been possible (though much more difficult) to arrive at these equations by non-path-integral methods. A formalism in which the path-integral nature of the method really comes to the foreground would open to us a vast area of unexplored physics, such as the simultaneous treatment of nucleation and spinodal decomposition [26]. On these lines we are continuing our research. ACKNOWLEDGMENTS I thank Bei-Lok Hu for many helpful discussions. This work originated while I was at the University of Maryland, with support from the National Science Foundation under Grant PHY 85-15689.

REFERENCES 1. “The Scientific Papers of J. Williard Gibbs,” p. 105, Dover, New York, 1961. J. W. CAHN, Trans. Metall. Sot. AIME 242 (1968), 166; Acta Metal. 14 (1966), 1685. A. H. GUTH, in “The Very Early Universe” (G. W. Gibbons, S. W. Hawking, and S. T. C. Siklos, Eds.), p. 171, Cambridge Univ. Press, London/New York, 1983; A. LINDE, ibid., p. 205. 4. For a comprehensive review see J. D. GUNTON, M. SAN MIGUEL, AND P. S. SAHNI, in “Phase Transitions and Critical Phenomena” (C. Domb and J. J. Kebowitz, Eds.), Vol. 8, Academic Press, San Diego, CA, 1983; J. D. GIJNTON AND M. DROZ, in “Lecture Notes in Physics,” Vol. 183, SpringerVerlag, New York/Berlin, 1983. 5. A. D. LINDE, Rep. Prog. Theor. Phys. 47 (1984), 925; R. BRANDENBERGER,Rev. Mod. Phys. 57 (1985), 1. 6. J. S. LANGER, M. BARON, AND H. D. MILLER, Phys. Rev. A 11 (1975), 1417; J. S. LANCER, in “Fluctuations, Instabilities and Phase Transitions” (T. Riste, Ed.), p. 19, Plenum, New York, 1975. 7. C. BILLOTET AND K. BINDER, 2. Phys. B 32 (1979), 195; K. BINDER, in “Systems Far from Equilibrium” (L. Garrido, Ed.), p. 76, Lecture Notes in Physics, Vol. 132, Springer-Verlag, New York/Berlin, 1980. 8. J. W. CAHN AND J. E. HILLIARD, J. Chem. Phys. 31 (1959), 688. 9. H. E. COOK, Acta Metal 18 (1970), 297. 10. J. S. LANCER, Ann. Phys. (N.Y.) 54 (1969), 258; 41 (1967), 108. Il. G. ZHOU, Z. Su, AND B. HAO, Phys. Rev. B 22 (1980), 3385. 12. V. P. SKRIPOV AND A. V. SKRIPOV, Usp. Fiz. Nauk 128 (1979), 193; Sov. Phys. Usp. 22 (1979), 389. 13. G. F. MAZENKO, W. G. UNRUH, AND R. M. WALD, Phys. Rev. D 31 (1985), 273. 14. A. ALBRECHT AND R. H. BRANDENBERGER, Phys. Rev. D 31 (1985), 1225; A. H. GUTH AND S. Y. PI, Phys. Rev. D 32 (1985), 1899. 15. P. AMSTERDAMSKI,Phys. Rev. D 35 (1987), 2323. 16. R. JACKIW, Phys. Rev. D 9 (1974), 1686; J. ILIOPOULOS,C. ITZYKSON, AND A. MARTIN, Reu. Mod. Phys. 47 (1975), 165. 2. 3.

58

ESTEBAN

CALZETTA

17. See E. CALZETTA AND B. L. Hu, Phys. Rev. D 35 (1987), 495, and references therein. 18. A. A. STAROBINSKY, in “Field theory, Quantum Gravity and Strings” (H. J. de Vega and N. Sanchez, Eds.), p. 107, Lecture Notes in Physics, Vol. 246, Springer-Verlag, New York/Berlin, 1986; J. BARDEEN AND G. BUBLIK, Class. Quantum Grav. 8 (1987) 573; S. J. REY, Nucl. Phys. B 284 (1987) 706. 19. J. SCHWINGER, J. Math. Phys. 2 (1961) 407; G. Z. ZHOU, Z. B. Su, B. L. HAO, AND L. Yu, Phys. Rep. C 118 (1985), 1. 20. J. M. CORNWALL, R. JACKIW, AND E. TOMBOLJLIS, Phys. Rev. D 10 (1978), 2428. 21. E. CALZETTA AND B. L. Hu, Phys. Rev. D 37 (1988), 2878; E. CALZETTA, in “Proceedings, 1987 CAP/NSERC Summer Institute on Quantum Field Theory” (F. Khanna, G. Kunstatter, H. C. Lee, and H. Umezawa, Eds.), p. 371, World Sci., Singapore, 1988. 22. G. MAZENKO, Phys. Rev. Left. 54 (1985), 2163; Phys. Rev, D 34 (1986). 2223. 23. K. BINDER, C. BILLOTET, AND P. MIROLD, Z. Phys. B 30 (1978). 183. 24. V. GEROLD AND G. KOSTORZ, J. Appl. Crystallogr. 11 (1978), 376. 25. K. BINDER. Phys. Rev. A 29 (1984) 341; J. S. LANCER, in “Systems Far From Equilibrium” (L. Garrido, Ed.), p. 12, Lecture Notes in Physics, Vol. 132, Springer-Verlag, New York/Berlin, 1980. 26. K. LEE AND E. WEINBERG, Nucl. Phys. B 267 (1986), 181; K. LEE, Nucl. Phys. B 282 (1987), 509. 27. L. PARKER, in “Asymptotic Structure of Space-Time” (F. P. Esposito and L. Witter, Eds.), p. 107, Plenum, New York,. 1977. 28. Similar equations for the evolution of the Higgs field have heen derived in A. Hosoya and M. Sakagami, Phys. Rev. D 29 (1984), 2228; M. MORIKAWA AND M. SASAKI, Prog. Theor. Phys. 72 (1984), 782; M. SAKAGAMI AND T. KUBATA, Prog. Theor. Phys. 76 (1986), 548; A. RINGWALD, 2. Phys. C 34 (1987) 481. 29. A. 0. BARVINSKY AND G. A. VILKOVISKY, Nucl. Phys. B 282 (1987). 163.