Stochastic quantization of real-time thermal field theory

Stochastic quantization of real-time thermal field theory

T. C. de Aguiar, N. F. Svaiter, and G. Menezes

Citation: J. Math. Phys. 51, 102304 (2010); doi: 10.1063/1.3492927 View online: http://dx.doi.org/10.1063/1.3492927

View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v51/i10 Published by the AIP Publishing LLC.

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Stochastic quantization of real-time thermal field theory

T. C. de Aguiar,1,a兲N. F. Svaiter,1,b兲and G. Menezes2,c兲 1

Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Rio de Janeiro 22290-180, Rio de Janeiro, Brazil

2

Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, São Paulo 01140-070, São Paulo, Brazil

共Received 11 May 2010; accepted 2 September 2010; published online 19 October 2010兲

We use the stochastic quantization method to obtain the free scalar propagator of a finite temperature field theory formulated in the Minkowski space-time. First, we use the Markovian stochastic quantization approach to present the two-point func-tion of the theory. Second, we assume a Langevin equafunc-tion with a memory kernel and a colored noise. The convergence of the Markovian and non-Markovian sto-chastic processes in the asymptotic limit of the fictitious time is obtained. Our formalism can be the starting point to discuss systems at finite temperature out of equilibrium. ©2010 American Institute of Physics. 关doi:10.1063/1.3492927兴

I. INTRODUCTION

In order to study perturbative quantum field theory at finite temperature, there are three established formalisms: the imaginary time formalism or the Matsubara formalism,1,2 the closed time path Green’s function formalism,3,4and the thermofield dynamics approach.5,6In the imagi-nary time formalism, the fundamental idea is to replace the time t by it in order to obtain the matrix elements of the equilibrium density matrix operator exp共−␤H兲. A perturbative calculation for equilibrium thermal field theory can be developed using this approach.7 For nonequilibrium quantum field systems, which must be formulated using the real time, the path ordered method or the thermofield dynamics must be used. The modern field theoretical approach in the closed time path Green’s function method, as functional technique and path integral representation, was intro-duced in the 1980s when it was developed as a unified framework to describe both the equilibrium and the nonequilibrium situations in thermal field theory. Finally, in the thermofield dynamics approach, some degrees of freedom associated with the vacuum in quantum field theory act as thermal degrees of freedom. This thermal field dynamics mechanism for the creation of noise in a pure state can be done by doubling each dynamical degree of freedom of the system. For a complete survey of this method, see, for example, Ref.8.

An alternative way of quantization is the stochastic quantization method. In this paper, we present an approach to study finite temperature field theory in the real-time formalism using the stochastic quantization method. The Markovian and the non-Markovian stochastic quantization procedures are presented in order to quantize a neutral free scalar field.9 The basic ideas of the non-Markovian Langevin equation with a memory kernel can be found in Refs.10–13.

The program of stochastic quantization, first proposed by Parisi and Wu,14and the stochastic regularization were carried out for systems described by fields defined in flat, Euclidean mani-folds. A brief introduction to stochastic quantization can be found in Refs.15–17and a complete review is given in Ref.18. Although the stochastic quantization method was formulated using the Euclidean space-time obtained by Wick’s rotation, several authors have attempted to formulate the stochastic quantization directly in Minkowski space-time 共see, for example, the discussion

pre-a兲_{Electronic mail: deaguiar@cbpf.br.}

b兲_{Electronic mail: nfuxsvai@cbpf.br.}

c兲_{Electronic mail: gsm@ift.unesp.br.}

51, 102304-1

sented by Hüffel and Rumpf19 and Gozzi20兲. In the first of these papers, the authors proposed a modification of the original Parisi–Wu scheme by introducing a complex drift term in the Lange-vin equation in order to implement the stochastic quantization in Minkowski space-time. Gozzi studied the spectrum of the non-self-adjoint Fokker–Planck Hamiltonian to justify this program 共see also Refs.21and22兲. We observe that the implementation of the stochastic quantization in the Minkowski space-time can be seen as a special case of the Euclidean formulation for systems with complex action since the drift term is purely imaginary in this case.

Even when the stochastic quantization is formulated in the Euclidean space-time, difficulties appear in the case of complex actions. A complex action leads to a nonpositive definite measure and the convergence of the stochastic process in the asymptotic limit of the fictitious time is not achieved. There are many examples in literature where Euclidean action is complex. We have, for example, quantum chromodynamics with nonvanishing chemical potential at finite temperature; for SU共N兲theories with N⬎2, the fermion determinant becomes complex and also the effective action. Complex terms can also appear in the Langevin equation for fermions, but a suitable kernel, which defines a kerneled Langevin equation, can circumvent this problem.23–25Finally, for topological field theories, the Euclidean action is also complex. The crucial question is the fol-lowing: what happens if the Langevin equation possesses a complex drift term? Some authors claimed that it is possible to obtain meaningful results out of the Langevin equations describing diffusion processes around a complex action. Parisi26and Klauder and Peterson27investigated the complex Langevin equation, where some numerical simulations in one-dimensional systems were presented共see also Refs.28and29兲. We would also like to mention the approach developed by Okamoto et al.,30 where the role of the kernel in the complex Langevin equation was studied. More recently, Guralnik and Pehlevan31constructed an effective potential for the complex Lange-vin equation on a lattice. These authors also investigated a complex LangeLange-vin equation and Dyson–Schwinger equations that appear in such situations.32

As we have pointed out, an important case that deserves our attention is the stochastic quan-tization of topological field theories in the Euclidean formulation. A topological action always contains a factor of i in the Euclidean space-time; therefore, the path integral measure for the topological theory always remains to beeiS _{even after the Wick rotation. Recently, Menezes and}

Svaiter33,34 implemented the stochastic quantization to study such systems with complex valued path integral weights, where we have a nonpositive definite measure. As mentioned, the conver-gence of the stochastic process in the asymptotic limit of the fictitious time is not achieved. To circumvent this problem, these authors assumed a Langevin equation with memory kernel and Einstein’s relations with colored noise. In the asymptotic limit of the fictitious time, the equilib-rium solution of such Langevin equation was analyzed. It was shown that for a large class of elliptic non-Hermitian operators, which define different models in quantum field theory, the solu-tion converges to the correct equilibrium state in the asymptotic limit of the fictitious time ␶ →⬁. In conclusion, in these situations for complex actions in Euclidean field theory, the non-Markovian approach may be more suitable to deal with nonconvergence problems than the Mar-kovian approach since it is a more general method.

The aim of this paper is to implement the stochastic quantization in the Minkowski space-time. The theory of real-time Green’s functions can be used to describe systems in equilibrium and also out of equilibrium. To include the interaction within this formalism, we can use the results presented in Refs.9and34. In Ref.9, the non-Markovian stochastic quantization for an interact-ing scalar field theory formulated in the Euclidean space was presented. Also usinteract-ing Ref.34, we see that the method proposed can be extended to interacting field theory with complex actions, where a consistent perturbation theory out of equilibrium can be developed.

The outline of this paper is the following. The introduction is given in Sec. I. In Sec. II, we present a brief review of the real-time formalism in quantum field theory. In Sec. III, we use the Markovian stochastic quantization method to study a thermal field theory formulated in the Minkowski space-time. The non-Markovian approach of the stochastic quantization applied to a thermal scalar field theory is developed in Sec. IV. Conclusions are given in Sec. V. In the Appendix, convergence conditions for the stochastic process are derived. Summation over re-peated indices will always be meant, unless otherwise stated. In this paper, we useប=c=kB= 1.

II. REAL-TIME FORMALISM IN FINITE TEMPERATURE QUANTUM FIELD THEORY

In this section, we give a brief survey of the formulation of field theory at finite temperature in Minkowski space-time. Unlike in the imaginary time formalism, in real-time formulation, sums over Matsubara frequencies are absent and there is no need to analytically extend the Green’s functions back to the Minkowski space-time. Moreover, the real-time formalism is the starting point for the development of the nonequilibrium quantum field theory since the investigation of dynamical properties of systems is more naturally performed in this formalism. The real-time formalism can describe nonequilibrium processes because the time variable plays a fundamental role and cannot be traded in for an equilibrium temperature.

For simplicity, we work with a neutral scalar field. The field operator in the Heisenberg picture is given by

␾共t,x兲₌eiHt␾共0,x兲e−iHt_, 共₁兲

where the time variablet is allowed to be complex. The main quantities to be computed are the thermal Green’s functionsGC共x1, . . . ,xN兲, defined as

GC共x1, . . . ,xN兲=具TC共␾共x1兲. . .␾共xN兲兲典␤, 共2兲

where the time ordering is taken along a complex time path, yet to be defined. Considering a parametrizationt=z共v兲of the path, the following expressions:

␪C共t−t

⬘

兲=␪共v−v

⬘

兲, 共3兲

␦C共t−t

_⬘

兲=

冉

⳵z

⳵v

冊

−1

␦共v−v

⬘

兲 共4兲

define the generalized ␪- and ␦-functions. The functional differentiation is also extended in the following way:

␦j共x兲

␦j共x

_⬘

兲=␦C共t−t

⬘

兲␦

3_共x

−x

_⬘

兲 共₅兲

for functions j共x兲 defined on pathC. The Green’s functions defined by Eq.共2兲 can be obtained from a generating functionalZC关␤;j兴through the expression

GC共x1, . . . ,xN兲=

共−i兲N ZC关␤;j兴

冏

␦N_Z C关␤;j兴 ␦j共x1兲¯␦j共xN兲

冏

j=0

. 共6兲

Z关␤;j兴= Tr

冋

e−␤HTCexp

冉

i

冕

C

d4xj共x兲␾共x兲

冊

册

=

冕

D_␾

_⬘

_具␾

_⬘

_{共x兲;t−i␤兩TCexp}

冉

_i

冕

C

d4xj共x兲␾共x兲

冊

兩␾

_⬘

共x兲;t典, 共7兲 where pathCmust go through all the arguments of the Green’s functions. It is also possible to note from this expression that pathCstarts from a timeti=tand ends at a timetf=t−i␤. We may recast

the generating functional into the form

ZC关␤;j兴=Nexp

再

−i

冕

C

d4xV

冉

␦

i␦j共x兲

冊

冎

ZC F_关

␤;j兴, 共8兲

whereN _{is a normalization parameter and the free generating functional is given by}

Z_CF关␤;j兴= exp

再

−1 2

冕

C

d4x

冕

C

d4y j共x兲D_CF共x−y兲j共y兲

冎

. 共9兲

In Eq.共9兲, the propagatorD_CF共x−y兲 is defined through the formula

D_CF共x−x

_⬘

兲=␪C共t−t

⬘

兲DC

⬎_共x

,x

_⬘

兲+␪C共t

⬘

−t兲DC

⬍_共x

,x

_⬘

兲, 共10兲

whereD_C⬎共x,x

_⬘

兲andD_C⬍共x,x

_⬘

兲 are, respectively,

D_C⬎共x,x

_⬘

兲=具␾共x兲␾共x

_⬘

兲典␤,

D_C⬍共x,x

_⬘

兲=具␾共x

_⬘

兲␾共x兲典␤. 共11兲

Since the propagatorD_CF共x−x

_⬘

兲is properly defined in the interval −␤ⱕIm共t−t

_⬘

兲ⱕ␤, one may conclude that the path considered must be such that the imaginary part of the time variable t is nonincreasing when the parameter v increases. Furthermore, since we are interested in Green’s

functions whose arguments are real, pathCmust contain the real axis. One possible choice for the contourCis described in the following:41

共1兲 Cstarts from a real valueti, large and negative.

共2兲 The contour follows the real axis up to the large positive value −ti. This part ofCis denoted

byC1.

共3兲 The path from −tito −ti−i

␤

2, along a vertical straight line. This is denoted byC3.

共4兲 The path follows a horizontal lineC₂going from −t_i−i␤₂ tot_i−i␤₂. 共5兲 Finally, the path follows a vertical lineC4 fromti−i

␤

2 toti−i␤.

Taking ti→−⬁, the free generating functional can be factorized as

Z_CF关␤;j兴=Z_C 1艛C2

F _关␤

;j兴Z_C 3艛C4

F _关␤

;j兴. 共12兲

The Green’s functions with real-time arguments can be deduced from Z_C 1艛C2

F _关

␤;j兴 only. The Z_C

3艛C4

F _关

␤;j兴generating functional can be considered a multiplicative constant. Choosingtandt

_⬘

to be real, running from −⬁to⬁, and label the sourcesj1共x兲=j共t,x兲andj2共x兲=j共t−i␤/2 ,x兲. Also,

one has␦j_a共x兲/_␦j_b共x

_⬘

兲=␦ab␦4共x−x

⬘

兲. With these expressions, one may rewrite the free generating

functional as

Z_CF关␤;j兴=N

_⬘

_exp

再

₋1

2

冕

d

4_x

冕

_d4_x

⬘

ja共x兲Dab F _j

b共x

⬘

兲

冎

, 共13兲

where, again, N

_⬘

_{is a normalization parameter. The components of the matricial propagator}

D₁₁F共x−x

_⬘

兲=DF共t−t

⬘

,x−x

⬘

兲,

D₂₂F共x−x

_⬘

兲=Dⴱ_F共t−t

_⬘

_,x−x

_⬘

_兲,

D₁₂F共x−x

_⬘

兲=D⬍共t−t

_⬘

+i␤/_2,x₋x

_⬘

兲_,

D₂₁F共x−x

_⬘

兲=D⬎共t−t

_⬘

−i␤/_2,x₋x

_⬘

兲_. 共₁₄兲

The effective generating functional can be written as

ZC关␤;j兴=

冕

D␾1D␾2exp

再

−

1 2

冕

d

4_xd4_x

⬘

␾a共x兲共DF −1_兲

ab共x−x

⬘

兲␾b共x

⬘

兲

冎

⫻exp

再

−i

冕

d4x共V共␾1兲−V共␾2兲兲+i

冕

d4xja共x兲␾a共x兲

冎

. 共15兲

The field ␾2 may be interpreted as a ghost field on the contour C2. This doubling of the field

degrees of freedom, which does not occur in imaginary time formulation, is unavoidable in the real-time formulation. For more details on this subject, the reader is referred to the original paper of Niemi and Semenoff42 or the Landsman and van Weert review.43

III. REAL-TIME FINITE TEMPERATURE QUANTUM FIELD THEORY: THE MARKOVIAN STOCHASTIC QUANTIZATION APPROACH

The real-time formalism is a framework to describe both equilibrium and nonequilibrium systems. Dynamical questions, as, for example, a weakly interacting Bose gas having a tempera-ture gradient, can be studied only in the real-time formalism, with a matrix structempera-ture of the propagator. Before we study the non-Markovian approach, in this section we will analyze the usual stochastic quantization of a finite temperature field theory formulated in the Minkowski space. In the Minkowski space, it is well known that the Langevin equation should be written as

⳵

⳵␶␾共x,␶兲=i

冏

␦S

␦␾共x兲

冏

_{␾共x兲=␾共x,␶兲}+␩共x,␶兲, 共16兲

whereS共␾兲is the action for a free scalar field

S共␾兲=

冕

d4x1 2兵⳵

␮_␾⳵

␮␾−m2−i⑀␾2其, 共17兲

and the correlation functions for the noise field are

具␩共x,␶兲典␩= 0, 共18兲

具␩共x,␶兲␩共x

⬘

,␶

_⬘

兲典␩= 2␦共兩␶−␶

⬘

兩兲␦4共x−x

⬘

兲. 共19兲

Owing to the presence of factor iin front of the drift term, the resulting stochastic field ␾共x,␶兲

becomes complex valued. Reference18emphasizes that the presence of the negative imaginary mass term −共i/₂兲_⑀␾2 _{in the action given by Eq.}共₁₇兲_{is necessary to obtain convergence for the}

stochastic process being considered. So, it means that we can only take the limit ⑀→_{0 after all}

calculations have been performed. It is straightforward to obtain the two-point correlation func-tions for this case as well as to develop a perturbative solution to the self-interacting situation. Although we do not wish to go into the details here, for the interested reader, we recommend Refs.

Now, let us focus our attentions to the finite temperature case in real-time formulation. As we have discussed in Sec. II, the doubling of the field degrees of freedom is unavoidable in the real-time formulation. So, we write the field as an isovector␾=

共

␾␾12

兲

. The action for the isovector

scalar field, in the free case, is given by

S= 1 2

冕

d

4_xd4_x

⬘

␾a共x兲共DF −1

兲ab共x−x

⬘

兲␾b共x

⬘

兲, 共20兲

where the components ofDFare given by Eq.共14兲. In the Fourier space,

共DF兲ab共k兲=共Ut兲ac共␪兲共D0兲cd共k兲共U兲db共␪兲, 共21兲

where

U共␪兲=

冉

cosh␪ sinh␪

sinh␪ cosh␪

冊

, 共22兲

with

cosh2␪= e ␤兩k0兩

e␤兩k₀兩_{− 1}, 共23兲

andD0is given by

D0共k兲=

冢

1

k2−m2+i⑀ 0

0 − 1

k2−m2−i⑀

冣

. 共24兲

So, we can split DF into two parts,DF共k兲=D0共k兲+D␤共k兲, whereD0共k兲 is temperature

inde-pendent and all temperature dependence appears inD␤共k兲, which is given by

D␤共k兲= −i⑀ 共k2_−m2_兲2_+⑀2

冉

2 sinh2␪ sinh 2␪

sinh 2␪ 2 sinh2_␪

冊

. 共25兲

Notice that in the limit⑀→_{0, we have}

⑀

共k2_−m2_兲2_+⑀2→␲␦共k 2

−m2兲. 共26兲

As in the zero temperature case, we have to use the above expressions with finite⑀and take the

⑀→_{0 limit after all the calculations have been done in order to obtain convergence in the limit} ␶→_{⬁. We also know that this is the case for the path integral formalism.}42

The Markovian Langevin equation for the nonequilibrium case is given by

⳵

⳵␶␾a共k,␶兲=i共DF −1_兲

ab共k兲␾b共k,␶兲+␩a共k,␶兲, 共27兲

where␾=

共

␾1

D_F−1共k兲=I共k,⑀兲

⫻

冢

− 1 k2−m2−i⑀−

i⑀

共k2

−m2兲2+⑀22 sinh

2_␪ i⑀

共k2

−m2兲2+⑀2sinh 2␪

i⑀

共k2−m2兲2+⑀2sinh 2␪

1 k2−m2+i⑀−

i⑀

共k2−m2兲2+⑀22 sinh 2_␪

冣

,

共28兲

where

I共k,⑀兲=

−共k2−m2兲2−⑀2_共

cosh4␪+ sinh4␪兲−⑀

2

2sinh

2

2␪

共共k2−m2兲2+⑀2_兲 . 共29兲

The noise correlation functions are given by

具␩a共k,␶兲典␩= 0, 共30兲

具␩a共k,␶兲␩b共k

⬘

,␶

⬘

兲典␩= 2共2␲兲4␦ab␦ 4_共k

+k

_⬘

兲␦共兩␶−␶

_⬘

兩兲, 共31兲

and, again, ␩=

共

␩1

␩2

兲

, a,b= 1 , 2. Here, we emphasize that since the drift term in the Langevin

equation共27兲is a complex quantity, each component␾a共k,␶兲 becomes complex,␾a=␾aR+i␾aI.

Maintaining a real noise, we obtain two Langevin equations,

⳵

⳵␶␾aR共k,␶兲= −Re关共DF −1_兲

ab共k兲兴␾bI共k,␶兲−Im关共DF −1_兲

ab共k兲兴␾bR共k,␶兲+␩a共k,␶兲 共32兲

and

⳵

⳵␶␾aI共k,␶兲=Re关共DF −1_兲

ab共k兲兴␾bR共k,␶兲−Im关共DF −1_兲

ab共k兲兴␾bI共k,␶兲. 共33兲

This separation into real and imaginary parts will be important when we discuss the Fokker– Planck approach. However, for the moment we will treat it as a single equation for the complex field␾a, i.e., we treat Eq.共27兲as a holomorphic one. So, its solution will be given by

␾a共k,␶兲=

冕

⬁

d␶

_⬘

gab共k,␶−␶

⬘

兲␩b共k,␶

⬘

兲, 共34兲

whereg共k,␶兲=eiD_F−1共k兲␶_{␪共␶兲is the Green’s function for the diffusion problem.}

Now, we should worry ourselves about the convergence of this stochastic process, i.e., we should analyze ifg共k,␶兲兩␶→⬁→0. In order to do so, we must first diagonalize the matrixiDF

−1_共k兲

. Doing such calculation, we get the matrixD

_⬘

共k兲, given by

D

_⬘

共k兲=iI共k,⑀兲

冉

␭+ 0 0 ␭−

冊

, 共35兲

where

␭⫾=

⫾

冑

共k2

−m2兲2−⑀2

sinh22␪−i⑀共1 + 2 sinh2␪兲

共k2−m2兲2+⑀2 . 共36兲

Eqs.共35兲and共36兲, if we take the⑀→_{0 limit in the beginning of the calculations, we should lose}

the convergence factor e−I共k,⑀兲⑀共1+2 sinh2␪兲_{. So, similar to the path integral formalism mentioned}

above, this limit should be taken after all the calculations have been done. In doing so, it is easy to see that we obtain convergence in the limit␶→_⬁.

Now, we are ready to calculate the two-point function 具␾a共k,␶兲␾b共k

⬘

,␶兲典␩. Proceeding with

similar calculations as in the zero temperature case, it is possible to show that the two-point correlation function is given by

具␾a共k,␶兲␾b共k

⬘

,␶兲典␩=共2␲兲4␦4共k+k

⬘

兲i共DF兲ac共k兲共1 −e2iDF

−1_共

k兲␶_兲

cb, 共37兲

so we see that in the limit ␶→⬁, we recover the usual result. We are interested now to see the effects of a memory kernel in this nonequilibrium quantum field theory. This is the subject of Sec. IV.

IV. REAL-TIME FINITE TEMPERATURE QUANTUM FIELD THEORY: THE NON-MARKOVIAN STOCHASTIC QUANTIZATION APPROACH

The aim of this section is to study finite temperature quantum field theory in the Minkowski time using the non-Markovian stochastic quantization approach. In the Minkowski space-time, the Langevin equation with memory kernel is written as

⳵

⳵␶␾共x,␶兲=i

冕

₀

␶

dsM⌳共␶−s兲

冏

␦S

␦␾共x兲

冏

_{␾共x兲=␾共x,s兲}+␩共x,␶兲, 共38兲

whereSis the action for the free scalar field, given by Eq.共20兲. The noise field distribution is such that its first and second momenta are given by

具␩a共x,␶兲典␩= 0, 共39兲

具␩a共x,␶兲␩b共x

⬘

,␶

⬘

兲典␩= 2␦abM⌳共兩␶−␶

⬘

兩兲␦4共x−x

⬘

兲, 共40兲

that is, we have a colored Gaussian distribution. We remind the reader that Eq. 共38兲 is to be understood as a matrix equation. Also, since the drift term in this non-Markovian Langevin equation is complex again, we have the same separation as before,␾_a=␾_aR+i␾_aIand we keep the noise as a real quantity.

Using a Fourier decomposition for the scalar and noise fields, given by

X共x,␶兲= 1 共2␲兲2

冕

d

4_keikx_X共k

,␶兲, 共41兲

where the field X represents either the noise field ␩ or the scalar field␾, we obtain that each Fourier mode␾共k,␶兲 satisfies a Langevin equation of the form

⳵

⳵␶␾a共k,␶兲=i

冕

₀

␶

dsM⌳共␶−s兲共D_F−1兲ab共k兲␾b共k,s兲+␩a共k,␶兲, 共42兲

whereD_F−1共k兲is the inverse ofDF共k兲, defined by Eq.共21兲. With this decomposition, we obtain the

following relations for the noise field Fourier components from Eqs.共39兲and共40兲:

具␩a共k,␶兲典␩= 0, 共43兲

具␩a共k,␶兲␩b共k

⬘

,␶

⬘

兲典␩= 2共2␲兲4␦abM⌳共兩␶−␶

⬘

兩兲␦4共k+k

⬘

兲. 共44兲

M共z兲=

冕

0

⬁

d␶M⌳共␶兲e−z␶, 共45兲

we obtain the solution for Eq.共42兲, subject to the initial condition␾a共k,␶= 0兲= 0,a= 1 , 2,

␾_a共k,␶兲=

冕

0

⬁

d␶

_⬘

G_ab共k,␶−␶

_⬘

兲␩b共k,␶

⬘

兲. 共46兲

In Eq. 共46兲, Gab共k,␶−␶

⬘

兲=⍀ab共k,␶−␶

⬘

兲␪共␶−␶

⬘

兲 is the retarded Green’s function for this

non-Markovian diffusion problem and each component of the⍀-matrix is defined through its Laplace transform,

⍀ab=关zI−iM共z兲DF −1_共k兲兴

ab −1

, 共47兲

whereI _{denotes a 2⫻2 identity matrix. Thus, the two-point correlation function in the Fourier}

representation is written as

具␾a共k,␶兲␾b共k

⬘

,␶

⬘

兲典␩=Dab共k,␶,␶

⬘

兲

=共2␲兲4_␦4_共k

+k

_⬘

兲

冕

0

␶ ds

冕

0

␶_⬘

ds

_⬘

关⍀共k,␶−␶

_⬘

兲⍀共k,␶−s

_⬘

兲兴abM⌳共兩s−s

⬘

兩兲.

共48兲

The two-dimensional Laplace transform of the above equation is given by

冕

0

⬁

d␶e−z␶

冕

0

⬁

d␶

_⬘

e−z⬘␶⬘

冕

0

␶ ds

冕

0

␶_⬘

ds

_⬘

关⍀共k,␶−␶

_⬘

兲⍀共k,␶−s

_⬘

兲兴abM⌳共兩s−s

⬘

兩兲

=关⍀共k,z兲⍀共k

_⬘

,z

_⬘

兲兴ab

冉

M共z兲+M共z

_⬘

兲

z+z

_⬘

冊

. 共49兲

Using Eq.共47兲, this expression becomes

冕

0

⬁

d␶e−z␶

冕

0

⬁

d␶

_⬘

e−z⬘␶⬘

冕

0

␶ ds

冕

0

␶_⬘

ds

_⬘

关⍀共k,␶−␶

_⬘

兲⍀共k,␶−s

_⬘

兲兴abM⌳共兩s−s

⬘

兩兲

=i

冉

⍀共k,z兲+⍀共k,z

⬘

兲

z+z

_⬘

−⍀共k,z兲⍀共k,z

⬘

兲

冊

_ac共DF兲cb共k兲. 共50兲

Applying the inverse transform, we obtain for the two-point function

Dab共k,␶,␶

⬘

兲= 2i共2␲兲4␦4共k+k

⬘

兲共⍀共k,兩␶

⬘

−␶兩兲−⍀共k,␶兲⍀共k,␶

⬘

兲兲ac共DF兲cb共k兲. 共51兲

In order to investigate the convergence of the above equation, we need to specify an expres-sion for the memory kernelM_⌳. We set

M⌳共␶兲=1 2⌳

2_e−⌳2兩␶兩_{. 共52兲}

Substituting the Laplace transform of Eq. 共52兲 into Eq.共47兲, we have that the ⍀-matrix is given by

⍀共k,␶兲=

冉

⍀11共k,␶兲 ⍀12共k,␶兲 ⍀21共k,␶兲 ⍀22共k,␶兲

冊

, 共53兲

where the components⍀ab共k,␶兲are given in the Appendix. So, we are in a position to present an

D_ab共k,␶,␶

_⬘

兲兩␶=␶_⬘→⬁=i共2␲兲4␦4共k+k

⬘

兲共DF兲ab共k兲 共54兲

so that, in the limit⑀→_{0, we have}

D_ab共k,␶,␶

_⬘

兲兩␶=␶_⬘→⬁;⑀→0=i共2␲兲4␦4共k+k

⬘

兲共DF兲ab共k兲兩⑀→0. 共55兲

A question still remains opened. What are, if any, the advantages of our non-Markovian method over the usual Markovian one? In order to answer such question, we shall apply a Fokker–Planck analysis. As we know from Euclidean stochastic quantization, correlation functions are introduced as averages over␩,

具␾共x1,␶1兲␾共x2,␶2兲¯␾共xn,␶n兲典␩

=N−1

冕

关d␩兴exp

冉

−1 4

冕

d

4_x

冕

_d␶␩2_共x

,␶兲

冊

␾共x1,␶1兲␾共x2,␶2兲¯␾共xn,␶n兲, 共56兲

where␾ obeys an Euclidean Langevin equation

⳵

⳵␶␾共x,␶兲= −

冏

␦S

␦␾共x兲

冏

_{␾共x兲=␾共x,␶兲}+␩共x,␶兲, 共57兲

andS is the Euclidean version of the action given by Eq.共17兲.Nis given by

N=

冕

关d␩兴exp

冉

−1 4

冕

d

4_x

冕

_d␶␩2_共x

,␶兲

冊

. 共58兲

An alternative way to write this average is to introduce the probability densityP关␾,␶兴, which is defined as44

P关␾,␶兴 ⬅

冕

关d␩兴exp

冉

−1 4

冕

d

4_x

冕

_d␶␩2_共x

,␶兲

冊

兿

y

␦共␾共y兲−␾共y,␶兲兲. 共59兲

In terms ofP, the correlation functions will read as

具␾共x1,␶1兲␾共x2,␶2兲¯␾共xn,␶n兲典␩=N

冕

关d␾兴␾共x1兲␾共x2兲¯␾共xn兲P关␾,␶兴. 共60兲

The free probability density Psatisfies the following Fokker–Planck equation:

⳵

⳵␶P关␾,␶兴=

冕

d

4_x ␦

␦␾共x兲冉 ␦ ␦␾共x兲+

␦S

␦␾共x兲

冊

P关␾,␶兴, 共61兲

with the initial condition

P关␾,0兴=

兿

y

␦共␾共y兲兲. 共62兲

The stochastic quantization says that we shall have

w. lim ␶→⬁P关

␾,␶兴= exp共−S关␾兴兲

兰关d␾兴exp共−S关␾兴兲, 共63兲

where the limit is supposed to be taken “weakly” in the sense of Ref.44.

corresponding Fokker–Planck equation. For the finite temperature real-time case, since we also have double field degrees of freedom, we have similar equations as the case worked out above. First, any stochastic expectation value can be written as

具F关␾1共x,␶兲,␾2共x,␶兲兴典␩=N

⬙

冕

兿

i=1 2

关d␩i兴exp

冉

−1 2

冕

d

4_x

冕

_d␶␩i共x

,␶兲␦ij␩j共x,␶兲

冊

F关␾1共x,␶兲,␾2共x,␶兲兴,

共64兲

where each component␾a共x,␶兲,a= 1 , 2, obeys Eq.共27兲and

N

_⬙

−1=

冕

兿

i=1 2

关d␩i兴exp

冉

−1 2

冕

d

4_x

冕

_d␶␩i共x

,␶兲␦ij␩j共x,␶兲

冊

. 共65兲

An usual procedure shows us that the distributional functional P关␾a,␶兴 obeys the following

Fokker–Planck equation in momentum space:

⳵

⳵␶P=

冕

d

4_k

冋

␦

2_P

␦␾aR共k兲␦␾aR共−k兲

+ ␦

␦␾aR共k兲

共共Re关共DF −1_兲

ab共k兲兴␾bI共k兲+Im关共DF −1_兲

ab共k兲兴␾bR共k兲兲P兲

+ ␦

␦␾aI共k兲

共共−Re关共DF −1_兲

ab共k兲兴␾bR共k兲+Im关共DF −1_兲

ab共k兲兴␾bI共k兲兲P兲

册

, 共66兲

withP=P关␾a,␶兴. Following similar steps as in Ref.45, we may get the relations

P关␾1,␾2,␶兴兩␶→⬁=Neq −1

exp

冉

−

冕

d4k␾aR共k兲I˜ab␾bR共−k兲

冊

⫻exp

冉

冕

d4k共␾aI共k兲共˜I+˜I3·共˜I2+R˜2兲−1兲ab␾bI共−k兲

− 2␾aR共k兲共˜I2·R˜−1兲ab␾bI共−k兲兲

冊

, 共67兲

whereR˜=Re关i共D_F−1兲_ab共k兲兴,˜I=Im关i共D−1_F兲_ab共k兲兴,共D_F−1兲_ab共k兲is the inverse of the共D_F兲_ab共k兲, given by Eq.共21兲, and

冏

⳵P关␾1,␾2,␶兴 ⳵␶

冏

_␶→⬁

→_O共e−I共k,⑀兲⑀共1+2 sinh2␪兲␶_兲

. 共68兲

In resemblance to the results found in Ref.45, in spite of the fact that the equilibrium distribution given by Eq.共67兲is not of the formeiS, it can be proved that the expectation values over this real probability distribution coincide with those obtained by the usual path integral formalism. Besides that, we see that the convergence factor found in Eq.共68兲depends onkvalues, so that it cannot be put outside thek integral that appears when one is calculating⳵P关␾,␶兴/⳵␶. So, in principle, the convergence in the limit␶→_{⬁ is a bit more cumbersome for this case.}

In our real-time non-Markovian case, we notice the similarity between our retarded Green’s functionGab共k,␶兲and the one found in Ref.9, we may follow similar steps to calculate the free

⳵

⳵␶P=

冕

d

4_k

冋

␦

2

␦␾aR共k兲␦␾bR共−k兲Kab共k,␶兲P−

␦

␦␾aR共k兲

冉

Re

冉

i

␦N

␦␾aR共k兲

冊

P

冊

+

− ␦

␦␾aI共k兲

冉

Im

冉

i ␦N

␦␾aI共k兲

冊

P

冊

册

, 共69兲

where

Kab共k,␶−s兲 ⬅2

冕

0

␶

dsM⌳共␶−s兲Gab共k,␶−s兲, 共70兲

withGab共k,␶−s兲 being the retarded Green’s function for the non-Markovian diffusion problem,

and

N=N关␾;k,␶−s兴 ⬅

冕

0

␶

dsM_⌳共␶−s兲S共␾共k兲兲, 共71兲

withS being the Fourier transform of Eq.共20兲. Again, following similar steps as those in Refs.9

and 45, it is easy to verify that an equilibrium solution to this non-Markovian Fokker–Planck equation will be of a similar form of Eq. 共67兲, apart from some possible unimportant constants. Therefore, the probability distribution P关␾,␶兴 will satisfy, up to constants, similar relations as obtained above. However, in our approach, the convergence factor will always be given bye−⌳2␶_, which does not depend onkor⑀values共remember that we consider⑀→_{0, even though this limit}

is only taken at the end of the calculations兲. It is worth remarking that even though the drift term in the Langevin equation is complex, the probability distribution obtained is real, and the complex character appears only in the field. Since⌳ is a free parameter of the model, it can always be adjusted so that we may get an improved convergence. Of course, these situations would be best analyzed in a numerical simulation, which is outside the scope of the present work.

V. CONCLUSIONS

To quantize a classical thermal field theory out of equilibrium using the stochastic quantiza-tion, we have to work in the Minkowski space-time, where, naturally, an imaginary drift term appears in the Langevin equation, which leads to problems with convergence of the stochastic process. The problems of nonconvergence of the Langevin equation in the stochastic quantization framework also appears if someone considers the stochastic quantization of classical fields defined in a generic curved manifold. For curved static manifolds, the implementation of the stochastic quantization is straightforward.46,47 Nevertheless, for nonstatic curved manifolds, we have to ex-tend the formalism beyond the Euclidean signature, i.e., to formulate the stochastic quantization in pseudo-Riemannian manifold, instead of formulating it in the Riemannian space, as was originally proposed.

The main difference between the implementation of the stochastic quantization in the Minkowski space-time and in the Euclidean space-time is the fact that in the latter case the approach to the equilibrium state is a stationary solution of the Fokker–Planck equation. In the Minkowski formulation, the Hamiltonian is non-Hermitian and the eigenvalues of such Hamil-tonian are in general complex. The real part of such eigenvalues is important to the asymptotic behavior at large Markov time, and the approach to the equilibrium is achieved only if we can show its positive semidefiniteness.

based on stochastic quantization using a non-Markovian Langevin equation proved to be well suited to quantize a classical free field out of equilibrium in the real-time formalism at finite temperature.

Although nontrivial, the method can be extended to interacting field theory where a consistent perturbation theory out of equilibrium can be developed. Being more precise, the reader may recall that for weakly coupled fields, the interacting theory is obtained by a perturbation series over the free Gaussian theory. In this situation, the Green’s function for the diffusion problem that is used to solve the Langevin equation is the same as the one in the Langevin equation for the noninter-acting field theory. Since the convergence in the limit of the fictitious time ␶→⬁ is analyzed through this Green’s function, we notice that the convergence criterion for the interacting case should be the same as the free case. Therefore, apart from some tedious algebra manipulations, the extension of our method to self-interacting theories may be straightforward even though it may not be a trivial task. The implementation of this non-Markovian Langevin equation to study interact-ing field theory out of equilibrium is under investigation by the authors.

ACKNOWLEDGMENTS

We would like to thank Gastão Krein for many helpful discussions. N. F. Svaiter would like to acknowledge the hospitality of the Instituto de Física Teórica, Universidade Estadual Paulista, where part of this paper was carried out. This paper was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico do Brazil共CNPq兲and Fundação de Amparo a Pesquisa de São Paulo共FAPESP兲.

APPENDIX: Calculation of componentsΩ-matrix

In this appendix, we derive the components⍀ab共k,␶兲. We can write the⍀-matrix as an inverse

of matrixA, whose components are given by

A=

冉

z−iM共z兲d

⬘

iM共z兲b

⬘

iM共z兲b

⬘

z−iM共z兲a

_⬘

冊

. 共A1兲

The quantities that appear in theA-matrix are defined by

a

_⬘

= a

ad−b2, 共A2兲

b

_⬘

= b

ad−b2, 共A3兲

d

_⬘

= d

ad−b2, 共A4兲

and

a= 1 k2−m2+i⑀−

i⑀

共k2

−m2兲2+⑀22 sinh 2_␪

, 共A5兲

b= −i⑀ 共k2

d= − 1 k2−m2−i⑀−

i⑀

共k2−m2兲2+⑀22 sinh 2_␪

. 共A7兲

So, we will have

共⍀兲ab共k,z兲=共A−1兲ab=

冉

⍀11共k,z兲 ⍀12共k,z兲

⍀21共k,z兲 ⍀22共k,z兲

冊

, 共A8兲

where

⍀11共k,z兲=

z−iM共z兲a

_⬘

共z−iM共z兲a

_⬘

兲共z−iM共z兲d

_⬘

兲+M2共z兲b

_⬘

2, 共A9兲

⍀12共k,z兲=⍀21共z兲=

−iM共z兲b

⬘

共z−iM共z兲a

⬘

兲共z−iM共z兲d

⬘

兲+M2共z兲b

_⬘

2, 共A10兲

⍀22共k,z兲=

z−iM共z兲d

_⬘

共z−iM共z兲a

_⬘

兲共z−iM共z兲d

_⬘

兲+M2共z兲b

_⬘

2. 共A11兲

The Laplace transform for the memory kernel关Eq.共52兲兴is given by

M共z兲=⌳

2

2 1

z+⌳2. 共A12兲

So, inserting this result into Eqs.共A9兲–共A11兲, we get

⍀11共k,z兲=

P共z,ta兲

Q共z兲 , 共A13兲

⍀12共k,z兲=⍀21共z兲=

−共tbz+tb⌳2兲

Q共z兲 , 共A14兲

⍀22共k,z兲=

P共z,td兲

Q共z兲 , 共A15兲

wheretj=i共j

⬘

⌳2/2兲,j=a,b,d, and

P共z,t_j兲=z3+ 2⌳2z2+共⌳4−t_j兲z−t_j⌳2, 共A16兲

Q共z兲=z4+ 2⌳2_z3

+共⌳4

−u兲z2−u⌳2_z

+v, 共A17兲

withu=i共a

_⬘

+d

_⬘

兲⌳2_{/2 andv=共b}

⬘

2_−a

⬘

d

_⬘

兲⌳4_{/4. From Eqs.共A2兲–共A4兲, we have that}

u= ⌳

2_{⑀共1 + 2 sinh}2_{␪兲共共k}2_−m2_兲2_+⑀2_兲

−共k2−m2兲2−⑀2_共

cosh4␪+ sinh4␪兲+⑀

2

2sinh

2

2␪

共A18兲

v= −

⌳4

4

共共k2−m2兲2+⑀2_兲

−共k2−m2兲2−⑀2_共

cosh4␪+ sinh4␪兲+⑀

2

2sinh

2

2␪

, 共A19兲

so u⬍0 and v⬎0. In order to get the inverse Laplace transform of each component of the

⍀-matrix, we must seek for the solutions of the quartic equationQ共z兲= 0. As it is well known, a general quartic equation is a fourth-order polynomial equation of the form

z4+a3z3+a2z2+a1z+a0= 0. 共A20兲

Using the familiar algebraic technique,48it is easy to show that the roots of Eq.共A20兲are given by

z1= −

1 4a3+

1 2R+

1

2D, 共A21兲

z₂= −1 4a3+

1 2R−

1

2D, 共A22兲

z3= −

1 4a3−

1 2R+

1

2E, 共A23兲

z4= −

1 4a3−

1 2R−

1

2E, 共A24兲

where

R⬅

冉

1 4a3

2

−a2+y1

冊

1/2

, 共A25兲

D⬅

再

共F共R兲+G兲

1/2

for R⫽0 共F共0兲+H兲1/2

for R= 0,

冎

共A26兲

E⬅

再

共F共R兲−G兲

1/2

for R⫽0

共F共0兲−H兲1/2 for R= 0,

冎

共A27兲 F共R兲 ⬅3

4a3

2

−R2− 2a2, 共A28兲

H⬅2共y₁2− 4a0兲1

/2

, 共A29兲

G⬅1

4共4a3a2− 8a1−a3

3_兲R−1_{, 共A30兲}

andy1is a real root of the following cubic equation:

y3−a2y2+共a1a3− 4a0兲y+共4a2a0−a1 2

−a₃2a0兲= 0. 共A31兲

For convenience, let us assume thatR, defined by Eq.共A25兲, does not vanish. Comparing Eqs. 共A17兲 and 共A20兲, we easily see that a3= 2⌳2, a2=⌳4−u, a1= −u⌳2, and a0=v. Therefore, the

⍀11共k,␶兲=

P共z1,ta兲

共z1−z2兲共z1−z3兲共z1−z4兲

ez1␶

+ P共z2,ta兲 共z2−z1兲共z2−z3兲共z2−z4兲

ez2␶

+ P共z3,ta兲 共z3−z1兲共z3−z2兲共z3−z4兲

ez₃␶

+ P共z4,ta兲 共z4−z1兲共z4−z2兲共z4−z3兲

ez4␶_, 共_A32兲

⍀12共k,␶兲=⍀21共k,␶兲= −

冋

tbz1+tb⌳2

共z1−z2兲共z1−z3兲共z1−z4兲

ez1␶

+ tbz2+tb⌳

2

共z2−z1兲共z2−z3兲共z2−z4兲

ez2␶

+ tbz3+tb⌳

2

共z3−z1兲共z3−z2兲共z3−z4兲

ez3␶

+ tbz4+tb⌳

2

共z4−z1兲共z4−z2兲共z4−z3兲

ez4␶

册

_, 共_A33兲

and finally,⍀22共k,␶兲=⍀11共k,␶;ta→td兲. The rootsziare given by

z1= −

⌳2

2 +

1 2i␴+

1

2i␥, 共A34兲

z2= −

⌳2

2 +

1 2i␴−

1

2i␥, 共A35兲

z3= −

⌳2

2 −

1 2i␴+

1

2i␥, 共A36兲

z4= −

⌳2

2 −

1 2i␴−

1

2i␥, 共A37兲

with␴=共兩u兩−y1兲1

/2_{and␥=共−⌳}4_+兩u兩+y 1兲1

/2_{being real quantities. Equations共A32兲and共A33兲can}

be rewritten in a simpler form as

⍀11共k,␶兲= −

冋

冉

cos

冉

共␴+␥兲 2 ␶

冊

+

⌳2

共␴+␥兲sin

冉

共␴+␥兲

2 ␶

冊冊

h1

+

冉

cos

冉

共␴−␥兲 2 ␶

冊

+

⌳2

共␴−␥兲sin

冉

共␴−␥兲

2 ␶

冊冊

h2

+ 8tasin

冉

␴␶

2

冊

sin

冉

␥␶

2

冊

+ 4ta⌳2

冉

g1sin

冉

共␴+␥兲

2 ␶

冊

+g2sin

冉

共␴−␥兲

2 ␶

冊冊

册

e共−⌳2/2兲␶

⍀12共k,␶兲=⍀21共k,␶兲=

t_b

2␴␥

冋

⌳2

共␴+␥兲sin

冉

共␴+␥兲

2 ␶

冊

− ⌳2

共␴−␥兲sin

冉

共␴−␥兲

2 ␶

冊

+

− 2 sin

冉

␴␶ 2

冊

sin

冉

␥␶

2

冊

册

e 共−⌳2/2兲␶

, 共A39兲

where

h1= −共␴+␥兲2−⌳4, 共A40兲

h2=共␴−␥兲2+⌳4, 共A41兲

g₁=i− 2⌳

2

共␴+␥兲, 共A42兲

g2=i−

2⌳2

共␴−␥兲, 共A43兲

and, as before, ⍀22共k,␶兲=⍀11共k,␶;ta→td兲. Let us consider the convergence of the stochastic

process. In order for our stochastic process to converge, the retarded Green’s function for the diffusion problem should obey Gab共k,␶兲兩␶→⬁→0. In other words, we must have ⍀ab共k,␶兲兩␶→⬁

→_{0. From the above expressions, it is easy to see that the stochastic process will converge if all}

roots of the polynomialQ共z兲are negative. Indeed, from the usual relations between the zeros of a polynomial and the coefficients of the polynomial equation, we get兿izi=a0=v⬎0. So, we have an

even combination of equal signs in the roots. In other words, it is possible to obtain four negative roots as long as we have

冑

−兩u兩+y1+

冑

⌳4−兩u兩−y1⬍ ⌳2. Another way to reach convergence is to

demand that all roots are complex and set each real part of the roots to be negative. Since, in our case, all the real parts are equal and negative, we have chosen to follow this way. So, we will have convergence if the quantities␴and␥are real, as imposed before. This leads us to the following conditions: 兩u兩−y₁⬎0 and 兩u兩+y₁−⌳4_{⬎0, or, combining those requirements, 兩u兩⬎max兵y}

1,⌳4

−y₁其.

Now let us present the quantityy1. As was stated before,y1is a real root of a cubic equation

z3+b2z2+b1z+b0= 0. 共A44兲

Comparing Eqs. 共A17兲, 共A20兲, 共A31兲, and 共A44兲, we have the following identifications: b2=u

−⌳4_,b

1= −2⌳4u− 4v, andb0= −4uv−u2⌳4. If we let

q=1 3b1−

1 9b2

2

, 共A45兲

r=1

6共b1b2− 3b0兲− 1 27b2

3

, 共A46兲

we will have that

q= −4 9⌳

4_u

− ⌳

8

9 − 4 3v−

u2

9 , 共A47兲

r=4 9⌳

8_u

+ 2 3⌳

4_v

+ 1 18⌳

4_u2

+4 3uv−

1 27⌳

12

+ 1 27u

3

. 共A48兲

So, writings1=共r+

冑

q3+r2兲1

/3

ands2=共r−

冑

q3+r2兲1

/3

y₁=共s₁+s₂兲+ ⌳

4

−u

3 . 共A49兲

As one can see,y1⬎0. Also, in the limit⑀→0,y1 becomes a polynomial of⌳.

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