D Boyanovsky

^{3}.

The emergence of an effective field theory out of equilibrium is studied in the case in which a light field—the system—interacts with very heavy fields in a finite temperature bath. We obtain the reduced density matrix for the light field, its time evolution is determined by an effective action that includes the

Article funded by SCOAP^{3}.

Effective field theory is a powerful organizational principle to describe phenomena below some energy scale, or alternatively on large spatio-temporal scales, and is ubiquitous across fields. Several applications of effective field theory have become the pillars of fundamentally important paradigms, for example: universality in critical phenomena emerges at long wavelengths after coarse graining over short wavelength degrees of freedom a lá

The influence action approach has recently been argued to provide an effective description of the dynamics of long-wavelength fluctuations when combined with a Wilsonian approach to coarse graining the short wavelength components [

_{1} decaying into the light field

We obtain the time evolution of the reduced density matrix and the non-equilibrium effective action by tracing over the ‘bath’ degrees of freedom to order ^{2} in terms of the various correlation functions of the ‘bath’, which are determined by the spectral density in all cases. For the case

From the non-equilibrium influence action we obtain a

We obtain the quantum master equation up to

We study the non-equilibrium effective action for the case of interacting scalar fields, spinor and vector fields may be included, while technically more involved their treatment follows without major conceptual difficulties. We consider a bosonic field

Specifically we consider the following cases:

Only one field

Two different fields

Two different fields with the same hierarchy as the previous case but now with

While we are ultimately interested in obtaining an effective quantum field theory by tracing out ‘heavy degrees of freedom’ in cosmology, in this study we focus on Minkowski space time and consider that the fields

Consider the initial density matrix at a time

The factorization of the initial density matrix is an assumption often explicitly or implicitly made in the literature, it can be relaxed by including initial correlations, we will not consider here this important case, relegating it to future study. The initial factorization entails that the time evolution of the system can be described as ‘switching-on’ the coupling between fields at the initial time. This will result in transient dynamics, however, we will focus on the long time evolution.

In the field basis the matrix elements of

The physical situation described by this initial initial density matrix is that of a field (or fields) in thermal equilibrium at a temperature

To obtain the effective quantum field theory out of equilibrium for the light field

The time evolution of the initial density matrix is given by

The field variables

The reduced density matrix for the light field

In the above path integral defining the influence action,

The

With the purpose of comparing the influence functional approach to the quantum master equation developed in section

In the term with

In the term with

In the term with

In the term with

We find

We note that

The dynamics and dissipative processes depend on the correlation functions of the environment and crucially on their spectral density. In appendix

The spectral densities for the different cases are obtained in detail in appendix

One

In this case it is straightforward to find (see appendix

Two

In appendix

The contribution

The contribution

The contribution

Considering the factorized initial density matrix (

In order to relate the long time behavior of the influence functional to the spectral representation of the bath correlation functions we define_{2} we introduce (for the spatial Fourier transforms)_{2}^{1}

Alternatively take

This interpretation becomes clear by considering the two cases (

If the currents

In this case the total effective action (

Effective vertex in case (a) for

The term _{0} at momentum

We note that as

As we are considering a ‘low energy’ effective quantum field theory of a light field after integrating out the heavy fields, all the correlation functions of the light field are restricted to feature transferred momenta well below the two particle continuum, in particular

Effective vertex in case (b) for

For the case

Effective vertex in case (b) for

Although we focused on the limits

In the case of

This local effective and

When _{0} becomes larger than multiparticle threshold the dissipative contribution to the influence action,

For

In this section we study new dynamical phenomena associated with the

In order to establish a clear relation to a stochastic description we study the case

Correlation functions of the fields

Therefore we consider the generating functional

We introduce the center of mass and relative variables

The Wigner transform leads to a quasiclassical description of the dynamics, it is the closest ‘proxy’ to a (semi) classical phase space distribution [

The inverse transform of the Wigner function is given by

In terms of the spatial Fourier transforms of the center of mass and relative variables introduced above (

The term quadratic in the relative variable

We now set the sources

The non-equilibrium generating functional can now be written in the following form

The functional integral over _{i} can now be done, resulting in a functional delta function, that fixes the boundary condition

Finally the path integral over the relative variable can be performed, leading to a functional delta function and the final form of the generating functional given by

The meaning of the above generating functional is the following: in order to obtain correlation functions of the center of mass Wigner variable

There are two different averages:

The average over the stochastic noise term, which up to this order is Gaussian. We denote the average of a functional

In the literature it is usually

While such an approximation may lead to an agreement with the long time dynamics in special cases, it is generally a crude approximation that merits scrutiny in each case.

The average over the initial conditions with the Wigner distribution function

The average in the time evolved full density matrix is therefore defined by

Calling the solution of (

This result is remarkably similar to the Martin–Siggia–Rose [

The solution of the Langevin equation (

The real time solution for

In the non-interacting theory the integrand in (^{2} the

The Breit–Wigner approximation neglects the perturbatively small contribution from the continuum along the branch cuts, describing the propagator in terms of simple resonances with complex poles near the real axis. The contribution from the continuum gives rise to power law long time tails dominated by the behavior of the spectral density near the thresholds, with Landau damping being the dominant contribution asymptotically for long time [

Introducing this solution into (

At

However, for

For

In the narrow width approximation

At

The physical reason by which the

The interaction Hamiltonian

Quantum kinetic interpretation of the decay width: the decay process

This kinetic equation may be written as

Since

This is one of the important results of this article: the decay of heavy fields in the medium into the light field leads to the thermalization of the light field with the heavy degrees of freedom. The dynamics of thermalization is non-unitary and is manifest in the dissipative kernels which are determined by the support of the spectral density of the bath on the mass shell of the light degree of freedom. This results in that the light field is described as a

The equivalence between the results from the stochastic description and the quantum kinetic equation confirms the arguments of [

The case

For a generic coupling we now implement a Kramers–Moyal expansion in the relative coordinate [

Again, the term quadratic in

From the time evolution of the Wigner function, or generating functional (

In the master equation approach [^{4} and can be neglected to second order. The same analysis can be applied to all the other terms in (

Therefore in the Markov approximation the quantum master equation becomes

The relation between this master equation in operator form and the effective action of the previous section which is cast in terms of functional integrals in the field basis is not

The factorization assumption (

In order to understand how a local effective field theory emerges from the quantum master equation, and to establish contact with the results of section _{0} and momentum

The equivalence with the influence function and the results of the stochastic Langevin equation in particular the dynamics of damping and thermalization become more clear by studying the case

Taking the long time limit at this stage using the results on the right-hand side of equation (^{2} with the result in equation (

For any interaction picture operator

Therefore in the long time limit we find

To establish the relation with the influence action result (^{2}. Furthermore, the solutions to the equations (^{2}. The renormalization of the frequency

The solutions (

However, keeping the time dependence of

On the other hand, in the case when ^{2} we can neglect the exponential terms in (

An important conclusion from the equivalence with the stochastic description is the identification between the ‘rotating wave approximation’ and the Breit–Wigner approximation for the propagator (

In conclusion, the ‘counterrotating’ terms always yield perturbatively small contributions that are bound in time. They are negligible in the case when the spectral density of the bath has support on the mass shell of the light field (resonant terms), and are perturbatively small in the case when it does not, in agreement with the perturbative corrections obtained from the influence action approach in this case, (see equation (

We have established the relation between the quantum master equation and the influence action and compared the evolution of expectation values and correlation functions from the two approaches. Each approach has advantages and disadvantages that merit a discussion.

In this article we studied the emergence of an effective field theory out of equilibrium from the open quantum system perspective where a light field

When the environment contains

For

We also obtain the quantum master equation up to second order in the interaction and show directly that its solution in the field basis is precisely the

While the influence function, Langevin and quantum master equation approaches all agree in the long time limit, they offer advantages and disadvantages which we discussed in detail. In particular, the influence action approach leads directly to the stochastic Langevin description, while the stochastic nature is not

These results offer a note of caution on the application of effective field theories in a finite temperature (and likely a finite density) environment such as the early Universe, as dissipative effects arising from the influence of heavy environmental fields lead to non-local and non-unitary stochastic dynamics of the light degrees of freedom.

Our ultimate goal is to study the emergence of effective field theories in cosmology, in particular during the inflationary stage, under the assumption that there are heavy degrees of freedom with mass or energy scales larger than the Hubble scale that are traced over leading to an effective description in terms of a single ‘light’ scalar field. In inflationary cosmology there are novel processes associated with the lack of a global time-like Killing vector [

The author thanks the N S F for partial support through grant PHY-1202227.

Because the initial density matrix

In terms of the spectral densities we find

This is the general form of the fluctuation dissipation relation, with _{0}.

The finite temperature correlation function for a single

Defining in four vector notation

In the first line in (_{2} can be simplified by relabeling _{2} in the first line becomes of the same form as that for _{1} but with the replacement _{2} similar to that of _{1} upon

Then, we have

For the integrals in (_{0} the curve

The analysis for (

We summarize this result as