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arXiv:hep-ph/0104210 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Dynamical behavior of spatially inhomogeneous relativistic λφ4 quantum field theory in the Hartree approximation

Luís M. A. Bettencourt, Karen Pao, J. G. Sanderson

Published Dec 26, 2001• 15 citations
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Abstract

We study the dynamics of a spatially inhomogeneous quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case and, unlike the latter, shows some universality at long times. Nevertheless, we show that, within a family of Gaussian spatial profiles for the mean field, the evolution fails to thermalize in a quantum canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasiclassical inhomogeneous configurations (e.g. topological defects) in a background of dynamical self-consistent quantum fluctuations.

DOI
10.1103/physrevd.65.025015

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We study the dynamics of a spatially inhomogeneous quantum \ensuremathλ\ensuremathφ4\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}\ensuremathλ\ensuremathφ4 field theory in 1+11+11+1 dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case and, unlike the latter, shows some universality at long times. Nevertheless, we show that, within a family of Gaussian spatial profiles for the mean field, the evolution fails to thermalize in a quantum canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasiclassical inhomogeneous configurations (e.g. topological defects) in a background of dynamical self-consistent quantum fluctuations.
The $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ model in a finite volume is studied within a non-Gaussian, time-dependent Hartree-Fock (TDHF) approximation both at equilibrium and out of equilibrium, with particular attention to the structure of the ground state and of certain dynamical features in the broken symmetry phase. The mean-field coupled time-dependent Schr\"odinger equations for the modes of the scalar field are derived and the suitable procedure to renormalize them is outlined. A further controlled Gaussian approximation of our TDHF approach is used in order to study the dynamical evolution of the system from nonequilibrium initial conditions characterized by a uniform condensate. We find that, during the slow rolling down, the long-wavelength quantum fluctuations do not grow to a macroscopic size but do scale with the linear size of the system, in accordance with similar results valid for the large-N approximation of the O(N)O(N)O(N) model. This behavior undermines in a precise way the Gaussian approximation within our TDHF approach, which therefore appears as a viable means to correct an unlikely feature of the standard HF factorization scheme, such as the so-called ``stopping at the spinodal line'' of the quantum fluctuations. We also study the dynamics of the system in infinite volume with particular attention to the asymptotic evolution in the broken symmetry phase. We are able to show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential.
We perform a detailed numerical investigation of the dynamics of a single component ``explicitly broken symmetry'' $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in 1+1 dimensions, using a Schwinger-Dyson equation truncation scheme based on ignoring vertex corrections. In an earlier paper, we called this the bare vertex approximation (BVA). We assume here that the initial state is described by a Gaussian density matrix peaked around some nonzero value of $〈\ensuremath{\varphi}(0)〉,$ and characterized by a single particle Bose-Einstein distribution function at a given temperature. We compute the evolution of the system using three different approximations: the Hartree approximation, the BVA, and a related two-particle irreducible (2PI) 1/N1/N1/N expansion, as a function of coupling strength and initial temperature. In the Hartree approximation, the static phase diagram shows that there is a first order phase transition for this system. As we change the initial starting temperature of the system, we find that the BVA relaxes to a new final temperature and exhibits behavior consistent with a second order phase transition. We find that the average fields equilibrate for arbitrary initial conditions in the BVA, unlike the behavior exhibited by the Hartree approximation, and we illustrate how $〈\ensuremath{\varphi}(t)〉$ and $〈\ensuremath{\chi}(t)〉$ depend on the initial temperature and on the coupling constant. The Fourier transform of the two-point functions at late times can be fitted by a Bose-Einstein distribution function whose temperature is independent of momentum. We interpret this as evidence for thermalization.
We consider the out-of-equilibrium evolution of a classical condensate field and its quantum fluctuations for a ${\ensuremath{\Phi}}^{4}$ model in 1+11+11+1 dimensions with a symmetric and a double well potential. We use the two-particle point-irreducible (2PPI) formalism and go beyond the Hartree approximation by including the sunset term. In addition to the mean field $\ensuremath{\varphi}(t)=〈\ensuremath{\Phi}〉$ the 2PPI formalism uses as a variational parameter a time dependent mass M2(t){\mathcal{M}}^{2}(t)M2(t) which contains all local insertions into the Green's function. We compare our results to those obtained in the Hartree approximation. In the symmetric ${\ensuremath{\Phi}}^{4}$ theory we observe that the mean field shows a stronger dissipation than the one found in the Hartree approximation. The dissipation is roughly exponential in an intermediate time region. In the theory with spontaneous symmetry breaking, i.e., with a double well potential, the field amplitude tends to zero, i.e., to the symmetric configuration. This is expected on general grounds: in (1+1)(1+1)(1+1)-dimensional quantum field theory there is no spontaneous symmetry breaking for T>0,T>0,T>0, and so there should be none at finite energy density (microcanonical ensemble), either. Within the time range of our simulations the momentum spectra do not thermalize and display parametric resonance bands.
For homogeneous initial conditions, Hartree (Gaussian) dynamical approximations are known to have problems with thermalization because of insufficient scattering. We attempt to improve on this by writing an arbitrary density matrix as a superposition of Gaussian pure states and applying the Hartree approximation to each member of such an ensemble. Particles can then scatter via their back reaction on the typically inhomogeneous mean fields. Starting from initial states that are far from equilibrium we numerically compute the time evolution of particle distribution functions and observe that they indeed display approximate thermalization on intermediate time scales by approaching a Bose-Einstein form. However, for very large times the distributions drift towards classical-like equipartition.
This work is devoted to the study of relaxation--dissipation processes in systems described by Quantum Field Theory. In the first part, I focus on the phi^4 scalar quantum field theory in finite volume in the large N limit. I find that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions, while the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones. I also introduce a non-gaussian Hartree-Fock approximation (tdHF) and I derive the mean-field coupled time-dependent Schroedinger equations for the modes of the scalar field. The dynamical evolution of a uniform condensate is studied in a further controlled gaussian approximation, for N=1. Results similar to the large N case show an internal inconsistency of this approximation. I also study the dynamics of the system in infinite volume in the broken symmetry phase. The data show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential. I finally study the dynamical evolution for the O(N+1) nonlinear sigma model in 1+1 dimensions in the large N limit. I first of all verify that the large coupling limit of the O(N+1) phi^4 model, which renders the model non linear, commutes with the large N limit, so that the O(infty) nonlinear sigma model is uniquely defined. I then study the evolution of several observables, like the spectrum of produced particles, finding no evidence for parametric resonance. I conclude with some remarks on the ``state of art'' in gauge theories and some comments about the open issues in the subject.