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arXiv:0809.0496 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Renormalized nonequilibrium quantum field theory: Scalar fields

Sz. Borsányi, Urko Reinosa

Published Dec 29, 2009• 28 citations
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Abstract

We discuss the renormalization of the initial value problem in quantum field theory using the two-particle irreducible (2PI) effective action formalism. The nonequilibrium dynamics is renormalized by counterterms determined in equilibrium. We emphasize the importance of the appropriate choice of initial conditions and go beyond the Gaussian initial density operator by defining self-consistent initial conditions. We study the corresponding time evolution and present a numerical example which supports the existence of a continuum limit for this type of initial conditions.

DOI
10.1103/physrevd.80.125029

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We discuss the renormalization of the initial value problem in quantum field theory using the two-particle irreducible (2PI) effective action formalism. The nonequilibrium dynamics is renormalized by counterterms determined in equilibrium. We emphasize the importance of the appropriate choice of initial conditions and go beyond the Gaussian initial density operator by defining self-consistent initial conditions. We study the corresponding time evolution and present a numerical example which supports the existence of a continuum limit for this type of initial conditions.
We discuss the renormalization of the initial value problem in nonequilibrium quantum field theory within a simple, yet instructive, example and show how to obtain a renormalized time evolution for the two-point functions of a scalar field and its conjugate momentum at all times. The scheme we propose is applicable to systems that are initially far from equilibrium and compatible with nonsecular approximation schemes which capture thermalization. It is based on Kadanoff-Baym equations for non-Gaussian initial states, complemented by usual vacuum counterterms. We explicitly demonstrate how various cutoff-dependent effects peculiar to nonequilibrium systems, including time-dependent divergences or initial-time singularities, are avoided by taking an initial non-Gaussian three-point vacuum correlation into account.
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and, in particular, when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature nnn-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a precondition to describe the vacuum correlations of the interacting theory within the nonequilibrium framework. In this paper, Kadanoff-Baym equations for non-Gaussian correlated initial states describing vacuum and thermal equilibrium are derived from the 2PI effective action. A diagrammatic method for the explicit construction of vacuum and thermal initial correlations from the 2PI effective action is provided. We present numerical solutions of Kadanoff-Baym equations for a real scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory, which take the thermal initial four-point correlation as the leading non-Gaussian correction into account. We find that this minimal non-Gaussian initial condition yields an approximation to the complete equilibrium initial state that is quantitatively and qualitatively significantly improved as compared to Gaussian initial states.
I review the use of the 2PI effective action in nonequilibrium quantum field theory. The approach enables one to find approximation schemes which circumvent long-standing problems of non-thermal or secular (unbounded) late-time evolutions encountered in standard loop or 1/N expansions of the 1PI effective action. It is shown that late-time thermalization can be described from a numerical solution of the three-loop 2PI effective action for a scalar ϕ4ϕ^4ϕ4--theory in 1+1 dimensions (with Jurgen Cox, hep-ph/0006160). Quantitative results far from equilibrium beyond the weak coupling expansion can be obtained from the 1/N expansion of the 2PI effective action at next-to-leading order (NLO), calculated for a scalar O(N) symmetric quantum field theory (hep-ph/0105311). Extending recent calculations in classical field theory by Aarts et al. (hep-ph/0007357) and by Blagoev et al. (hep-ph/0106195) to N>1N>1N>1 we show that the NLO approximation converges to exact (MC) results already for moderate values of NNN (with Gert Aarts, hep-ph/0107129). I comment on characteristic time scales in scalar quantum field theory and the applicability of classical field theory for sufficiently high initial occupation numbers.
We consider the time evolution of nonequilibrium quantum scalar fields in the O(N) model, using the next-to-leading order 1/N expansion of the two-particle irreducible effective action. A comparison with exact numerical simulations in 1+1 dimensions in the classical limit shows that the 1/N expansion gives quantitatively precise results already for moderate values of N. For sufficiently high initial occupation numbers the time evolution of quantum fields is shown to be accurately described by classical physics. Eventually the correspondence breaks down due to the difference between classical and quantum thermal equilibrium.
A framework to describe the real-time evolution of interacting fermion-scalar field models is set up. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type to the set of one-body dynamical variables. A time-dependent projection technique is used then to generate a nonperturbative mean-field expansion leading to a set of self-consistent equations of motion for these observable, where the lowest order corresponds to the Gaussian approximation. As an application, we consider an uniform system of relativistic spin-1/2 fermion field coupled, through a Yukawa term, to a scalar field in 3+1 dimensions, known as quantum scalar plasma. The renormalizability for the Gaussian mean-field equations, both static and dynamical, are examined and initial conditions discussed. We also investigate solutions for the gap equation and show that the energy density has a single minimum.
We derive the time-evolution equations appropriate to initial-value problems in \ensuremath{\lambda}(${\ensuremath{\varphi}}_{\ensuremath{\alpha}}$${\ensuremath{\varphi}}_{\ensuremath{\alpha}}$)2{)}^{2})2 field theory at large N. The Heisenberg equations of motion for this theory are compared to the Schr\"odinger equation for a wave functional constrained to be in a Gaussian state [the time-dependent Hartree-Fock (TDHF) ansatz]. The TDHF ansatz corresponds to a special choice of initial conditions for the general large-N Heisenberg equations of motion. The renormalization of the theory is discussed in both approaches and a simple method is given to arrive at finite differential equations suitable for numerical integration forward in time. A necessary and sufficient general condition for these equations to be finite is that the initial state contain a finite average number of particles and/or correlated particle pairs per unit volume.