Divergences in Real-Time Classical Field Theories at Non-Zero Temperature

Abstract

The classical approximation provides a non-perturbative approach to time-dependent problems in finite temperature field theory. We study the divergences in hot classical field theory perturbatively. At one-loop, we show that the linear divergences are completely determined by the classical equivalent of the hard thermal loops in hot quantum field theories, and that logarithmic divergences are absent. To deal with higher-loop diagrams, we present a general argument that the superficial degree of divergence of classical vertex functions decreases by one with each additional loop: one-loop contributions are superficially linearly divergent, two-loop contributions are superficially logarithmically divergent, and three- and higher-loop contributions are superficially finite. We verify this for two-loop SU(N) self-energy diagrams in Feynman and Coulomb gauges. We argue that hot, classical scalar field theory may be completely renormalized by local (mass) counterterms, and discuss renormalization of SU(N) gauge theories.

arXiv comment31 pages with 7 eps figures

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• Classical approximation to quantum cosmological correlations
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• Low-cost fermions in classical field simulations
• Thermalization in a Hartree Ensemble Approximation to Quantum Field dynamics
• Non-renormalizability of the classical statistical approximation
• Spectral function at high temperature in the classical approximation
• Initial state and thermalization in the Color Glass Condensate framework
• Topological defects as inhomogeneous condensates in Quantum Field Theory: Kinks in (1+1) dimensional $\la ψ^4$ theory
• Diagrammar In Classical Scalar Field Theory
• Transversality of the logarithmic divergences in the Classical Finite Temperature SU(N) Self-Energy