Causality and Renormalization in Finite-Time-Path Out-of-Equilibrium $\phi^3$ QFT

TL;DR

This paper develops a formalism for out-of-equilibrium quantum field theory using finite-time-path and renormalization techniques, addressing causality and energy conservation issues in short-time phenomena like heavy ion collisions.

Contribution

It introduces a method to repair causality and energy conservation in out-of-equilibrium QFT within the finite-time-path formalism, using dimensional renormalization and retarded/advanced Green functions.

Findings

Vertices after divergent loops do not conserve energy initially.

Energy conservation is restored by vertex repair while keeping dimensionality below four.

Finite, oscillating tadpole contributions vanish at large times.

Abstract

Our aim is to contribute to quantum field theory (QFT) formalisms useful for descriptions of short time phenomena, dominant especially in heavy ion collisions. We formulate out-of-equilibrium QFT within the finite-time-path formalism (FTP) and renormalization theory (RT). The potential conflict of FTP and RT is investigated in $g \phi^3$ QFT, by using the retarded/advanced ($R/A$) basis of Green functions and dimensional renormalization (DR). For example, vertices immediately after (in time) divergent self-energy loops do not conserve energy, as integrals diverge. We "repair" them, while keeping $d<4$, to obtain energy conservation at those vertices. Already in the S-matrix theory, the renormalized, finite part of Feynman self-energy $\Sigma_{F}(p_0)$ does not vanish when $|p_0|\rightarrow\infty$ and cannot be split to retarded and advanced parts. In the Glaser--Epstein approach, the causality is repaired in the composite object $G_F(p_0)\Sigma_{F}(p_0)$. In the FTP approach, after repairing the vertices, the corresponding composite objects are $G_R(p_0)\Sigma_{R}(p_0)$ and $\Sigma_{A}(p_0)G_A(p_0)$. In the limit $d\rightarrow 4$, one obtains causal QFT. The tadpole contribution splits into diverging and finite parts. The diverging, constant component is eliminated by the renormalization condition $\langle 0|\phi|0\rangle =0$ of the S-matrix theory. The finite, oscillating energy-nonconserving tadpole contributions vanish in the limit $t\rightarrow \infty $.