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

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 $.