Alternative Dimensional Reduction via the Density Matrix

TL;DR

This paper introduces a new method for deriving a dimensionally-reduced effective action from the density matrix of scalar and gauge fields at finite temperature, avoiding spurious infinities and enabling calculations at any temperature.

Contribution

It presents graphical rules to construct the density matrix and a dimensionally-reduced effective action that is free of spurious divergences, improving thermal field theory analysis.

Findings

The DREA encodes all thermal matrix elements.

It replaces spurious infinities with calculable ln T terms.

Computed the phase transition temperature at one-loop order.

Abstract

We give graphical rules, based on earlier work for the functional Schrodinger equation, for constructing the density matrix for scalar and gauge fields at finite temperature T. More useful is a dimensionally-reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expansions; it encodes all thermal matrix elements. The DREA is useful because it gives a dimensionally-reduced field theory usable at any T including infinity, where it yields the usual dimensionally-reduced field theory (DRFT). However, it cannot and does not have spurious infinities which sometimes occur in the density matrix or the DRFT; these come from ln T factors at infinite temperature. An example of spurious divergences in the DRFT occurs in d 2+1 $\phi^4$ theory dimensionally reduced to d=2. We show that the rules for the DREA replace these "wrong" divergences in physical parameters by calculable powers of ln T; we also compute the phase transition temperature of this theory at one loop order.