N-quantum approach to quantum field theory at finite T and $\mu$: the NJL model

TL;DR

This paper extends the N-quantum approach to quantum field theory at finite temperature and chemical potential, applying it to the NJL model to recover known results and explore beyond mean field approximations.

Contribution

It introduces a generalized Bogoliubov transformation within the N-quantum framework for finite T and μ, and discusses methods to go beyond mean field approximation.

Findings

Mean field results are recovered with first Haag expansion term.

Mean field approximation cannot diagonalize the Hamiltonian at finite T in broken symmetry phase.

Including diquark channels can lower the vacuum energy density.

Abstract

We extend the N-quantum approach to quantum field theory to finite temperature ($T$) and chemical potential ($\mu$) and apply it to the NJL model. In this approach the Heisenberg fields are expressed using the Haag expansion while temperature and chemical potential are introduced simultaneously through a generalized Bogoliubov transformation. Known mean field results are recovered using only the first term in the Haag expansion. In addition, we find that at finite T and in the broken symmetry phase of the model the mean field approximation can not diagonalize the Hamiltonian. Inclusion of scalar and axial vector diquark channels in the SU(2)$_{rm f}$ $otimes$ SU(3)$_{\rm c}$ version of the model can lead to a lowering of the vacuum energy density. We discuss how to go beyond the mean field approximation by including higher order terms in the Haag expansion.