The auxiliary field method in quantum mechanical four-fermi models: a study towards chiral condensation in QED

Abstract

A study for checking the validity of the auxiliary field method (AFM) is made in quantum mechanical four-fermi models which act as prototypes of models for chiral symmetry breaking in quantum electrodynamics. It has been shown that AFM, defined by an insertion of Gaussian identity to path integral formulae and by the loop expansion, becomes more accurate when taking higher order terms into account under the bosonic model with a quartic coupling in 0- and 1-dimensions as well as the model with a four-fermi interaction in 0-dimension. The case is also confirmed in terms of two models with the four-fermi interaction among N species in 1-dimension (the quantum mechanical four-fermi models): higher order corrections lead us towards the exact energy of the ground state for a whole region including the weak as well as the strong coupling. It is found that the second model belongs to a ‘WKB exact class’ that has no higher order corrections other than the lowest correction. Discussions are also made for unreliability on the continuous time representation of path integration and for a new model of QED as a suitable probe for chiral symmetry breaking.