The Auxiliary Field Method in Quantum Mechanical Four-Fermi Models -- A Study Toward Chiral Condensation in QED

A study for checking validity of the auxiliary field method (AFM) is made in quantum mechanical four-fermi models which act as a prototype 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 formulas 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 toward the exact energy of the ground state. 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.