Fermionic Integral Research Articles

Regularized functional integral for fermions and anomalies

The regularization of fermionic functional integrals is investigated with the method of finite-dimensional approximations. The conditions for preserving gauge invariance in these approximations are stated. This regularization is applied to describe the divergences of vector and axial-vector currents.

Fermionic coordinates and supersymmetry in quantum mechanics

We describe the quantum mechanics of particles with fermionic degrees of freedom, both in the Schrödinger wave function and Feynman path integral formalism. In particular we derive an exact expression for fermionic path integrals in (0 + 1) dimensions. Under suitable circumstances the theories we consider can exhibit supersymmetry. As an application we analyze in detail how the spontaneous breaking of supersymmetry, occurring in certain models recently discussed by Witten, can be derived as an instanton effect in the path integral formalism.

Generalized two-dimensional QED and functional determinants

We obtain a closed expression for the functional determinant of the Dirac operator in two-dimensional QED by confronting results on cluster decomposition with direct fermion integration in the presence of Atiyah-Singer zero modes. The result is shown to be identical with the one obtained from the zeta-function definition and the use of the modified anomaly equation.

Vacuum energy density in large orders of perturbation theory for the scalar Yukawa 2 field theory

An expression for the vacuum energy density of scalar Y 2 in large orders of perturbation theory is derived using a generalization of the WKB method to fermionic integrals and the method of steepest descents.

Axial anomaly and Atiyah-Singer theorem

The fermion integration for gauge theories requires the study of the Euclidean c number Dirac equation in arbitrary external gauge fields. Winding of Aμ leads to a trapping for the Euclidean ψ's. We complete our previous discussion by relating this problem to the Atiyah-Singer index theory. For gauge theories a suitably modified axial anomaly relation provides a new proof of this theorem.