Wigner transformation for the determinant of Dirac operators

TL;DR

This paper introduces a systematic and conceptually simple method using Wigner transformation and zeta-function regularization to define and compute the determinant of Dirac operators, capturing anomalies and extending to complex cases.

Contribution

It provides a new, unambiguous, and extendable approach to evaluate Dirac operator determinants using Wigner representation and explicit Seeley-DeWitt coefficients.

Findings

Explicit closed-form Seeley-DeWitt coefficients to all orders.

Demonstrates gauge invariance and correct anomalies of the determinant.

Method avoids separate real and imaginary part definitions and complex rotations.

Abstract

We use the $\zeta$-function regularization and an integral representation of the complex power of a pseudo differential operator, to give an unambiguous definition of the determinant of the Dirac operator. We bring this definition to a workable form by making use of an asymmetric Wigner representation. The expression so obtained is amenable to several treatments of which we consider in detail two, the inverse mass expansion and the gradient expansion, with concrete examples. We obtain explicit closed expressions for the corresponding Seeley-DeWitt coefficients to all orders. The determinant is shown to be vector gauge invariant and to posses the correct axial and scale anomalies. The main virtue of our approach is that it is conceptually simple and systematic and can be extended naturally to more general problems (bosonic operators, gravitational fields, etc). In particular, it avoids defining the real and imaginary parts of the effective action separately. In addition, it does not reduce the problem to a bosonic one to apply heat kernel nor performs further analytical rotations of the fields to make the Dirac operator Hermitian. We illustrate the flexibility of the method by studying some interesting cases.