Wigner Transformation for the Determinant of Dirac Operators

We use theζ-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 formalism is amenable to the study of several problems of which we consider in detail two, the inverse mass expansion and the gradient expansion. We obtain explicit closed-form 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 in order to apply heat kernel expansions, nor performs further analytical rotations of the fields in order to make the Dirac operator Hermitian. We illustrate the flexibility of the method by studying some interesting cases.