The Zeta-Function

Abstract

The spectral theorem enables one to define complex powers of elliptic operators, and on taking traces of such complex powers one obtains the eta-and zeta-functions used in Riemannian geometry and in quantum field theory. The zeta-function may be viewed as a particular case of eta-function, and is here defined for elliptic, self-adjoint, positive-definite operators. It admits an analytic continuation to the complex plane as a meromorphic function which is regular at the origin. Remarkably, this property is stable under a smooth variation of the given elliptic operator. In quantum field theory, the ζ(0) value yields both the scaling properties of the amplitudes, and the one-loop divergences of the theory. Three methods for evaluating ζ(0) are then described in detail: the Laplace transform of the heat equation, the Moss algorithm, which relies on the explicit knowledge of the uniform asymptotic expansion of basis functions, and a more recent method that elucidates the general structure of ζ(s) in quantum field theory. The latter uses again the uniform asymptotic expansion of such basis functions, but is more powerful, since it is independent of a detailed knowledge of the polynomials occurring in the uniform asymptotics of basis functions. A linear combination of ζ(0) and ζ’(0) yields the one-loop effective action.