Zero modes, gauge fixing, monodromies, $\zeta$-functions and all that

TL;DR

This paper investigates the calculation of a specific differential operator's determinant using gauge fixing, monodromy, heat kernel, and zeta-function methods, motivated by quantum cosmology applications.

Contribution

It introduces a comprehensive approach combining gauge fixing, monodromy, and zeta-function techniques for evaluating the determinant of a special differential operator in quantum cosmology.

Findings

Derived the period dependence of the determinant using heat kernel and zeta-function methods.

Clarified the gauge-fixed path integral representation for the operator.

Applied the methods to problems in quantum cosmology, including initial conditions and cosmological constant.

Abstract

We discuss various issues associated with the calculation of the reduced functional determinant of a special second order differential operator $\boldmath${F}$ =-d^2/d\tau^2+\ddot g/g$, $\ddot g\equiv d^2g/d\tau^2$, with a generic function $g(\tau)$, subject to periodic and Dirichlet boundary conditions. These issues include the gauge-fixed path integral representation of this determinant, the monodromy method of its calculation and the combination of the heat kernel and zeta-function technique for the derivation of its period dependence. Motivations for this particular problem, coming from applications in quantum cosmology, are also briefly discussed. They include the problem of microcanonical initial conditions in cosmology driven by a conformal field theory, cosmological constant and cosmic microwave background problems.