Ghost Fields and Gauge Modes in One-Loop Quantum Cosmology

Abstract

At first we briefly review the geometric formulation of gauge theories and the problems one faces in trying to quantize them. We then apply the extended phase-space method of Batalin, Fradkin and Vilkovisky to the spin-1 field, which is described by a constrained Hamiltonian system with first-class constraints. The charge Q and the gauge-fixed action are derived. The Lorentzian path integral is restricted to the trajectories of the extended phase space which satisfy the boundary conditions which project on the original phase space. This path integral is independent of gauge fixing provided the charge Q remains nilpotent on quantization, and the Faddeev-Popov formula is formally derived from the path integral involving the gauge-fixed action.

We then study the effect of ghost fields and gauge-fixing terms in the path integral for linearized gravity. The quantum theory is only well-defined when expressed in terms of its physical degrees of freedom, the transverse-traceless modes. One can formally show that a suitable measure exists such that the gauge-invariant form of the path integral for the ground-state wave function is equal to the one expressed in terms of the physical degrees of freedom only. However, it remains to be seen whether this result can be generalized to the boundary-value problems which occur in quantum cosmology. In fact different quantization techniques for gauge fields have led to discrepancies in the literature. As an example, we show how to obtain the Moss and Poletti result, according to which magnetic and electric boundary conditions for electromagnetic fields lead to the same result for ζ(0), thus correcting the PDF values of chapter three. This is obtained by using an indirect technique, where ζ(0) is expressed in terms of functions of the geometrical objects of the problem, multiplied by coefficients which take into account the contribution of all degrees of freedom (PDF and gauge) and of the ghost field.