A manifestly gauge-invariant approach to quantum theories of gauge fields

Summary

Abstract

In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be extended to face these kinematical non-linearities squarely. We first present a pedagogical account of this problem and then suggest an avenue for its resolution.

Introduction

As is well-known, for over 40 years, quantum field theory has remained in a somewhat peculiar situation. On the one hand, perturbative treatments of realistic field theories in four space-time dimensions have been available for a long time and their predictions are in excellent agreement with experiments. It is clear therefore that there is something “essentially right” about these theories. On the other hand, their mathematical status continues to be dubious in all cases (with interactions), including QED. In particular, it is generally believed that the perturbation series one encounters here can be at best asymptotic. However, it is not clear what exactly they are asymptotic to.

This overall situation is in striking contrast with, for example, non-relativistic quantum mechanics. There, we know well at the outset what the Hilbert space of states is and what the observables are. In physically interesting models, we can generally construct the Hamiltonian operator and show that it is self-adjoint. We take recourse to perturbation theory mainly to calculate its eigenvalues and eigenvectors.