Mathematical Problems of Gauge Quantum Field Theory: A Survey on the Schwinger Model

TL;DR

This survey explores the mathematical challenges in quantizing gauge systems using the Schwinger model, highlighting issues like indefinite Wightman functions and the structure of the state space, and introduces new results on indefinite inner product spaces.

Contribution

It provides an introductory overview of gauge quantum field theory problems with new insights into Pontrjagin space structures in indefinite inner product spaces.

Findings

Analysis of the indefinite structure of Wightman functions in the Schwinger model

Reconstruction of the state space for gauge theories

New results on quasi-positive Pontrjagin spaces

Abstract

This extended write-up of a talk gives an introductory survey of mathematical problems of the quantization of gauge systems. Using the Schwinger model as an exactly tractable but nontrivial example which exhibits general features of gauge quantum field theory, I cover the following subjects: The axiomatics of quantum field theory, formulation of quantum field theory in terms of Wightman functions, reconstruction of the state space, the local formulation of gauge theories, indefiniteness of the Wightman functions in general and in the special case of the Schwinger model, the state space of the Schwinger model, special features of the model. New results are contained in the Mathematical Appendix, where I consider in an abstract setting the Pontrjagin space structure of a special class of indefinite inner product spaces - the so called quasi-positive ones. This is motivated by the indefinite inner product space structure appearing in the above context and generalizes results of Morchio and Strocchi [J. Math. Phys. 31 (1990) 1467], and Dubin and Tarski [J. Math. Phys. 7 (1966) 574].