The Utility of Affine Variables and Affine Coherent States

TL;DR

This paper explores the use of affine variables and affine coherent states as an alternative to canonical variables, demonstrating their effectiveness in quantizing nonrenormalizable scalar fields and formulating quantum gravity.

Contribution

It introduces affine variables as a viable and advantageous alternative to canonical variables for classical and quantum studies, including quantum field theory and gravity.

Findings

Successful quantization of a nonrenormalizable scalar quantum field theory using affine techniques

Development of a framework for quantum gravity favoring affine variables

Demonstration of the close connection and advantages of affine over canonical variables

Abstract

Affine coherent states are generated by affine kinematical variables much like canonical coherent states are generated by canonical kinematical variables. Although all classical and quantum formalisms normally entail canonical variables, it is shown that affine variables can serve equally well for many classical and quantum studies. This general purpose analysis provides tools to discuss two major applications: (1) the completely successful quantization of a nonrenormalizable scalar quantum field theory by affine techniques, in complete contrast to canonical techniques which only offer triviality; and (2) a formulation of the kinematical portion of quantum gravity that favors affine kinematical variables over canonical kinematical variables, and which generates a framework in which a favorable analysis of the constrained dynamical issues can take place. All this is possible because of the close connection between the affine and the canonical stories, while the few distinctions can be used to advantage when appropriate.