The Utility of Coherent States and other Mathematical Methods in the Foundations of Affine Quantum Gravity

TL;DR

This paper explores advanced mathematical techniques like coherent states and reproducing kernel Hilbert spaces to address foundational issues in affine quantum gravity, focusing on metric positivity and quantum constraints.

Contribution

It introduces novel applications of mathematical methods such as coherent states and functional integrals to the complex framework of affine quantum gravity.

Findings

Coherent state representations aid in understanding quantum constraints.

Reproducing kernel Hilbert spaces provide a new perspective on state spaces.

Unique integration measures are crucial for the quantization process.

Abstract

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate first- and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity.