Metric and Curvature in Gravitational Phase Space

TL;DR

This paper explores the geometric structure of gravitational phase space, showing it has constant scalar curvature similar to higher-dimensional Poincare planes, which aids in constructing suitable Hilbert spaces in Metrical Quantization.

Contribution

It introduces a specific metric on gravitational phase space inspired by Affine Quantum Gravity, resulting in geometries with constant scalar curvature useful for quantization.

Findings

Phase space geometries have constant scalar curvature.

These geometries are higher-dimensional analogs of the Poincare plane.

The results facilitate the construction of high-dimensional Hilbert spaces.

Abstract

At a fixed point in spacetime (say, x_0), gravitational phase space consists of the space of symmetric matrices F^{ab} [corresponding to the canonical momentum pi^{ab}(x_0) and of symmetric matrices {G_{ab}}[corresponding to the canonical metric g_{ab}(x_0), where 1 \leq a,b \leq n, and, crucially, the matrix {G_{ab}} is necessarily positive definite, i.e. \sum u^a G_{ab}u^b > 0 whenever \sum (u^a)^2 > 0. In an alternative quantization procedure known as Metrical Quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase space metrics, guided by a recent study of Affine Quantum Gravity, leads to gravitational phase space geometries which possess constant scalar curvature and may be regarded as higher dimensional analogs of the Poincare plane, which applies when n=1. This result is important because phase spaces endowed with such symmetry lead naturally via the procedures of Metrical Quantization to acceptable Hilbert spaces of high dimension.