Quantizing Regge calculus

TL;DR

This paper develops a discretized canonical gravity model in (3+1) dimensions, analyzing its structure and demonstrating the impossibility of direct quantization using complex variables due to constraints on the metric.

Contribution

It introduces the Liouville form and Poisson brackets into the discretized gravity model and examines the implications for quantization.

Findings

Quantization with complex variables is not feasible due to non-hermitian area variables.

The model's second class constraints prevent defining a suitable quantum metric.

Explicit parametrization reveals the nature of variables under constraints.

Abstract

A discretized version of canonical gravity in (3+1)-d introduced in a previous paper is further developed, introducing the Liouville form and the Poisson brackets, and studying them in detail in an explicit parametrization that shows the nature of the variables when the second class constraints are imposed. It is then shown that, even leaving aside the difficult question of imposing the first class constraints on the states, it is impossible to quantize the model directly, using complex variables and leaving the second class constraints to fix the metric of the quantum Hilbert, because one cannot find a metric which makes the area variables hermitean.