New commutation relations for quantum gravity

TL;DR

This paper introduces a novel set of non-canonical commutation relations for quantum gravity, using unimodular dreibein components and SL(3,R) generators, with explicit representations constructed.

Contribution

It presents a new group-theoretic framework for quantum gravity with explicit unitary representations, differing from canonical approaches.

Findings

Constructed infinite-dimensional irreducible representations in metric and dreibein variables.

Dreibein components commute among themselves, unlike momentric variables.

The new relations may explain the metric nature of our universe.

Abstract

A new set of fundamental commutation relations (CR) for quantum gravity is presented. The basic variables are the eight components of the unimodular part of the spatial dreibein and eight SL(3,R) generators which correspond to Klauder's momentric variables. The commutation relations are not canonical, but they have well-defined group-theoretic meanings. Explicit unitary irreducible infinite-dimensional representations are constructed both in the metric as well as dreibein representations. The dreibein components commute among themselves; but momentric, unlike momenta, do not. This differs starkly from the usual canonical CR that allow wave functionals to be realized in either of the conjugate representations; and it may explain why our universe seems fundamentally and intuitively `metric' in nature, and not `conjugately realized'.