New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis

TL;DR

This paper develops a classical Hamiltonian framework for higher-dimensional gravity theories, generalizing loop quantum gravity to any dimension without the time gauge, facilitating future quantization of supergravity and string theories.

Contribution

Introduces a new higher-dimensional connection formulation of gravity that extends the ADM phase space without requiring the time gauge, suitable for quantization.

Findings

Constructed a Yang-Mills phase space for D > 1

Identified a new simplicity constraint in higher dimensions

Preferred gauge group for quantization is SO(D+1)

Abstract

Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D+1 = 4 spacetime dimensions. However, interesting String theories and Supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional Supergravity loop quantisations at one's disposal in order to compare these approaches. In this series of papers, we take first steps towards this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG, which does not require the time gauge and which generalises to any dimension D > 1. The result is a Yang-Mills theory phase space subject to Gauss, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1,D) or SO(D+1) and the latter choice is preferred for purposes of quantisation.