Symplectic Structure of Intrinsic Time Gravity

Eyo Ita,Amos Kubeka

doi:10.3390/universe2030018

Abstract

The Poisson structure of intrinsic time gravity is analysed. With the starting point comprising a unimodular three-metric with traceless momentum, a trace-induced anomaly results upon quantization. This leads to a revision of the choice of momentum variable to the (mixed index) traceless momentric. This latter choice unitarily implements the fundamental commutation relations, which now take on the form of an affine algebra with SU(3) Lie algebra amongst the momentric variables. The resulting relations unitarily implement tracelessness upon quantization. The associated Poisson brackets and Hamiltonian dynamics are studied.

Highlights

A crucially important question in the quantization of gravity in 3+1 dimensions, as for any theory, is the choice of the fundamental dynamical variables of the classical theory, which upon quantization become promoted to quantum operators
From intractable non polynomial phase space functions, as they appear in the Arnowitt Deser Misner (ADM) theory [2], into polynomial form at the expense of an additional set of constraints related to the SU(2) gauge symmetry inherent in the theory
It is hoped that the polynomial form of the constraints in Loop Quantum Gravity (LQG) make the constraints more tractable for quantization and the construction of a physical Hilbert space

Summary

Introduction

A crucially important question in the quantization of gravity in 3+1 dimensions, as for any theory, is the choice of the fundamental dynamical variables of the classical theory, which upon quantization become promoted to quantum operators. Starting from this canonically conjugate pair, let us define as fundamental classical variables the following barred quantities qij , a unimodular metric with detqij = 1, and a traceless momentum variable π ij via the relations [7,8]. (16) is the same as the barred contribution to (6), with the difference that the tracelessness of π ij has been implicitly enforced due to a unimodular metric This calculation demonstrates that extreme care must be exercised when extracting Poisson brackets from a symplectic two form, particular when the index structure of the fundamental variables has implicit symmetries. It is the unique choice of unimodular and traceless variables, which makes this the case, which admits a complete quantization of these variables

A Prelude into the Quantum Theory

Conclusions