Asymptotically Flat Boundary Conditions for the U(1)3 Model for Euclidean Quantum Gravity

Sepideh Bakhoda,Hossein Shojaie,Thomas Thiemann

doi:10.3390/universe7030068

Sepideh Bakhoda, Hossein Shojaie + Show 1 more

Open Access

https://doi.org/10.3390/universe7030068

Copy DOI

Abstract

A generally covariant U(1)3 gauge theory describing the GN→0 limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we the study boundary conditions and asymptotic symmetries of the U(1)3 model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that the non-Abelian part of the SU(2) Gauss constraint, which is absent in the U(1)3 model, plays a crucial role in obtaining boost and rotation generators.

Highlights

In the framework of the Ashtekar variables in terms of which General Relativity (GR) is formulated as a SU(2) gauge theory [1,2,3], attempts to find an operator corresponding to the Hamiltonian constraint of the Lorentzian vacuum canonical GR led to the result [4] that the Lorentzian Hamiltonian constraint can be written in terms of the Euclidean Hamiltonian constraint and the volume operator
Since the latter is under much control in the context of Loop Quantum Gravity (LQG) [5,6,7,8], an essential step towards quantising the Lorentzian Hamiltonian constraint is the quantisation of the Euclidean Hamiltonian constraint
Due to the fact that the GN → 0 limit of Euclidean general relativity is an interesting toy model for Lorentzian GR, this paper is devoted to studying its boundary conditions, yielding a well-defined symplectic structure and finite and integrable charges associated with the asymptotic symmetries

Summary

Introduction

In the framework of the Ashtekar variables in terms of which General Relativity (GR) is formulated as a SU(2) gauge theory [1,2,3], attempts to find an operator corresponding to the Hamiltonian constraint of the Lorentzian vacuum canonical GR led to the result [4] that the Lorentzian Hamiltonian constraint can be written in terms of the Euclidean Hamiltonian constraint and the volume operator. The GN → 0 limit of Euclidean gravity introduced by Smolin [10] is one of these models which is described by a U(1) gauge theory This model contains three Gauss constraints, three spatial diffeomorphism constraints and a Hamiltonian constraint whose constraint algebra for the Hamiltonian and diffeomorphism constraints is isomorphic to that of general relativity. Observing that they are not well-defined and functionally differentiable, we revisit the results and reasoning, presented in [30] and in [31,32], to improve the constraints using suitable boundary terms to well-defined generators of Poincaré and ISO(4) group for Lorentzian and Euclidean GR respectively. In the last section we conclude with a brief summary

Background

Conclusions