Hidden Quantum Group Structure in Einstein's General Relativity

TL;DR

This paper introduces a q-deformed Poincare` group framework for General Relativity, revealing a hidden quantum group structure that preserves the classical metric and Einstein-Hilbert action across a family of theories.

Contribution

It presents a novel formalism replacing the Poincare` group with a q-deformed version, maintaining Einstein's equations while incorporating noncommutative fields and a one-parameter Hamiltonian family.

Findings

Classical Einstein gravity is recovered at q=1.

A one-parameter family of Hamiltonian formalisms is constructed.

The constraints remain polynomial, but Poisson brackets lose skew-symmetry for q≠1.

Abstract

A new formal scheme is presented in which Einstein's classical theory of General Relativity appears as the common, invariant sector of a one-parameter family of different theories. This is achieved by replacing the Poincare` group of the ordinary tetrad formalism with a q-deformed Poincare` group, the usual theory being recovered at q=1. Although written in terms of noncommuting vierbein and spin-connection fields, each theory has the same metric sector leading to the ordinary Einstein-Hilbert action and to the corresponding equations of motion. The Christoffel symbols and the components of the Riemann tensor are ordinary commuting numbers and have the usual form in terms of a metric tensor built as an appropriate bilinear in the vierbeins. Furthermore we exhibit a one-parameter family of Hamiltonian formalisms for general relativity, by showing that a canonical formalism a` la Ashtekar can be built for any value of q. The constraints are still polynomial, but the Poisson brackets are not skewsymmetric for q different from 1.