A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements

TL;DR

This paper introduces a generalized Hamiltonian constraint operator in loop quantum gravity that incorporates arbitrary SU(2) representations, leading to a family of operators with the same classical limit and exploring their properties and implications.

Contribution

It proposes a new generalized operator in loop quantum gravity using arbitrary SU(2) representations, expanding the quantization ambiguity and analyzing its properties.

Findings

Derived the action of the generalized operator on trivalent states.

Established the relation between the generalization and crossing symmetry.

Demonstrated the classical limit consistency of the new operators.

Abstract

We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry.