The Extended Loop Representation of Quantum Gravity

TL;DR

This paper introduces an extended loop representation for quantum gravity, simplifying the constraints and enabling new solutions, while addressing regularization issues present in previous loop-based approaches.

Contribution

It develops a novel extended loop group framework that generalizes the loop representation, simplifying constraints and facilitating new solutions in quantum gravity.

Findings

Constraints are linear in momenta in the new representation.

Nondegenerate solutions from the loop representation remain valid.

A new solution to the Wheeler-DeWitt equation is found.

Abstract

A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop dependent wavefunctions. The extended representation generalizes the loop representation and contains this representation as a particular case. The resulting diffeomorphism and hamiltonian constraints take a very simple form and allow to apply functional methods and simplify the loop calculus. In particular we show that the constraints are linear in the momenta. The nondegenerate solutions known in the loop representation are also solutions of the constraints in the new representation. The practical calculation advantages allows to find a new solution to the Wheeler-DeWitt equation. Moreover, the extended representation puts in a precise framework some of the regularization problems of the loop representation. We show that the solutions are generalized knot invariants, smooth in the extended variables, and any framing is unnecessary.