The physical hamiltonian in nonperturbative quantum gravity

TL;DR

This paper defines a finite, diffeomorphism-invariant quantum Hamiltonian for nonperturbative quantum gravity using loop representation techniques, addressing the evolution of the gravitational field with respect to a scalar field.

Contribution

It introduces a regularization method for the quantum Hamiltonian in loop quantum gravity, reducing its construction to a combinatorial problem and providing explicit operators for the Hamiltonian constraint and spatial volume.

Findings

The quantum Hamiltonian is finite and diffeomorphism invariant.

A combinatorial approach simplifies the Hamiltonian's construction.

Explicit operators for the Hamiltonian constraint and volume are developed.

Abstract

A quantum hamiltonian which evolves the gravitational field according to time as measured by constant surfaces of a scalar field is defined through a regularization procedure based on the loop representation, and is shown to be finite and diffeomorphism invariant. The problem of constructing this hamiltonian is reduced to a combinatorial and algebraic problem which involves the rearrangements of lines through the vertices of arbitrary graphs. This procedure also provides a construction of the hamiltonian constraint as a finite operator on the space of diffeomorphism invariant states as well as a construction of the operator corresponding to the spatial volume of the universe.