Piecewise linear loop quantum gravity

TL;DR

This paper introduces a piecewise linear modification of loop quantum gravity (plLQG), demonstrating its equivalence to standard LQG at the diffeomorphism invariant level and enabling exact embedding of loop quantum cosmology into it.

Contribution

The paper defines plLQG with piecewise linear graphs and proves its equivalence to standard LQG, facilitating the embedding of loop quantum cosmology into the framework.

Findings

plLQG is unitarily equivalent to standard LQG.

A Hamiltonian and Master constraint can be defined in plLQG.

LQC can be exactly embedded into plLQG.

Abstract

We define a modification of LQG in which graphs are required to consist in piecewise linear edges, which we call piecewise linear LQG (plLQG). At the diffeomorphism invariant level, we prove that plLQG is equivalent to standard LQG, as long as one chooses the class of diffeomorphisms appropriately. That is, we exhibit a unitary map between the diffeomorphism invariant Hilbert spaces that maps physically equivalent operators into each other. In addition, using the same ideas as in standard LQG, one can define a Hamiltonian and Master constraint in plLQG, and the unitary map between plLQG and LQG then provides an exact isomorphism of dynamics in the two frameworks. Furthermore, loop quantum cosmology (LQC) can be exactly embedded into plLQG. This allows a prior program of the author to embed LQC into LQG at the dynamical level to proceed. In particular, this allows a formal expression for a physically motivated embedding of LQC into LQG at the diffeomorphism invariant level to be given.