Propagation in Polymer Parameterised Field Theory

TL;DR

This paper demonstrates that in a polymer quantization of 2D Parameterised Field Theory, ultralocal Hamiltonian constraints can still lead to propagation effects in physical states, challenging prior assumptions about ultralocality in quantum gravity.

Contribution

It provides explicit evidence that ultralocal Hamiltonian constraints in a polymer framework can produce propagation effects, informing loop quantum gravity models.

Findings

Propagation effects are encoded in physical states, not in repeated actions of the Hamiltonian.

The structure of the finite triangulation Hamiltonian is crucial for propagation.

Imposing the continuum limit of the adjoint Hamiltonian as a constraint is essential.

Abstract

The Hamiltonian constraint operator in Loop Quantum Gravity acts ultralocally. Smolin has argued that this ultralocality seems incompatible with the existence of a quantum dynamics which propagates perturbations between macroscopically seperated regions of quantum geometry. We present evidence to the contrary within an LQG type `polymer' quantization of two dimensional Parameterised Field Theory (PFT). PFT is a generally covariant reformulation of free field propagation on flat spacetime. We show explicitly that while, as in LQG, the Hamiltonian constraint operator in PFT acts ultralocally, states in the joint kernel of the Hamiltonian and diffeomorphism constraints of PFT necessarily describe propagation effects. The particular structure of the finite triangulation Hamiltonian constraint operator plays a crucial role, as does the necessity of imposing (the continuum limit of) its kinematic adjoint as a constraint. Propagation is seen as a property encoded by physical states in the kernel of the constraints rather than that of repeated actions of the finite triangulation Hamiltonian constraint on kinematic states. The analysis yields robust structural lessons for putative constructions of the Hamiltonian constraint in LQG for which ultralocal action co-exists with a description of propagation effects by physical states.