Can Hamiltonians be boundary observables in Parametrized Field Theories?

TL;DR

This paper investigates whether Hamiltonians can serve as boundary observables in parametrized field theories, finding that they cannot, which suggests these theories lack holographic properties due to the absence of useful Gauss Laws.

Contribution

It demonstrates that parametrized field theories do not possess non-trivial boundary Hamiltonians, clarifying their limitations in holographic contexts and the role of diffeomorphism invariance.

Findings

Hamiltonian vanishes on shell even with boundaries

Parametrized theories lack useful Gauss Laws

No holography in these theories

Abstract

It has been argued that holography in gravitational theories is related to the existence of a particularly useful Gauss Law that allows energy to be measured at the boundary. The present work investigates the extent to which such Gauss Laws follow from diffeomorphism invariance. We study parametrized field theories, which are a class of diffeomorphism-invariant theories without gravity. We find that the Hamiltonian for parametrized field theories vanishes on shell even in the presence of a boundary and under a variety of boundary conditions. We conclude that parametrized theories have no useful Gauss Law, consistent with the absence of holography in these theories.