A tale of two derivatives: phase space symmetries and Noether charges in diffeomorphism invariant theories

TL;DR

This paper explores the interplay between symmetries, conservation laws, and phase space in diffeomorphism-invariant theories like general relativity, using differential forms and double complexes to derive exact conserved quantities at finite distances.

Contribution

It develops a double differential complex framework to relate symmetries and conserved charges, enabling exact calculations of quantities like mass and angular momentum at finite radii.

Findings

Conserved quantities can be computed exactly at finite distances.

The double complex captures the cohomology relevant to invariants.

The approach applies to gravitational theories and quantum anomalies.

Abstract

For a field theory that is invariant under diffeomorphisms there is a subtle interplay between symmetries, conservation laws and the phase space of the theory. The natural language for describing these ideas is that of differential forms and both differential forms on space-time and differential forms on the infinite dimensional space of solutions of the equations of motion of the field theory play an important role. There are exterior derivatives on both spaces and together they weave a double differential complex which captures the cohomology of the theory. This is important in the definition of invariants in general relativity, such as mass and angular momentum and is also relevant to the study of quantum anomalies in gauge theories. We derive the structure of this double complex and show how it relates to conserved quantities in gravitational theories. One consequence of the construction is that conserved quantities can be calculated exactly at finite distance --- for example it is not necessary to go to asymptotic regimes to calculate the mass or angular momentum of a stationary solution of Einstein's equations, they can be obtained exactly by an integration over any sphere outside the mass even at finite radius.