Generally covariant theories: the Noether obstruction for realizing certain space-time diffeomorphisms in phase space

TL;DR

This paper investigates the limitations of representing certain spacetime diffeomorphisms in phase space within generally covariant theories, revealing an obstruction that affects the algebraic structure of gauge generators.

Contribution

It demonstrates an obstruction to projectability of tangent-space diffeomorphisms to phase space, clarifying the connection to the soft algebra of gauge generators in General Relativity.

Findings

Certain tangent-space diffeomorphisms cannot be projected to phase space.

The algebra of gauge generators in General Relativity is a soft algebra, not a Lie algebra.

Choosing metric-dependent diffeomorphisms affects the algebraic structure of constraints.

Abstract

Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms to be projectable to phase-space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of General Relativity, or other generally covariant theories, only closes as a soft algebra and not a a Lie algebra. The deep relationship between these two issues is clarified. In particular, we see that the second one may be understood as a side effect of the procedure to solve the first. It is explicitly shown how the adoption of specific metric-dependent diffeomorphisms, as a way to achieve projectability, causes the algebra of gauge generators (constraints) in phase space not to be a Lie algebra --with structure constants-- but a soft algebra --with structure {\it functions}.