Beables/Observables in Classical and Quantum Gravity

ABSTRACT

Observables `are observed' whereas beables just `are'. Beables have more scope in the cosmological and quantum domains. Both are entities that form `brackets' with `the constraints' that are `equal to' zero. Moreover, a variety of notions of brackets, constraints and equalities are to be considered. I.e. Poisson, Dirac, commutator, histories, Schouten-Nijenhuis, double and Nambu brackets; first-class, gauge, linear and effective constraints; plain alias strong, weak and weak-effective equalities. The Dirac-Bergmann distinction in notions of gauge leads to further notions of beable, and is tied to some diffeomorphism-specific subtleties. Thus we cover a wide range of notions of beables/observables: Dirac, Kuchar, effective, Bergmann, histories, multisymplectic, master, Nambu and bi-. We also cover a wide range of classical and quantum theories of gravity: general relativity, loop quantum gravity, histories theory, supergravity and M-theory.