Beables/Observables in Classical and Quantum Gravity

Abstract

Observables 'are observed' whereas beables just 'are'. This gives beables more scope in the cosmological and quantum domains. Both observables and beables are entities that form 'brackets' with 'the constraints' that are 'equal to' zero. We explain how depending on circumstances, these could be, e.g., Poisson, Dirac, commutator, histories, Schouten-Nijenhuis, double or Nambu brackets, first-class, gauge, linear or effective constraints, and strong, weak or weak-effective equalities. The Dirac-Bergmann distinction in notions of gauge leads to further notions of observables or beables, and is tied to a number of diffeomorphism-specific subtleties. Thus we cover a wide range of notions of observables or beables that occur in classical and quantum gravitational theories: Dirac, Kuchař, effective, Bergmann, histories, multisymplectic, master, Nambu and bi-. Indeed this review covers a representatively wide range of such theories: general relativity, loop quantum gravity, histories theory, supergravity and M-theory.

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• Configuration Spaces in Fundamental Physics
• TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)
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• A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging
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• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
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• A Local Resolution of the Problem of Time. IX. Spacetime Constructability
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• A Local Resolution of the Problem of Time. VII. Constraint Closure
• Geometry from Brackets Consistency
• Chaos, Dirac observables and constraint quantization
• Monopoles of Twelve Types in 3-Body Problems
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
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• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Problem of Time and Background Independence: classical version's higher Lie Theory
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
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• Problem of Time: Temporal Relationalism Compatibility of Other Local Classical Facets Completed
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Topological Shape Theory
• Nijenhuis-type variants of Local Theory of Background Independence
• Nambu variant of Local Resolution of Problem of Time and Background Independence
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Lie Theory suffices to understand, and Locally Resolve, the Problem of Time
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Comparative Theory of Background Independence
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• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems