A Local Resolution of the Problem of Time. VIII. Assignment of\n Observables

2020/01/13 by Edward Anderson, Anderson, Edward
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Relativity and Gravitational Theory

paper · pdf · doi:10.48550/arxiv.2001.04423

openalex publication_date 2020/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a state space, Assignment of Observables involves Taking Function\nspaces Thereover. At the classical level, the state space in question is phase\nspace or configuration space. This assignment picks up nontrivialities when\nwhichever combination of constraints and the quantum apply.\n For Finite Theories, weak observables equations are inhomogeneous-linear\nfirst-order PDE systems. Their general solution thus splits into complementary\nfunction plus particular integral: strong and nontrivially-weak observables\nrespectively. We provide a PDE analysis for each of these. In the case of\nsingle observables equations - corresponding to single constraints - the Flow\nMethod readily applies. Finding all the observables requires free\ncharacteristic problems. For systems, this method can be applied sequentially,\ndue to integrability conferred by Frobenius' Theorem. In each case, the first\npart of this approach is Lie's Integral Theory of Geometrical Invariants, or\nthe physical counterpart thereof. The second part finds the function space\nthereover, giving the entire space of (local) observables.\n We also outline the Field Theory counterpart. Here one has functional\ndifferential equations. Banach (or tame Fr 'echet) Calculus is however\nsufficiently standard for the Flow Method and free characteristic problems to\nstill apply. These calculi support the Lie-theoretic combination of machinery\nthat our Local Resolution of the Problem of Time requires, by which Field\nTheory and GR are included.\n

Citations

• The Problem of Time and Quantum Cosmology in the Relational Particle\n Mechanics Arena
• Problem of Time and Background Independence: the Individual Facets
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• Geometry from Brackets Consistency
• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging
• A Local Resolution of the Problem of Time. I. Introduction and Temporal\n Relationalism
• A Local Resolution of the Problem of Time. VII. Constraint Closure
• A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories
• A Local Resolution of the Problem of Time. IX. Spacetime\n Constructability
• Canonical Quantum Gravity

Related

• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion
• A Local Resolution of the Problem of Time. I. Introduction and Temporal\n Relationalism
• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• A general perspective on time observables