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

Abstract

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

arXiv comment25 pages including 3 figures

References in corpus (32)

• The Dynamics of General Relativity
• Partial and Complete Observables for Hamiltonian Constrained Systems
• Problem of Time in Quantum Gravity
• Observables in effective gravity
• Some remarks on the notions of general covariance and background independence
• Canonical Quantum Gravity
• The Problem of Time in Quantum Gravity
• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Beables/Observables in Classical and Quantum Gravity
• Problem of Time and Background Independence: the Individual Facets
• Explicit partial and functional differential equations for beables or observables
• A Local Resolution of the Problem of Time
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• On Types of Observables in Constrained Theories
• TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)
• Theory of Transformation Groups, by S. Lie and F. Engel (Vol. I, 1888). Modern Presentation and English Translation
• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• Shape Theory. III. Comparative Theory of Backgound Independence
• 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 Relationalism
• A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• A Local Resolution of the Problem of Time. IX. Spacetime Constructability
• Geometry from Brackets Consistency
• Chaos, Dirac observables and constraint quantization
• A Local Resolution of the Problem of Time. VII. Constraint Closure
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• 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