A Local Resolution of the Problem of Time. VII. Constraint Closure

Abstract

We now set up Constraint Closure in a manner consistent with Temporal and Configurational Relationalism. This requires modifying the Dirac Algorithm - which addresses the Constraint Closure Problem facet of the Problem of Time piecemeal - to the TRi-Dirac Algorithm. This is a member of the wider class of Dirac-type algorithms that enjoys the property of being Temporal Relationalism implementing (TRi). Constraint algebraic structures ensue. We include examples of types of constraint, outcomes of the Dirac Algorithm and different kinds of Constraint Closure Problems. Enough new Principles of Dynamics is required to support this venture that an Appendix on it is provided: differential Hamiltonians, anti-Routhians, and the brackets, state spaces and morphisms corresponding to these.

arXiv comment23 pages including 7 figures. v2 improves concepts for, and presentation of, our temporal-relationally modified Dirac Algorithm

References in corpus (25)

• The Dynamics of General Relativity
• Problem of Time in Quantum Gravity
• Canonical Quantum Gravity
• The Problem of Time in Quantum Gravity
• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
• Revisiting observables in generally covariant theories in the light of gauge fixing methods
• Constraints and gauge transformations: Dirac's theorem is not always valid
• Beables/Observables in Classical and Quantum Gravity
• Problem of Time and Background Independence: the Individual Facets
• Classical Machian Resolution of the Spacetime Reconstruction Problem
• Does relationalism alone control geometrodynamics with sources?
• A Local Resolution of the Problem of Time
• On Types of Observables in Constrained Theories
• TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)
• 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. IV. Quantum outline and piecemeal Conclusion
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• 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 Constructability
• Geometry from Brackets Consistency
• Shape Theories. II. Compactness Selection Principles
• Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric

Cited by in corpus (11)

• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• 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 Relationalism
• Geometry from Brackets Consistency
• 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
• Nijenhuis-type variants of Local Theory of Background Independence
• Nambu variant of Local Resolution of Problem of Time and Background Independence
• Lie Theory suffices to understand, and Locally Resolve, the Problem of Time
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables
• Comparative Theory of Background Independence