Foundations of Relational Particle Dynamics

Abstract

Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural mechanics on these spaces. I also show that these coincide with Barbour's indirectly-constructed relational dynamics by performing a full reduction on the latter. Then the identification of the configuration spaces as n-spheres and complex projective spaces, for which spaces much mathematics is available, significantly advances the understanding of Barbour's relational theory in spatial dimensions 1 and 2. I also provide the parallel study of a new theory for which positon and scale are purely relative but orientation is absolute. The configuration space for this is an n-sphere regardless of the spatial dimension, which renders this theory a more tractable arena for investigation of implications of scale invariance than Barbour's theory itself.

Minor typos corrected; references updated

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Cited by in corpus (49)

• Problem of Time in Quantum Gravity
• The Link between General Relativity and Shape Dynamics
• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• Records Theory
• Does relationalism alone control geometrodynamics with sources?
• Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
• Triangleland. II. Quantum Mechanics of Pure Shape
• Machian Time Is To Be Abstracted From What Change?
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• On the Semiclassical Approach to Quantum Cosmology
• Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics
• Configuration Spaces in Fundamental Physics
• Relational mechanics of shape and scale
• 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
• Kendall's Shape Statistics as a Classical Realization of Barbour-type Timeless Records Theory approach to Quantum Gravity
• Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories
• Implementing Mach's Principle Using Gauge Theory
• Geometry from Brackets Consistency
• On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics
• Two new versions of Heron's Formula
• Relational Motivation for Conformal Operator Ordering in Quantum Cosmology
• Monopoles of Twelve Types in 3-Body Problems
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric
• Shape Theories. II. Compactness Selection Principles
• Maximal Angle Flow on the Shape Sphere of Triangles
• Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities
• Quantum Cosmological Metroland Model
• Scaled Triangleland Model of Quantum Cosmology
• Six New Mechanics corresponding to further Shape Theories
• Shape Space Methods for Quantum Cosmological Triangleland
• Relational Quadrilateralland. Analogues of Isospin and Hypercharge
• Shape Quantities for Relational Quadrilateralland
• 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
• Quantum Cosmological Relational Model of Shape and Scale in 1-d
• Topological Shape Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Relational Quadrilateralland Interpretation of CP^2 and Quotients
• A Novel Approach to Quantum Gravity in the Presence of Matter without the Problem of Time
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Relational Quadrilateralland. II. The Quantum Theory