The relation between classical and quantum mechanics

Thesis

This thesis examines the relation between classical and quantum mechanics from philosophical, mathematical and physical standpoints.

It first presents arguments in support of "conjectural realism" in scientific theories distinguished by explicit contextual structure and empirical testability; and it analyses intertheoretic reduction in terms of weakly equivalent theories over a domain of applicability.

Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to spacetime geometry and the constraints this places on the dynamics.

Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localization in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localization and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in tum is applied to yield a topological theory of bound and scattering states:

Finally, the thesis questions the validity of "classical limits" and "quantisations" in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.

Authors: Taylor, P

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Keywords: V-asymptotic state; Momentum kinematic property; Automorphism; Hepp; Phase-space boundedness; D-bimeasurable; Q-flow; Gaussian coherent state; Levy-Leblond; Symplectomorphism; Weyl-Wigner-Moyal correspondence; Schrödinger’s non-invariance theorem; Perspectivity equivalence class; Co-ordinative definition; Duhamel formula; WKB-Maslov; Inter-theoretic reduction; Lie algebra; Diffeomorphism; Transition-probability space; Axiom of choice; Isomorphism; Hilbert subspace; Hilbert-Schmidt operator; Orthocomplementation; Bijection; Hagedorn; Intrinsic probability; Riemannian manifold; Lagrangian foliation; Galilei group; Gaussian; Metaplectic; Q-propagator; Superposition set; Avoidable probability; Leray-Schauder-Tychonoff theorem; Hamilton-Jacobi theory; Quantum mechanics; Poincaré recurrence theorem; Borel function; Riemann-Lebesgue lemma; Intertheoretic reduction; Reductionism; Weak modularity; Phase-space localisation; Hilbertian; Von Neumann’s ergodic theorem; Borel; Moyal bracket; Banach space; System of imprimitivity; Ehrenfest’s theorem; Perspectivity class; Rellich’s criterion; Schwarzschild’s theorem; Hausdorff’s Maximality Principle; Weyl; Morphism; Quantum no-capture; Superposition principle; Lebesgue; λ-scaled function; C-system; Classical mechanics; Q-realisation; Souriau-Kostant program; Piron’s Theorem; Annihilator set; Q-system; Cotangent bundle; Analytic problem of reduction; Hamiltonian vector; Gleason’s Theorem; Symplectic manifold; Lie group; Riesz’s criterion; Instrumentalism; Liouville measure; Paley-Wiener theorem; Primas; Birkhoff’s theorem; Bolzano-Weierstrass theorem; Orthomodularity; Symplectic structure; Borel set; Hilbert space