The relation between classical and quantum mechanics

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.

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
Advisor: Davies, B