Mathematics of the classical and the quantum

TL;DR

This paper explores a unified geometric framework that connects classical Newtonian and quantum Schrödinger dynamics within a single Hilbert space, revealing new insights into their relationship and measurement processes.

Contribution

It introduces a novel geometric formulation that embeds classical and quantum dynamics into a single framework, showing that quantum mechanics extends classical mechanics uniquely within this setting.

Findings

Classical dynamics are a constrained subset of quantum dynamics.

Quantum observables correspond to vector fields on the state space.

The Born rule emerges from classical measurement distributions.

Abstract

Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail, and applied to the process of measurement of spin and position observables.