From probabilistic mechanics to quantum theory

TL;DR

This paper demonstrates that quantum theory can be derived as an extension of classical probabilistic physics by introducing phase space functions for observables, leading to quantum-like equations and relations.

Contribution

It extends classical Hamiltonian mechanics by defining phase space functions for all observables, enabling a derivation of quantum theory within a classical probabilistic framework.

Findings

Derives Schrödinger's equation from extended classical structures

Shows how quantum operators and relations emerge from classical probabilistic physics

Provides a new perspective on the interpretation of quantum mechanics

Abstract

We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton's function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus we have all of the above entities generated not only by Hamilton's function but by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a "quantum-like" form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud.:Math. Found. (2018) 5:219-227]. We obtain the most important relations of QT, namely the form of operators, Schr\"odinger's equation, eigenvalue equations, commutation relations, expectation values, and Born's rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.