From probabilistic mechanics to quantum theory

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 also 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 5:219-227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger'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.Keywords Quantum-classical relation · Ensemble theory · Quantization · Derivation of quantum theory Mathematics Subject Classification 81P05 · 81S05 · 82C03 · 70H20 IntroductionGeneral agreement regarding the meaning of the quantum-mechanical formalism has not been achieved so far. This lack of clarity is closely related to a lack of clarity regarding the relation between quantum theory (QT) and classical physics. The present-day ideas on the quantum-classical interface have been established more than 90 years ago. They have never undergone a critical reexamination, despite the fact that a wealth of new information, both experimentally and theoretically, has been obtained since then.Einstein's claim that QT must be an ensemble theory, and not a theory about individual particles, is neither generally accepted

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