Klein, Ulf (2019) From probabilistic mechanics to quantum theory. Quantum Studies: Mathematics and Foundations.

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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.

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Item Type:	Published Article or Volume
Creators:	Creators Email ORCID Klein, Ulf ulf.klein@jku.at
Keywords:	quantum-classical relation, ensemble theory, quantization, derivation of quantum theory
Subjects:	General Issues > Determinism/Indeterminism General Issues > History of Philosophy of Science Specific Sciences > Probability/Statistics Specific Sciences > Physics > Quantum Mechanics General Issues > Reductionism/Holism
Depositing User:	Dr. Ulf Klein
Date Deposited:	07 Nov 2019 02:01
Last Modified:	07 Nov 2019 02:01
Item ID:	16609
Journal or Publication Title:	Quantum Studies: Mathematics and Foundations
Publisher:	Springer
DOI or Unique Handle:	https://doi.org/10.1007/s40509-019-00201-w
Subjects:	General Issues > Determinism/Indeterminism General Issues > History of Philosophy of Science Specific Sciences > Probability/Statistics Specific Sciences > Physics > Quantum Mechanics General Issues > Reductionism/Holism
Date:	August 2019
URI:	https://philsci-archive.pitt.edu/id/eprint/16609

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