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Topics in Koopman-von Neumann Theory
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Topics in Koopman-von Neumann Theory
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In this thesis we study several features of the operatorial approach to classical mechanics pionereed by Koopman and von Neumann (KvN) in the Thirties. In particular in the first part we study the role of the phases of the KvN states. We analyze, within the KvN theory, the two-slit experiment and the Aharonov-Bohm effect and we make a comparison between the classical and the quantum case. In the second part of the thesis we study the extension of the KvN formalism to the space of forms and Jacobi fields. We first show that all the standard Cartan calculus on symplectic spaces can be performed via Grassmann variables or via suitable combinations of Pauli matrices. Second we study the extended Hilbert space of KvN which now includes forms and prove that it is impossible to have at the same time a positive definite scalar product and a unitary evolution. Clear physical reasons for this phenomenon are exhibited. We conclude the thesis with some work in progress on the issue of quantization.
Forward citations
Cited by 2 Pith papers
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The History of Hilbert-Space Formulations of Classical Physics
Classical wave functions with ρ = |ψ|² were introduced after Koopman and von Neumann, probably by Schönberg, so calling them 'KvN waves' is a misattribution.
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Classical and quantum mechanics across representations: an operational reading of the Wigner Weyl correspondence
The robust classical–quantum boundary across Wigner–Weyl representations is noncommutative star-multiplication, not negativity or the choice of phase-space versus Hilbert-space language.
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