Classical Representations of Quantum Mechanics Related to Statistically Complete Observables

Abstract

We present a reformulation of quantum mechanics in terms of probability measures and functions on a general classical sample space and in particular in terms of probability densities and functions on phase space. The basis of our proceeding is the existence of so-called statistically complete observables and the duality between the state spaces and the spaces of the observables, the latter holding in the quantum as well as in the classical case. In the phase-space context, we further discuss joint position-momentum observables, Hilbert spaces of infinitely differentiable functions on phase space, and dequantizations. Finally, the relation of quantum dynamics to the classical Liouville dynamics is investigated.

118 pages, no figures. This work was published 1997 as a book by Wissenschaft und Technik Verlag Berlin. Since it has proved to be of interest to some researchers, it is published here for an easier access

Cited by in corpus (13)

• An introduction to the tomographic picture of quantum mechanics
• Necessity of negativity in quantum theory
• Optimal measurements in quantum mechanics
• On the coexistence of position and momentum observables
• Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics
• Tasks and premises in quantum state determination
• A pedagogical presentation of a $C^{\star }$-algebraic approach to quantum tomography
• Positive Operator Valued Measures and Feller Markov Kernels
• Semispectral Measures and Feller markov Kernels
• Joint measurability through Naimark's theorem
• Exact Quantum Correlations of Conjugate Variables From Joint Quadrature Measurements
• Quantization and noiseless measurements
• Error and unsharpness in approximate joint measurements of position and momentum