On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed.
On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω = H × H , where H is the real Hilbert space. Statistical states are given by Gaussian measures on Ω having zero mean value and dispersion of very small magnitude α (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps f : Ω → R (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point ψ vacuum ≡ 0) of variables f : Ω → R with respect to the small parameter √α The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space Ω. A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on Ω. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.
We show that QM can be represented as a natural projection of a classical statistical model on the phase space Ω=H×H, where H is the real Hilbert space. Statistical states are given by Gaussian measures on Ω having zero mean value and dispersion of very small magnitude α (which is considered as a small parameter of the model). Such statistical states can be interpreted as fluctuations of the background field, cf. with SED and Nelson's mechanics. Physical variables (e.g., energy) are given by maps f:Ω→R (functions of classical fields). The conventional quantum representation of our prequantum classical statistical model is constructed on the basis of the Taylor expansion (up to the terms of the second order at the vacuum field point ψ_vacuum≡0) of variables f:Ω→R with respect to the small parameter α. The complex structure of QM is induced by the symplectic structure on the infinite-dimensional phase space Ω. A Gaussian measure (statistical state) is represented in QM by its covariation operator. Equations of Schrödinger, Heisenberg and von Neumann are images of Hamiltonian dynamics on Ω. The main experimental prediction of our prequantum model is that experimental statistical averages can deviate from ones given by QM.
We consider transition from the classical statistical model to the quantum statistical model through ignorance (of huge volume) of information in process of construction of a wave function { a complex probability amplitude. Our approach clarifles relation between classical and quantum statistical models (and hence relation between classical and quantum information theories). In particular, it can be considered as a step toward demystiflcation of quantum theory. The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities.In our approach it is forbidden to consider abstract context independent probabilities: \flrst context and then probability. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. Our formalism can be applied to various domains outside quantum physics, e.g., cognitive sci- ences, psychology, economics, chemistry: roughly speaking in any domain in that it is impossible to extract the whole information about an ontic (or to say realistic) model and hence the epis- temological (experimental or to say observational) model is constructed through huge reduction of information volume. We introduced a notion of prespace { space of states containing complete information about systems under consideration. Classical phase space and quantum Hilbert space are obtained as projections of such a prespace.
We show that the probabilistic formalism of QM can be obtained as a special projection of classical statistical mechanics for systems with an infinite number of degrees of freedom. Such systems can be interpreted as classical fields. Thus in our approach QM is a projection of (prequantum) classical statistical field theory (PCSFT). This projection is based on the Taylor expansion of classical physical variables – maps f: Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and negligibly small dispersion. On one hand, the creation of such a prequantum model strongly supports attempts (first of all by Schrödinger and Einstein) to create purely field model of QM. On the other hand, it gives the possibility to go beyond QM. The main experimental prediction is that averages calculated in the mathematical formalism of QM (von Neumann’s trace formula) are only approximative averages. If predictions of PCSFT are correct then it would be possible to find deviations of experimental averages from quantum ones. 1
A comparison of structural features of quantum and classical physical theories, such as the information capacity of systems subject to these theories, requires a common formal framework for the presentation of corresponding concepts (such as states, observables, probability, entropy). Such a framework is provided by the notion of statistical model developed in the convexity approach to statistical physical theories. Here we use statistical models to classify and survey all possible types of embedding and extension of quantum probabilistic theories subject to certain reasonable constraints. It will be shown that the so-called canonical classical extension of quantum mechanics is essentially the only `good' representation of the quantum statistical model in a classical framework. All quantum observables are thus identified as fuzzy classical random variables.