Andrei Khrennikov is a professor of applied mathematics at Linnaueus University (Vaxjo, South-East Sweden) and the director of the multidisciplinary research center at this university, the International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science.
Introduction Author's views on quantum foundations Prequantum classical statistical field theory: introduction Where is discreteness? Devil in detectors? On experiments to tests the Euclidean model Conventional quantum theory: fundamentals Postulates Quantization Interpretations of Wave Function Vaxjo interpretation of quantum mechanics Short introduction to classical probability theory Quantum Conditional Probability Interference of Probabilities in Quantum Mechanics Two slit experiment Corpuscular interference Interference of probabilities in cognitive science Fundamentals of Prequantum Classical Statistical Field Theory Noncomposite systems Composite systems Stochastic process corresponding to Schrodinger's evolution Correlations of the components of the prequantum field PCSFT-formalism for classical electromagnetic field-1 Discussion of a possible experimental verification of PCSFT Photonic field Correlation between polarization vectors of entangled photons Functionals of prequantum fields corresponding to operators of photon polarization Classical representation of Heisenberg's uncertainty relation Towards violation of Born's rule: description of a simple experiment Why Gaussian? On correspondence between quantum observables and classical variables Prequantum Dynamics from Hamiltonian Equations on the Infinite-dimensional Phase Space Hamiltonian mechanics Symplectic representation of Schrodinger dynamics Classical and quantum statistical models Measures on Hilbert spaces Lifting of pointwise dynamics to spaces of variables and measures Dispersion preserving dynamics Dynamics in the space of physical variables Probabilistic dynamics Detailed analysis of dispersion preserving dynamics Quantum Mechanics as Approximation of Statistical Mechanics of Classical Fields The Taylor approximation of averages for functions of random variables Quantum model: finite-dimensional case Prequantum --> quantum correspondence: finite dimensional case Prequantum phase space: infinite-dimensional case Gaussian measures corresponding to pure quantum states Illustration of the prequantum -->quantum coupling in the case of qubit mechanics Prequantum classical statistical field theory (PCSFT) PCSFT-formalism for classical electromagnetic field-2 Asymptotic expansion of averages with respect to electromagnetic random field Interpretation Simulation of quantum-like behavior for the classical electromagnetic field Maxwell equations as Hamilton equations or as Schrodinger equation Quadratic variables without quantum counterpart Generalization of quantum mechanics Coupling between the time scale and dispersion of a prequantum random signal Supplementary Mathematical Considerations Dispersion preserving dynamics with nonquadratic Hamilton functions Formalism of rigged Hilbert space Quantum pure and mixed states from the background field Classical model for unbounded quantum observables Mathematical Presentation for Composite Systems Derivation of basic formulas Vector and operator realizations of the tensor product Operation of the complex conjugation in the space of self-adjoint operators The basic operator equality for arbitrary (bounded) self-adjoint operators Operator representation of reduced density operators Classical random field representation of quantum correlations Infinite-dimensional case Correlations in triparticle systems PCSFT-representation for a mixed state Phenomenological Detection Model 271 Finite-dimensional model Position measurement for the prequantum field Field's energy detection model Coupling between probabilities of detection of classical random fields and quantum particles Deviation from predictions of quantum mechanics Averages Local measurements Measurement of observables with discrete spectra Classical field treatment of discarding of noise contribution in quantum detectors Quantum channels as linear filters of classical signals Quantum individual events Classical random signals: ensemble and time representations of averages Discrete-counts model for detection of classical random signals Quantum probabilities from threshold type detectors The case of an arbitrary density operator The general scheme of threshold detection of classical random signals Probability of coincidence Stochastic process description of detection
The present wave of interest in quantum foundations is caused by the tremendous development of quantum information science and its applications to quantum computing and quantum communication. It has become clear that some of the difficulties encountered in realizations of quantum information processing have roots at the very fundamental level. To solve such problems, quantum theory has to be reconsidered. This book is devoted to the analysis of the probabilistic structure of quantum theory, probing the limits of classical probabilistic representation of quantum phenomena.