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.
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
V¨axj¨o 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 Schr¨odinger’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 Schrödinger 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 Schrödinger 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
Biography
Andrei Khrennikov is a professor of applied mathematics at Linnaueus University (Växjö, 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.
