A state reduct,ion is the stat1e change caused by a lueasurement on a quan- tum system conditional upon the outcome.A rigorous theorv of the state reduction is developed with nlathematiCa1 formalism, I)hysiCa1 interpret.ation,and nιodels .A special em1JhasiS is on the pure state reduction which trans- forms a pure 1) riorStatCe to the pure posterior state for every outcome.Mathe- 1natiCa1 structure of general pure state reductions is discussed and it is 1) roVed that every pure state reduction is decomposed into just two types, called tlle von Neumann-Davies (ND)tyl)e and the Gordon-Louisell (GL) type; a state reduction $\psirightarrow\psi_{x}$ ' is of the ND type if the 1IlaP1in ) gψ ) ↦P(x∣ψ)1/2?l′x ' is linear, where P(x∣ψ′) is the probability density of the outcome x , and of the GL tyl ) e if ψx depends only on the outcome x (independent of the prior state )) .
We explored the stochastic analysis on a complex Hilbert space to show that one of the cornerstones of quantum mechanics (QM), namely Heisenberg's uncertainty relation, can be derived in the classical probabilistic framework. We created a new mathematical representation of quantum averages: as averages with respect to classical random fields. The existence of a classical stochastic model matching with Heisenberg's uncertainty relation makes the connection between classical and quantum probabilistic models essentially closer. In real physical situations, random fields are valued in the L2-space. Hence, although we model QM and not QFT, the classical systems under consideration have an infinite number of degrees of freedom. And in our modeling, infinite-dimensional stochastic analysis is the basic mathematical tool.
The book collects a series of papers centered on two main streams: Feynman path integral approach to Quantum Mechanics and statistical mechanics of quantum open systems. Key authors discuss the state-of-the-art within their fields of expertise. In addition, the volume includes a number of contributed papers with new results, which have been thoroughly refereed. The contributions in this volume highlight emergent research in the area of stochastic analysis and mathematical physics, focusing, in particular on Feynman functional integral approach and, on the other hand, in quantum probability. The book is addressed to an audience of mathematical physicists, as well as specialists in probability theory, stochastic analysis and operator algebras.