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Quantum stochastic equation for the low density limit
Full text linkA new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown
Quantum theory and its stochastic limit
Full text linkWell suited as a textbook in the emerging field of stochastic limit, which is a new mathematical technique developed for solving nonlinear problems in quantum theory
Non--linear extensions of classical and quantum stochastic calculus and essentially infinite dimensional analysis
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Non-commutative (quantum) probability, master fields and stochastic bosonization
Full text linkIn this report we discuss some results of non--commutative (quantum) probability theory relating the various
notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and physics including: --deformed and free central limit theorems; the description of the master (i.e. central limit) field in matrix models along the recent Singer suggestion to relate it to Voiculescu's results on the freeness of the large limit of random matrices; quantum stochastic differential equations for the gauge master field in QCD; the theory of stochastic limits of quantum fields and its applications to stochastic bosonization of
Fermi fields in any dimensions; new structures in QED such as a nonlinear
modification of the Wigner semicircle law and the interacting Fock space: a natural explicit example of a self--interacting quantum field which exhibits the non crossing diagrams of the Wigner semicircle law
A stochastic golden rule and quantum Langevin equation for the low density limit
Full text linkA rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups
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