Quantum Ito’s formula and stochastic evolutions. (English) Zbl 0546.60058

Summary: Using only the Boson canonical commutation relations and the Riemann- Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.


60H05 Stochastic integrals
60F05 Central limit and other weak theorems
81P20 Stochastic mechanics (including stochastic electrodynamics)
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