Communications in Mathematical Physics

, Volume 93, Issue 3, pp 301-323

First online:

Quantum Ito's formula and stochastic evolutions

  • R. L. HudsonAffiliated withMathematics Department, University of Nottingham
  • , K. R. ParthasarathyAffiliated withIndian Statistical Institute

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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.