Abstract
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.
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Communicated by H. Araki
Parts of this work were completed while the first author was a Royal Society-Indian National Science Academy Exchange Visitor to the Indian Statistical Institute, New Delhi, and visiting the University of Texas supported in part by NSF grant PHY81-07381, and part while the second author was visiting the Mathematics Research Centre of the University of Warwick
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Hudson, R.L., Parthasarathy, K.R. Quantum Ito's formula and stochastic evolutions. Commun.Math. Phys. 93, 301–323 (1984). https://doi.org/10.1007/BF01258530
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DOI: https://doi.org/10.1007/BF01258530