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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References
Accardi, L.: On the quantum Feynman-Kac formula. Rend. Sem. Mat. Fis. Milano48, 135–180 (1980)
Applebaum, D., Hudson, R.L.: Fermion diffusions. J. Math. Phys. (to appear)
Barnett, C., Steater, R.F., Wilde, I.: The Ito-Clifford integral. J. Funct. Anal.48, 172–212 (1982)
Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups ofn-level systems. J. Math. Phys.17, 821–5 (1976)
Hudson, R.L., Ion, P.D.F., Parthasarathy, K.R.: Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae. Commun. Math. Phys.83, 261–80 (1982)
Hudson, R.L., Karandikar, R.L., Parthasarathy, K.R.: Towards a theory of noncommutative semimartingales adapted to Brownian motion and a quantum Ito's formula; and Hudson, R.L., Parthasarathy, K.R.: Quantum diffusions. In: Theory and applications of random fields, Kallianpur, (ed.). Lecture Notes in Control Theory and Information Sciences 49, Berlin, Heidelberg, New York, Tokyo: Springer 1983
Hudson, R.L., Parthasarathy, K.R.: Construction of quantum diffusions. In: Quantum probability and applications to the quantum theory of irreversible processes, Accardi (ed.) (to appear)
Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Math. Applicandae (to appear)
Ikeda, N., Watenabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981
Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119–30 (1976)
Liptser, R.S., Shiraev, A.N.: Statistics of random processes. I. General theory. Berlin, Heidelberg, New York: Springer 1977
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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