On Quantum Ito Algebras and their decompositions

Details On Quantum Ito Algebras and their decompositions On Quantum Ito Algebras and their decompositions

2005-12-21

arxiv.org/abs/math-ph/0512071v1

Letters in Mathematical Physics 45 (1998) 131--145

Physics

Mathematical Physics

13 pages. See related papers at http://www.maths.nott.ac.uk/personal/vpb/research/cha_noi.html http://www.maths.nott.ac.uk/p

Scientific paper

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.

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