Symmetry in Nonlinear Mathematical Physics - 2009

Viacheslav Belavkin (School of Mathematics, University of Nottingham, UK)

Quantum LÚvy-It˘ fields over modular algebras

      An algebraic characterization of the general (infinite dimensional, noncommutative) It˘ algebra is given and a fundamental matrix realization for such algebras in the Krein space and its exponential in Fock space is described. The notion of four-normed It˘ B*-algebra, generalizing the C*-algebra is introduced to include the Banach infinite dimensional It˘ algebras of noncommutative Brownian and LÚvy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. The first ones posses a vacuum vector in the Krein space, the second ones are described by a generalized Hilbert modular algebra
      It is proved that every It˘ algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum LÚvy (Poisson) algebra. In particular, every quantum thermal field is the orthogonal sum of a quantum Gaussian field and a quantum Poisson field as it is stated by the LÚvy-Khinchin theorem in the classical stochastic case corresponding to the commutative It˘ algebras.The well known LÚvy-Khinchin classification of the classical noise can be reformulated in purely algebraic terms as the decomposability of any commutative It˘ algebra into Wiener (Brownian) and Poisson (LÚvy) orthogonal components. In the general case we shall show that every It˘ ⋆-algebra is also decomposable into a quantum Brownian, and a quantum LÚvy orthogonal components.
      Thus classical stochastic calculus developed by It˘, and its quantum stochastic analog, given by Hudson and Parthasarathy in 2, has been unified in a ⋆-algebraic approach to the operator integration in Fock space 3, in which the classical and quantum calculi become represented as two extreme commutative and noncommutative cases of a generalized It˘ calculus.