Quantum independent increment processes on superalgebras
type: Item
instance of: scholarly article
title: Quantum independent increment processes on superalgebras
zbMATH Open document ID: 0627.60014
DOI: 10.1007/BF01162868
author: L. Accardi
author: Michael Schürmann
author: Wilhelm von Waldenfels
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173938
review text: Starting from the concept of quantum stochastic processes in the sense of the first author, \textit{A. Frigerio} and \textit{J. T. Lewis} [Publ. Res. Inst. Math. Sci. 18, 97-133 (1982; Zbl 0498.60099)], the notion of quantum independent increment processes on superalgebras is introduced. A reconstruction theorem is proved establishing a one-to-one correspondence between these processes and their infinitesimal generators, which are conditionally positive linear functionals on an involutive bialgebra. (Bialgebras sometimes are called Hopf-algebras.) In particular a new technique for constructing continuous tensor products of \(Z_ 2\)-graded involutive algebras not based on the representations of the canonical (anti-)commutation relations is provided. A quantum version of the Lévy-Khintchine formula and a full classification of the ``continuous trajectories'' quantum processes with independent stationary increments are obtained. It is proved that solutions of quantum stochastic differential equations on the Fock space over \(L_ 2(R_ +)\) in the sense of \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058)], give rise to quantum independent increment processes as defined in the paper. A formula for the generators of these special processes is derived.
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 81S05
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 4020082
zbMATH Keywords: quantum stochastic processes
zbMATH Keywords: quantum independent increment processes on superalgebras
zbMATH Keywords: Lévy-Khintchine formula
zbMATH Keywords: quantum stochastic differential equations
MaRDI profile type: Publication
cites work: Q3933122
cites work: Nonrelativistic quantum mechanics as a noncommutative Markov process
cites work: A martingale characterization of canonical commutation and anticommutation relations
cites work: Quantum stochastic processes
cites work: The strong Markov property for fermion Brownian motion
cites work: Fermion Ito's formula and stochastic evolutions
cites work: Quantum mechanical Wiener processes
cites work: An algebraic version of the central limit theorem
cites work: Q4153376
cites work: Q3685776
cites work: Quantum Ito's formula and stochastic evolutions
cites work: On the structure of Hopf algebras
cites work: Positive definite kernels, continuous tensor products, and central limit theorems of probability theory
cites work: Q3704028
cites work: Q3025933
cites work: Q5590089
cites work: Q5648966
cites work: An algebraic central limit theorem in the anti-commuting case
cites work: Q3314679
description: scientific article; zbMATH DE number 4020082