Accardi, Luigi; Parthasarathy, K. R. A martingale characterization of canonical commutation and anticommutation relations. (English) Zbl 0642.60032 J. Funct. Anal. 77, No. 1, 211-231 (1988). Using a martingale condition and some restrictions on moments up to fourth order the characterization problem of boson, fermion, and classical Brownian motions is studied from a unified point of view entirely within the framework of elementary operator theory. Global commutation and anticommutation rules turn out to be consequences of corresponding commutation and anticommutation rules between past and future observables. Reviewer: Sh.Ayupov Cited in 1 ReviewCited in 8 Documents MSC: 60G44 Martingales with continuous parameter 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 60J65 Brownian motion Keywords:martingale condition; characterization problem; commutation and anticommutation rules PDF BibTeX XML Cite \textit{L. Accardi} and \textit{K. R. Parthasarathy}, J. Funct. Anal. 77, No. 1, 211--231 (1988; Zbl 0642.60032) Full Text: DOI OpenURL References: [1] Hudson, R.L; Parthasarathy, K.R, Quantum Ito’s formula and stochastic evolutions, Commun. math. phys., 93, 301-323, (1984) · Zbl 0546.60058 [2] Hudson, R.L; Parthasarathy, K.R, Unification of fermion and boson stochastic calculus, Commun. math. phys., 104, 457-470, (1986) · Zbl 0604.60063 [3] Ikeda, N; Watanabe, S, Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam · Zbl 0495.60005 [4] Nelson, E, Dynamical theories of Brownian motion, (1972), Princeton Univ. Press Princeton, NJ [5] Parthasarathy, K.R; Sinha, K.B, Boson-fermion relations in several dimensions, Pramāna, 27, 105-116, (1986) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.