Accardi, Luigi; Quaegebeur, J. The Ito algebra of quantum Gaussian fields. (English) Zbl 0669.60099 J. Funct. Anal. 85, No. 2, 213-263 (1989). The notion of mutual quadratic variation (square bracket) is extended to a quantum probabilistic framework. The mutual quadratic variations of the annihilation, creation, and number fields in a Gaussian representation are calculated, in both the Boson and the Fermion case, in the strong topology on a common invariant domain. It is proved that the corresponding Ito table closes at the second order. The Fock representation is characterized, among the Gaussian ones, by the property that its Ito table closes at the first order. Cited in 8 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 81P20 Stochastic mechanics (including stochastic electrodynamics) 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 60H99 Stochastic analysis Keywords:quantum Gaussian fields; finitely additive measure; mutual quadratic variation; annihilation; creation; Fock representation PDFBibTeX XMLCite \textit{L. Accardi} and \textit{J. Quaegebeur}, J. Funct. Anal. 85, No. 2, 213--263 (1989; Zbl 0669.60099) Full Text: DOI Link References: [2] Accardi, L., Mathematical theory of quantum noise, (Proceedings, 1st World Conference of the Bernoulli Society. Proceedings, 1st World Conference of the Bernoulli Society, Tashkent, 1986 (1987), VNU Science Press) · Zbl 1031.60064 [5] Applebaum, D.; Hudson, R. L., Fermion Ito formula and stochastic evolutions, Comm. Math. Phys., 96, 473 (1987) · Zbl 0572.60052 [6] Barnett, C.; Streater, R. F.; Wilde, I. F., Quasi-free quantum stochastic integrals for the CCR and CAR, J. Funct. Anal., 52, 19-47 (1983) · Zbl 0513.60063 [7] Chung, K. L.; Williams, R. J., Introduction to Stochastic Integration (1983), Birkhäuser: Birkhäuser Basel · Zbl 0527.60058 [8] Dellacherie, M.; Meyer, P. A., Probabilities et potentiel (1975), Hermann: Hermann Paris [9] Haken, H., Laser Theory, (Handbook of Physics, Vol. 25 (1970), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0439.93050 [10] Hudson, R. L.; Parthasarathy, K. R., Quantum Ito’s formula and stochastic evolutions, Comm. Math. Phys., 93, 301-323 (1984) · Zbl 0546.60058 [12] Meyer, P. A., Elements de probabilités quantiques, (Azema, J.; Yor, M., Séminaire de Probabilités XX. Séminaire de Probabilités XX, 1985. Séminaire de Probabilités XX. Séminaire de Probabilités XX, 1985, Lecture Notes in Mathematics, Vol. 1204 (1986), Springer-Verlag: Springer-Verlag New York/Berlin), 186-312 [13] Nelson, E., Dynamical Theories of Brownian Motion (1972), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ [16] Sakai, S., \(C^∗\)-Algebras and \(W^∗\)-Algebras (1971), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0219.46042 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.