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Covariant Poisson Fields In Fock Space

David Applebaum

Published Jan 1, 1996• 2 citations

Abstract

As I type these words, I have in front of me a collection of notes from a seminar in Quantum Probability run by Luigi Accardi in Roma in 1984-5 where I was at the time based as a research fellow. In those days I was rather meticulous in ordering my jottings and on turning to page 131 in my file I find twenty pages entitled “Representations of Current Groups” by K.R. Parthasarathy. I remember being fascinated by these lectures (I think there were two) in which Partha gave an account of how the Lévy-Khinchine formula arises, naturally from the construction of continuous tensor products of unitary group representations (see e.g. [St], [Ar], [PS]) and then revealed the beautiful Fock space decomposition of infinitely divisible random variables as sums of creation, conservation and annihilation operators. This is in fact one of the many perspectives through which the historian can see the “birth” of quantum probability — as a student of Robin Hudson, I had seen the subject evolve from a different point of view, through the search for quantum central limit theorems and non-commuting analogues of Brownian motion. The marriage of these two viewpoints turned out to be a wonderful thing, leading as it did to the creation of quantum stochastic calculus.

DOI: 10.1007/978-93-80250-87-8_1

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Previous article Next article A Representation of Lévy-Khinchine Type in Quantum Probability TheoryA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1132015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. S. Kholevo, Infinitely divisible measurements in quantum probability, Theory Probab. Appl., 32 (1986), 493–497 0623.60005 Google Scholar[2] A. Barchielli, , L. Lanz and , G. M. Prosperi, Statistics of continuous trajectories in quantum mechanics: operation-valued stochastic processes, Found. Phys., 13 (1983), 779–812 87b:81006 CrossrefGoogle Scholar[3] A. Barchielli and , G. Lupieri, Dilations of operation valued stochastic processesQuantum probability and applications, II (Heidelberg, 1984), Lecture Notes in Math., Vol. 1136, Springer, Berlin, 1985, 57–66 87g:46101 0572.60046 CrossrefGoogle Scholar[4] Erik Christensen and , David E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. Soc. (2), 20 (1979), 358–368 81f:46082 0448.46040 CrossrefGoogle Scholar[5] E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976x+171 58:8853 0388.46044 Google Scholar[6] Alain Guichardet, Symmetric Hilbert spaces and related topics, Springer-Verlag, Berlin, 1972v+197, Lecture Notes in Math., 26 58:12416 0265.43008 CrossrefGoogle Scholar[7] K. R. Parthasarathy and , K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Springer-Verlag, Berlin, 1972vi+107, Lecture Notes in Math., 272 58:29849 0237.43005 CrossrefGoogle Scholar[8] K. R. Parthasarathy, One parameter semigroups of completely positive maps on groups arising from quantum stochastic differential equations, 1985, Indian Statist. Inst., New Delhi, preprint Google Scholar[9] Shôichirô Sakai, C*-algebras and W*-algebras, Springer-Verlag, New York, 1971xii+253 56:1082 CrossrefGoogle Scholar[10] W. Forrest Stinespring, Positive functions on $C\sp *$-algebras, Proc. Amer. Math. Soc., 6 (1955), 211–216 16,1033b 0064.36703 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Generalized Bochner theorem and Levy-Khinchine representation for completely positive definite generalized Toeplitz kernelsIntegral Equations and Operator Theory, Vol. 27, No. 4 Cross Ref On the Lévy–Khinchin Formula in Noncommutative Probability TheoryA. S. Kholevo17 July 2006 | Theory of Probability & Its Applications, Vol. 38, No. 4AbstractPDF (1196 KB)Convolution Semigroups in Noncommutative Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 3AbstractPDF (636 KB)Infinitely Divisible Markov Mappings in Quantum Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 2AbstractPDF (521 KB) Volume 32, Issue 1| 1988Theory of Probability & Its Applications History Submitted:01 April 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132015Article page range:pp. 131-136ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

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