Abstract
In the framework of von Neumann’s description of measurements of discrete quantum observables we establish a one-to-one correspondence between symmetric statistical operators W of quantum mechanical systems and classical point processes κ W , thereby giving a particle picture of indistinguishable quantum particles. This holds true under irreducibility assumptions if we fix the underlying complete orthonormal system. The method of the Campbell measure is developed for such statistical operators; it is shown that the Campbell measure of a statistical operator W coincides with the Campbell measure of the corresponding point process κ W . Moreover, again under irreducibility assumptions, a symmetric statistical operator is completely determined by its Campbell measure. Themethod of the Campbell measure then is used to characterize Bose-Einstein and Fermi-Dirac statistical operators. This is an elementary introduction into the work of Fichtner and Freudenberg [10, 11] combined with the quantum mechanical investigations of [2] and the corresponding point process approach of [30]. It is based on the classical work of von Neumann [22], Segal, Cook and Chaiken [7, 8, 28] as well as Moyal [18].
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References
W. Arveson, An Invitation to C*-algebras (Springer, New-York, 1976).
A. Bach, Indistinguishable Classical Particles (Lecture notes in physics, Springer, 1997).
F. A. Berezin, The Method of Second Quantization (Academic Press, New York, 1966).
H. Boerner, Darstellungen von Gruppen (Springer, Berlin, 1955).
O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer, Berlin, 1997).
G. Cassinelli, N. Zanghi, “Conditional probabilities in quantum theory I - conditioning with respect to a single event”, Nuovo cimento, 73, 237–245, 1983).
J. M. Chaiken, “Finite-particle representations and states of the canonical commutation relations”, Ann. Phys., 42, 23–80, 1967).
J. M. Cook, “The mathematics of second quantization”, Transactions AMS, 74, 222–245, 1953).
E. B. Davies, Quantum Theory of Open Systems (Academic Press, NY, 1976.
K. H. Fichtner, W. Freudenberg, “Point processes and the position distribution of infinite Boson systems”, J. Statist. Phys., 47, 959–978, 1987).
K. H. Fichtner, W. Freudenberg, “Characterizations of states of infinite Boson systems”, Commun. Math. Phys., 137, 315–357, 1991).
K. H. Fichtner, G. Winkler, “Generalized Brownian motion, point processes and stochastic calculus for random fields”, Math.Nachr., 161, 291–307, 1993).
V. Fock, “Konfigurationsraum und zweite Quantelung”, Zeitschrift fu¨ r Physik, 75, 622–647, 1932).
G. A. Goldin, U. Moschella, T. Sakuraba, “Self-similar processes and infinite-dimensional configuration spaces”, American Inst. Physics. Physics of atomic nuclei, 68, 1615–1684, 2005).
K. Krickeberg, Point Processes. A Random Radon Measure Approach (Walter Warmuth Verlag, Nächst Neuendorf, 2014).
V. Liebscher, “Using weights for the description of states of Boson systems”, Commun. Stoch. Anal., 3, 175–195, 2009).
J. Mecke, “Stationäre Maße auf lokal-kompakten Abelschen Gruppen”, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 9, 36–58, 1967).
J. E. Moyal, “Particle populations and number operators in Quantum Theory”, Adv. Appl. Prob., 4, 39–80, 1972).
B. Nehring, Construction of Classical and Quantum Gases. The method of cluster expansions, (Walter Warmuth Verlag, Nächst Neuendorf, 2013).
B. Nehring, H. Zessin, “The Papangelou Process. A concept for Fermion and Boson Processes”, Journal of ContemporaryMathematical Analysis, 46, 49–66, 2011).
B. Nehring, H. Zessin, “A representation of the moment measures of the general ideal Bose gas”, Math. Nachr., 285, 878–888, 2012.
J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer Verlag, Berlin, 1968).
M. Ozawa, “Conditional probability and a posteriori states in quantum mechanics”, Publ. RIM SKyotoUniv., 21, 279–295, 1985).
K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser, Basel, 1992).
M. Rafler, Gaussian Loop- and Pólya Processes. A point process approach (Universitätsverlag, Potsdam, 2009).
M. Rafler, “The Pólya sum process: Limit theorems for conditioned random fields”, J. Theoretical Probab., 26, 1097–1116, 2013).
M. Rafler, H. Zessin, “The logical postulates of Böge, Carnap and Johnson in the context of Papangelou processes”, J. Theoret. Probab., DOI 10.1007/s10959-014-0543-2, 2014.
I. E. Segal, “Mathematical characterization of the physical vacuum for a linear Bose-Einstein field”, Illinois J.Math., 6, 500–523, 1962).
H. Tamura, K.R. Ito, “A canonical ensemble approach to the Fermion/Boson random point processes and its applications”, Commun. Math. Phys., 263, 353–380, 2006).
H. Zessin, Classical Symmetric Point Processes (ICIMAF, La Habana, Cuba, 2010).
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Original Russian Text © A. Bach, H. Zessin, 2017, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2017, No. 1, pp. 3-25.
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Bach, A., Zessin, H. The particle structure of the quantum mechanical Bose and Fermi gas. J. Contemp. Mathemat. Anal. 52, 14–29 (2017). https://doi.org/10.3103/S1068362317010034
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DOI: https://doi.org/10.3103/S1068362317010034