Abstract
We introduce the quadratic Fermi algebra, which is a Lie algebra, and calculate the vacuum distributions of the associated Hamiltonians. In order to emphasize the difference with the Bose case, we apply a modification of the method used in the above calculation to obtain a simple and straightforward classification of the 1-dimensional Meixner laws in terms of homogeneous quadratic expressions in the Bose creation and annihilation operators. There is a huge literature of the Meixner laws but this, purely quantum probabilistic, derivation seems to be new. Finally we briefly discuss the possible multidimensional extensions of the above results.
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References
L. Accardi, I. Ya. Aref’eva and I. V. Volovich, Non isomorphism of the Bose and Fermi realization of sl(2,R). in preparation (2014).
L. Accardi, H.-H. Kuo and A. Stan, Moments and commutators of probability measures, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Quantum Probability Communications, 10 (2007) 591–612.
L. Accardi, A. Boukas and U. Franz, Renormalized powers of quantum white noise, IDA-QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), 9(1) (2006), 129–147, MR2214505, DOI: 10.1142/S0219025706002263 Preprint Volterra n. 597 (2006).
L. Accardi and Andreas Boukas, Higher Powers of q-Deformed White Noise, Methods of Functional Analysis and Topology, 12(3) (2006), 205–219.
Luigi Accardi and Andreas Boukas, White noise calculus and stochastic calculus, in: Stochastic Analysis: Classical and Quantum, T. Hida, K. Saito (eds.) World Scientific (2005), 260-300. Proceedings International Conference on Stochastic analysis: classical and quantum, Perspectives of white noise theory, Meijo University, Nagoya, 1-5 November 2004, Preprint Volterra n. 579 (2005).
L. Accardi, U. Franz and M. Skeide, Renormalized squares of white noise and other non-Gaussian noises as Levy processes on real Lie algebras, Comm. Math. Phys., 228 (2002), 123–150. Preprint Volterra, N. 423 (2000).
L. Accardi, Y. G. Lu and I. V. Volovich, White noise approach to classical and quantum stochastic calculi, Lecture Notes of the Volterra—CIRM International School with the same title, Trento, Italy, 1999, Volterra Preprint N. 375 July (1999).
L. Accardi, M. Schurmann and W. vonWaldenfels, Quantum independent increment processes on superalgebras, Math. ZeitSchr., 198(4) (1988), 451–477. Preprint, Heidelberg, Stochastische Mathematische Modelle, January (1987).
F. A. Berezin, The Method of Second Quantization, Pure Appl. Phys., 24, Academic Press, New York, 1966.
K. O. Friedrichs, Mathematical aspects of the quantum theory of fields, Commu Pure Appl. Math., 4 (1951), 161–224, 5(1-56) (1952), 349-411, 6(1953), 1-72, (Collectively reissued: Interscience, New York, 1953).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition Hardcover by Alan Jeffrey (Editor), Daniel Zwillinger (Editor), Academic Press (2007).
B. Grigelionis, Generalized z-distributions and related stochastic processes, Lithuanian Math. J., 41 (2001), 303–319.
B. Grigelionis, Processes of Meixner type, Lithuanian Math. J., 39 (1999), 33–41.
J. Meixner, Orthogonal Polynomsysteme mit einer besonderen gestalt der erzeugenden funktion, (Orthogonal polynomial systems with a generating function of a special form), J. London Math. Soc., 9 (1934), 6–13 (Zbl. 7, 307).
K. R. Parthasarathy and K. Schmidt, Positive definite kernels continuous tensor products and central limit theorems of probability theory, Springer Lecture Notes in Mathematics no. 272 (1972).
M. Schürmann, White noise on bialgebras, Springer LNM 1544 (1993).
P. Śniady, Quadratic bosonic and free white noises, Commun. Math. Phys., 211(3) (2000), 615–628, Preprint (1999).
Gabriela Popa and Aurel I. Stan, Two-dimensional Meixner random vectors and their semi-quantum operators, preprint November 2014.
J. Schwinger, U.S. Atomic Energy Commission Report NYO-3071, 1952 or D. Mattis, The Theory of Magnetism, Harper and Row, (1982).
R. Anishetty, M. Mathur and I. Raychowdhury, Irreducible SU(3) Schwinger Bosons, arXiv:0901.0644.–A Wolfram Web Resource. http://mathworld.wolfram.com/ GammaDistribution.html.
http://mathworld.wolfram.com/NegativeBinomialDistribution.html.
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Dedicated to Prof. Kalyan B. Sinha on occasion of his 70th birthday.
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Accardi, L., Aref’eva, I.Y. & Volovich, I.V. Fermionic Meixner classes, Lie algebras and quadratic Hamiltonians. Indian J Pure Appl Math 46, 517–538 (2015). https://doi.org/10.1007/s13226-015-0150-7
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DOI: https://doi.org/10.1007/s13226-015-0150-7