Fermionic Meixner classes, Lie algebras and quadratic Hamiltonians
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.
KeywordsMeixner probability distributions Lie algebra quantum Fermi and Bose hamiltonians
Unable to display preview. Download preview PDF.
- 1.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).Google Scholar
- 5.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).Google Scholar
- 7.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).Google Scholar
- 9.F. A. Berezin, The Method of Second Quantization, Pure Appl. Phys., 24, Academic Press, New York, 1966.Google Scholar
- 11.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).Google Scholar
- 18.Gabriela Popa and Aurel I. Stan, Two-dimensional Meixner random vectors and their semi-quantum operators, preprint November 2014.Google Scholar
- 19.J. Schwinger, U.S. Atomic Energy Commission Report NYO-3071, 1952 or D. Mattis, The Theory of Magnetism, Harper and Row, (1982).Google Scholar
- 20.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.Google Scholar
- 22.http://mathworld.wolfram.com/NegativeBinomialDistribution.html.Google Scholar