Abstract
We define the notion of ‘generalized statistics’ and give some examples. In particular, we consider the relationsa i a * j -q ij a * j a i =δ ij for - 1 ⩽q ij =q ji ⩽ + 1 and we prove the existence of a Fock space representation of these relations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araki, H., On the connection of spin and commutation relations between different fields,J. Math. Phys. 2, 267–270 (1961).
Bożejko, M. and Speicher, R., An example of a generalized Brownian motion,Comm. Math. Phys. 137, 519–531 (1991).
Bożejko, M. and Speicher, R., An example of a generalized Brownian motion II, in L. Accardi (ed),Quantum Probability and Related Topics VII, World Scientific, Singapore, 1992, pp. 67–77.
Cushen, C. D. and Hudson, R. L., A quantum-mechanical central limit theorem,J. Appl. Probab. 8, 454–469 (1971).
Fivel, D., Interpolation between Fermi and Bose statistics using generalized commutators,Phys. Rev. Lett. 65, 3361–3364 (1990).
Giri, N. and von Waldenfels, W., An algebraic version of the central limit theorem.Z. Wahrscheinlichkeitstheorie verw. 42, 129–134 (1978).
Goderis, D., Verbeure, A., and Vets, P., Theory of fluctuations and small oscillations for quantum lattice systems,J. Math. Phys. 29, 2581–2585 (1988).
Green, H. S., A generalized method of field quantization.Phys. Rev. 90, 270–273 (1953).
Greenberg, O. W., Particles with small violations of Fermi or Bose statistics,Phys. Rev. D 43, 4111–4120 (1991).
Greenberg, O. W. and Messiah, A. M. L., High-order limit of para-Bose and para-Fermi fields,J. Math. Phys. 6, 500–504 (1965).
Kreweras, G., Sur les partitions non croisees d'un cycle,Discrete Math. 1, 333–350 (1972).
Speicher, R., A new example of ‘independence’ and ‘white noise’.Probab. Theory Related Fields 84, 141–159 (1990).
Speicher, R., A non-commutative central limit theorem,Math. Z. 209, 55–66 (1992).
Speicher, R., The lattice of admissible partitions, SFB-Preprint 654, Heidelberg, 1991.
Stanciu, S., The energy operator for infinite statistics,Comm. Math. Phys. 147, 211–216 (1992).
Streater, R. F. and Wightman, A. S.,PCT, Spin and Statistics, and all that, Benjamin, New York, 1964.
von Waldenfels, W., An algebraic central limit theorem in the anti-commuting case,Z. Wahrscheinlichkeitstheorie verw. 42, 135–140 (1978).
Zagier, D., Realizability of a model in infinite statistics,Comm. Math. Phys. 147, 199–210 (1992).
Author information
Authors and Affiliations
Additional information
This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123).