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arXiv:hep-th/9401168 · Journal of Physics A Mathematical and General

Remarks on the number operator for generalized quons

Miroslav Dorešić

Published Jan 7, 1995• 1 citation
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Abstract

We construct the number operator for particles obeying infinite statistics, defined by a generalized q-deformation of the Heisenberg algebra, and prove the positivity of the norm of linearly independent state vectors.
DOI
10.1088/0305-4470/28/1/021

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The momentum operator representation of nonrelativistic anyons is developed in the Chern-Simons formulation of fractional statistics. The connection between anyons and the qqq-deformed bosonic algebra is established.
The necessary and sufficient condition of the existence of the Fock-like space for quons with a fixed number q is proven, and the uniqueness theorem of the Fock-like space is given. The general-q operators, which satisfy the q-mutation relation a_ja_k - qa_ka_j =δ_jk, areconstructed by using the q = 0 operators as standard building blocks. The Fock-like spaces for quons with q∈(-1,1) prove to be the same as the one with q = 0, which manifestly is the direct sum of all the tensor product powers of the single particle state subspace, hence allow only the Boltzmann statistics .
After a brief mention of Bose and Fermi oscillators and of particles which obey other types of statistics, including intermediate statistics, parastatistics, paronic statistics, anyon statistics and infinite statistics, I discuss the statistics of ``quons'' (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases. To accomplish this for parabose, parafermi and quon operators, I introduce parabose, parafermi and quon Grassmann numbers, respectively. I also discuss interactions of non-relativistic quons, quantization of quon fields with antiparticles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields. I conclude with the bound on the parameter qqq for electrons due to the Ramberg-Snow experiment.
I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or qqq-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the TCP\mathrm{TCP}TCP theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.