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arXiv:quant-ph/9909015

q-deformed fermion oscillators, zero-point energy and inclusion-exclusion principle

P. Narayana Swamy

Published Sep 3, 1999• 1 citation
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Abstract

The theory of Fermion oscillators has two essential ingredients: zero-point energy and Pauli exclusion principle. We devlop the theory of the statistical mechanics of generalized q-deformed Fermion oscillator algebra with inclusion principle (i.e., without the exclusion principle), which corresponds to ordinary fermions with Pauli exclusion principle in the classical limit $q \to 1$. Some of the remarkable properties of this theory play a crucial role in the understanding of the q-deformed Fermions. We show that if we insist on the weak exclusion principle, then the theory has the expected low temperature limit as well as the correct classical q-limit.

DOI
10.48550/arxiv.quant-ph/9909015

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The theory of Fermion oscillators has two essential ingredients: zero-point energy and Pauli exclusion principle. We devlop the theory of the statistical mechanics of generalized q-deformed Fermion oscillator algebra with inclusion principle (i.e., without the exclusion principle), which corresponds to ordinary fermions with Pauli exclusion principle in the classical limit q→1q \to 1q→1. Some of the remarkable properties of this theory play a crucial role in the understanding of the q-deformed Fermions. We show that if we insist on the weak exclusion principle, then the theory has the expected low temperature limit as well as the correct classical q-limit.
Just as for the ordinary quantum harmonic oscillators, we expect the zero-point energy to play a crucial role in the correct high temperature behavior. We accordingly reformulate the theory of the statistical distribution function for the q-deformed boson oscillators and develop an approximate theory incorporating the zero-point energy. We are then able to demonstrate that for small deformations, ln q≪1, the theory reproduces the correct limits both for very high temperatures and for very low temperatures. The deformed theory thus reduces to the undeformed theory in these extreme cases.
A manifestation of the Pauli Exclusion Principle is observed when fermions are trapped in the ground state of a 2D harmonic oscillator trap at very low temperatures. This non-interaction of fermions results in the formation of Pauli crystals. This work introduces a statistical mechanical interpretation of the principles that led to the observation of Pauli crystals by calculating the energy, radius and other parameters of the non-interacting arrangement of ultra-cold fermions. This model approaches the problem from two different directions, namely fermion degeneracy and harmonic oscillator treatment. Unifying the two different approaches gives a more comprehensive and robust description of the various parameters of Pauli crystals.
All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.
We study the thermostatistics of q-deformed bosons and fermions obeying the symmetric (q<-->q(-1)) algebra and show that it can be built on the formalism of q calculus. The entire structure of thermodynamics is preserved if ordinary derivatives are replaced by an appropriate Jackson derivative. In this framework, we derive the most important thermodynamic functions describing the q-boson and q-fermion ideal gases in the thermodynamic limit. We also investigate the semiclassical limit and the low-temperature regime and demonstrate that the nature of the q deformation gives rise to pure quantum statistical effects stronger than undeformed boson and fermion particles.
qqq-Deformed Classical Lie Algebras and their Anyonic Realization
All classical Lie algebras can be realized \\`a la Schwinger in terms of fermionic oscillators. We show that the same can be done for their qqq-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two dimensional lattice. The deformation parameter qqq is a phase related to the anyonic statistical parameter. A crucial r\\^ole in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the underformed ones. The entire procedure works even on one dimensional chains; in such a case qqq can also be real.