Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 106, 17 pages      arXiv:0909.3697      https://doi.org/10.3842/SIGMA.2009.106

Wigner Quantization of Hamiltonians Describing Harmonic Oscillators Coupled by a General Interaction Matrix

Gilles Regniers and Joris Van der Jeugt
Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium

Received September 22, 2009, in final form November 20, 2009; Published online November 24, 2009

Abstract
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix. The quantization of a Hamiltonian describing such a system has been done in the canonical case. In this paper, we take a more general approach and look at the system as a Wigner quantum system. Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations. Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n). We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail. We also make the connection with the well-known canonical case.

Key words: Wigner quantization; solvable Hamiltonians; Lie superalgebra representations.

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