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

Gilles Regniers
2009 Symmetry, Integrability and Geometry: Methods and Applications  
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.
more » ... nberg 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.
doi:10.3842/sigma.2009.106 fatcat:cnesxhkgnvh7vob66xpnp5gv5y