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

Gilles Regniers ; Joris Van der Jeugt

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

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 $\mathfrak{gl}(1|n)$ and $\mathfrak{osp}(1|2n)$. 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.

Détail

Keywords: Wigner quantization; solvable Hamiltonians; Lie superalgebra representations.

Bibliographie

[1] Cramer M., Eisert J., “Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices”, New J. Phys., 8 (2006), 71.1–71.24 ; quant-ph/0509167 | DOI | MR

[2] Regniers G., Van der Jeugt J., “Analytically solvable Hamiltonians for quantum systems with a nearest-neighbour interaction”, J. Phys. A: Math. Theor., 42 (2009), 125301, 16 pp., ages ; arXiv:0902.2308 | DOI | MR | Zbl

[3] Wigner E. P., “Do the equations of motion determine the quantum mechanical commutation relations?”, Phys. Rev., 77 (1950), 711–712 | DOI | MR | Zbl

[4] Kamupingene A. H., Palev T. D., Tsavena S. P., “Wigner quantum systems. Two particles interacting via a harmonic potential. I. Two-dimensional space”, J. Math. Phys., 27 (1986), 2067–2075 | DOI | MR | Zbl

[5] Green H. S., “A generalized method of field quantization”, Phys. Rev., 90 (1953), 270–273 | DOI | MR | Zbl

[6] Palev T. D., “Wigner approach to quantization. Noncanonical quantization of two particles interacting via a harmonic potential”, J. Math. Phys., 23 (1982), 1778–1784 | DOI | MR | Zbl

[7] Palev T. D., Stoilova N. I., “Many-body Wigner quantum systems”, J. Math. Phys., 38 (1997), 2506–2523 | DOI | MR | Zbl

[8] Błasiak P., Horzela A., Kapuścik E., “Alternative hamiltonians and Wigner quantization”, J. Opt. B: Quantum Semiclass. Opt., 5 (2003), S245–S260 | DOI | MR

[9] Lievens S., Stoilova N. I., Van der Jeugt J., “Harmonic oscillators coupled by springs: discrete solutions as a Wigner quantum system”, J. Math. Phys., 47 (2006), 113504, 23 pp., ages ; hep-th/0606192 | DOI | MR | Zbl

[10] Lievens S., Stoilova N. I., Van der Jeugt J., “Harmonic oscillator chains as Wigner quantum systems: periodic and fixed wall boundary conditions in $\mathfrak{gl}(1|n)$ solutions”, J. Math. Phys., 49 (2008), 073502, 22 pp., ages ; arXiv:0709.0180 | DOI | MR | Zbl

[11] Lievens S., Van der Jeugt J., “Spectrum generating functions for non-canonical quantum oscillators”, J. Phys. A: Math. Theor., 41 (2008), 355204, 20 pp., ages | DOI | MR | Zbl

[12] Cohen-Tannoudji C., Diu B., Laloë F., Quantum mechanics, Vol. 1, Wiley, New York, 1977

[13] Brun T. A., Hartle J. B., “Classical dynamics of the quantum harmonic chain”, Phys. Rev. D, 60 (1999), 123503, 20 pp., ages ; quant-ph/9905079 | DOI | MR

[14] Audenaert K., Eisert J., Plenio M. B., Werner R. F., “Entanglement properties of the harmonic chain”, Phys. Rev. A, 66 (2002), 042327, 14 pp., ages ; quant-ph/0205025 | DOI

[15] Gould M. D., Zhang R. B., “Classification of all star irreps of $\mathfrak{gl}(m|n)$”, J. Math. Phys., 31 (1990), 2552–2559 | DOI | MR | Zbl

[16] Ganchev A. Ch., Palev T. D., “A Lie superalgebraic interpretation of the para-Bose statistics”, J. Math. Phys., 21 (1980), 797–799 | DOI | MR | Zbl

[17] Lievens S., Stoilova N. I., Van der Jeugt J., “The paraboson Fock space and unitary irreducible representations of the Lie superalgebra $\mathfrak{osp}(1|2n)$”, Comm. Math. Phys., 281 (2008), 805–826 ; arXiv:0706.4196 | DOI | MR | Zbl

[18] King R. C., Stoilova N. I., Van der Jeugt J., “Representations of the Lie superalgebra $\mathfrak{gl}(1|n)$ in a Gelfand–Zetlin basis and Wigner quantum oscillators”, J. Phys. A: Math. Gen., 39 (2006), 5763–5785 ; hep-th/0602169 | DOI | MR | Zbl

[19] Macdonald I. G., Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995 | MR | Zbl

[20] Wybourne B. G., Symmetry principles and atomic spectroscopy, Wiley, New York, 1970 | MR

[21] Palev T. D., $\mathfrak{sl}(3|N)$ Wigner quantum oscillators: examples of ferromagnetic-like oscillators with noncommutative, square-commutative geometry, hep-th/0601201