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Diagonalization of the quadratic boson Hamiltonian with zero modes

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  • Colpa, J.H.P.

Abstract

The necessary mathematical equipment is developed for an analysis of the zero modes connected with hamiltonians quadratic in boson creation and annihilation operators. A “cross-matrix” being a matrix of a certain partitioned form, we consider the group G of conjuctivity transformations T, where T is a 2m-square cross-matrix subjected to the restriction T†ÎT = Î, Î : = diag(I, -I) (I is the m-square unit matrix). With respect to this group the concept of standard form is introduced for the set of positive-semidefinite hermitian cross-matrices D. It is proved that any such matrix D can be transformed into a matrix E of the standard form according to T†DT = E, T ⊆ G. An algorithm, suitable as a basis for computer programs, is given for finding standardized and standardizing matrices E and T for any given D.

Suggested Citation

  • Colpa, J.H.P., 1986. "Diagonalization of the quadratic boson Hamiltonian with zero modes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 134(2), pages 377-416.
    • Handle: RePEc:eee:phsmap:v:134:y:1986:i:2:p:377-416
    DOI: 10.1016/0378-4371(86)90056-7
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    References listed on IDEAS

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    1. Gaaff, A. & Hijmans, J., 1975. "Symmetry relations in the sixteen-vertex model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 80(2), pages 149-171.
    2. Gaaff, A. & Hijmans, J., 1978. "Interpretation of the symmetry group of the homogeneous 16-vertex model in terms of Lorentz similarity transformations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 94(2), pages 192-210.
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