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Journal of Mathematical Physics

Transformation brackets for generalized Bogolyubov–boson transformations

Yoel Tikochinsky

Published Jan 1, 1978• 13 citations
Publisher

Abstract

We consider the transformation bi=Σsj=1 (λijaj+μijaj) between two sets of boson operators, with λij and μij complex. Closed formulas are derived for the transformation brackets connecting base states of the two sets. As an application, the general quadratic Hamiltonian in one dimension is diagonalized and two examples, involving time-dependent real transformation and time-independent complex transformation, are worked out.

DOI
10.1063/1.523548

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We consider the transformation bi=Σsj=1 (λijaj+μijaj) between two sets of boson operators, with λij and μij complex. Closed formulas are derived for the transformation brackets connecting base states of the two sets. As an application, the general quadratic Hamiltonian in one dimension is diagonalized and two examples, involving time-dependent real transformation and time-independent complex transformation, are worked out.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
The homogeneous canonical transformation of boson operators is studied from the standpoint of the SU(1, 1) quasi-spin formalism. It is shown that this transformation can be regarded as a rotation in the SU(1, 1) quasi-spin space. This situation is just analogous to the SU(2) quasi-spin formalism of many-fermion problem, in which fermion Bogolyubov transformation can be regarded as a rotation in the SU(2) quasi-spin space. The transformation law of state vectors under this boson Bogolyubov transformation is then written down explicitly in terms of the SU(1, 1) d-functions obtained in the preceding paper. For clarity, we treat the simplest nontrivial example of one-dimensional harmonic oscillator or, equivalently, a system consisting of one kind of bosons. Generalization to many-boson problem and its possible applications to Bose fluid and nuclear structure are briefly discussed.