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Progress of Theoretical Physics

Boson Bogolyubov Transformation and theSU(1, 1) Group

Haruo Ui

Published Sep 1, 1970• 9 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.44.703

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Progress of Theoretical Physics•Dec 1, 1998•0 citations
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.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.
The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.
A method for describing a boson system interacting with an external harmonic oscillator, which was proposed recently by the present authors, is extended to the case of a statistically mixed state. The basic idea comes from the su(2, 1)-boson model, the general form of which was recently investigated by the present authors. In terms of the interweaving of the phase space doubling and the su(1, 1)-spin-like behavior in the su(2)-spin system, the extension is performed in the framework of a canonical form of classical mechanics. The results obtained with regard to the present idea are interpreted from various viewpoints, for example, in relation to linear response theory of non-equilibrium statistical mechanics.