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Journal of Physics A Mathematical and General

Generalized Bogolyubov transformation

Fan Hong-Yi, Jack Vanderlinde

Published Nov 7, 1990• 10 citations
Publisher

Abstract

The Bogolyubov transformation (1958) frequently used in dealing with pairing interactions of fermions, is extended to allow for binary coupling between any (even) number of fermions by using a matrix transformation 'coefficient'. The corresponding quasiparticles and quasi-vacuum state are derived. The normally ordered expansion of the unitary generalized Bogolyubov operator is obtained by application of the technique of integration within an ordered product (IWOP) for the fermion system.

DOI
10.1088/0305-4470/23/21/008

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The Bogolyubov transformation (1958) frequently used in dealing with pairing interactions of fermions, is extended to allow for binary coupling between any (even) number of fermions by using a matrix transformation 'coefficient'. The corresponding quasiparticles and quasi-vacuum state are derived. The normally ordered expansion of the unitary generalized Bogolyubov operator is obtained by application of the technique of integration within an ordered product (IWOP) for the fermion system.
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.
By the properties of the coherent state and the technique of performing integrations within ordered products as in Ref. [1], we derive some new useful operator ordering identities. Their applications, especially in studying the Boglyubov transformations both in one and two dimensional cases, are presented. The explicit non-unitary operators which turn the coordinate or momentum eigen-state to the corresponding coherent state are also given. Moreover, by using the Schwinger angular momentum theory and the technique mentioned above we find the normal product form of the rotation operator, with which the coefficient of the rotation D-matrix can be easily obtained.