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Journal of Physics A Mathematical and General

Permutation operators in Hilbert space gained via the IWOP technique-fermion case

Fan Hong-Yi

Published Aug 21, 1989• 8 citations
Publisher

Abstract

For the case of fermions, permutation operators in Hilbert space are derived in terms of the fermion coherent state and the technique of integration within ordered products with the integration variables being Grassmann numbers. These operators are shown to be quantum maps imaged by a certain permutation transformation of the numbers. Some new Fermi operator identities are also obtained with the use of the same technique.

DOI
10.1088/0305-4470/22/16/031

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For the case of fermions, permutation operators in Hilbert space are derived in terms of the fermion coherent state and the technique of integration within ordered products with the integration variables being Grassmann numbers. These operators are shown to be quantum maps imaged by a certain permutation transformation of the numbers. Some new Fermi operator identities are also obtained with the use of the same technique.
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.
The normally ordered Hilbert space image of the complex linear similarity transformation in Grassmann number phase space is derived in fermionic coherent state representation. Though keeping the anticommutator of Fermi operators invariant, the quantum mechanical images of classical transformations are generally not unitary. Remarkably, although kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, they still form a complete basis. The derivation of the normally ordered operator which engenders the similarity transformation is facilitated by virtue of the technique of integration within order products.