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Journal of Mathematical Physics

Normally ordered Fermi operator realization of the SUn group

Hong-Yi Fan

Published Feb 1, 1990• 12 citations
Publisher

Abstract

The mapping of the SUn group transformation in Grassmann number space into unitary normally ordered and antinormally ordered Fermi operator realizations in Hilbert space are investigated. The unitary Fermi operators are evaluated in fermion coherent state representation using the ‘‘integration within ordered product’’ technique for a fermionic system. Some new Fermi operator identities are thus obtained.

DOI
10.1063/1.528908

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Modern Physics Letters B•Nov 20, 1998•0 citations
The mapping of the SUn group transformation in Grassmann number space into unitary normally ordered and antinormally ordered Fermi operator realizations in Hilbert space are investigated. The unitary Fermi operators are evaluated in fermion coherent state representation using the ‘‘integration within ordered product’’ technique for a fermionic system. Some new Fermi operator identities are thus obtained.
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.
We introduce the generalized fermionic Wigner operator with an s parameter. Based on its remarkable properties, we establish one-to-one mapping between fermion operators and their s-parametrized pseudo-classical correspondence, which may involve fermionic Weyl pseudo-classical correspondence, P-representation and Q-representation in a unified way. Furthermore, starting with the projector of the fermionic coherent state, we obtain the s-ordered operator expansion formula of fermionic density operators, which includes normally ordered, antinormally ordered and Weyl ordered product of operators for different values of s. Applications in calculating some Fermi operators' s-ordered expansions are presented.