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Modern Physics Letters A

INVARIANCE OF WEYL ORDERING OF FERMI OPERATORS UNDER SIMILAR TRANSFORMATIONS

Hong-Yi Fan

Published Mar 28, 2006• 10 citations
Publisher

Abstract

A Weyl–Wigner quantization scheme for Fermi system which is parallel to the bosonic case is established. We prove a new theorem: Weyl ordering of Fermi operators is invariant under fermionic similar transformations. Comparing with the preceding paper,1 we see the Bose–Fermi supersymmetry in this aspect. As an application of the theorem, we construct a generalized fermionic squeezed state.

DOI
10.1142/s0217732306019025

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A Weyl–Wigner quantization scheme for Fermi system which is parallel to the bosonic case is established. We prove a new theorem: Weyl ordering of Fermi operators is invariant under fermionic similar transformations. Comparing with the preceding paper,1 we see the Bose–Fermi supersymmetry in this aspect. As an application of the theorem, we construct a generalized fermionic squeezed state.
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.
By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.
By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ω k ( p , q ) with a real k parameter and can unify the P—Q , Q—P , and Weyl ordering of operators in k = 1, − 1, 0, respectively, we find the mutual transformations between δ ( p — P ) δ ( q — Q ), δ ( q — Q ) δ ( p — P ), and Ω k ( p, q ), which are, respectively, the integration kernels of the P—Q , Q—P , and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The — and — ordered forms of Ω k ( p , q ) are also derived, which helps us to put the operators into their — and — ordering, respectively.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator ?k(p,q) with a real k parameter and can unify the P–Q, Q–P, and Weyl ordering of operators in k = 1,-1, 0, respectively, we find the mutual transformations between δ(p- P)δ(q- Q), δ(q- Q)δ(p- P), and Ωk(p,q), which are, respectively, the integration kernels of the P–Q, Q–P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The P- and Q- ordered forms of Ωk(p,q) are also derived,which helps us to put the operators into their P- and Q- ordering, respectively.