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arXiv:0909.3580 · Chinese Physics B

Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and thes-ordered operator expansion formula of density operators

Fan Hong-Yi

Published May 1, 2010• 23 citations
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Abstract

By introducing the s -parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s -ordered product of operators (which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s -ordered operator expansion (denoted by ) formula of density operators is derived, which is . The s -parameterized quantization scheme is thus completely established.

DOI
10.1088/1674-1056/19/5/050303

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By introducing the s -parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s -ordered product of operators (which considers normally ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s -ordered operator expansion (denoted by ) formula of density operators is derived, which is . The s -parameterized quantization scheme is thus completely established.
By extending the usual Wigner operator to the s-parameterized one as with s being a real parameter, we propose a generalized Weyl quantization scheme which accompanies a new generalized s -parameterized ordering rule. This rule recovers P-Q ordering, Q-P ordering, and Weyl ordering of operators in s = 1, - 1, 0 respectively. Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering, antinormal ordering, and Weyl ordering. We also show that in this scheme the s -parameter plays the role of correlation between two quadratures Q and P . The formula that can rearrange a given operator into its new s -parameterized ordering is presented.
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.
Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys. A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator Δ ( q',p' ) (q-number transform) in phase space quantum mechanics, and its inverse where Q , P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q–P ordering, P–Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.
We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.