Generalized Positive-Definite Operator in Quantum Phase Space Obtained by Virtue of the Weyl Quantization Rule

Liyun Hu, Fan Hong‐Yi

We introduce a generalized positive-definite operator Δ g (q,p) by smoothing out the Wigner operator Δ w (q,p) and by averaging over the “coarse graining" function. The function is then regarded as the classical Weyl correspondence of the operator Δ g (q,p); in this way we can easily identify a quantum state |Φ〉 such that Δ g (q,p) = |Φ〉 〈Φ|, and |Φ〉 turns out to be a new kind of squeezed coherent state. Correspondingly, the generalized distribution function for any state |Ψ〉 is 〈Ψ|Δ g (q,p)|Ψ〉 = |〈Φ|Ψ〉| 2 , which is obviously positive-definite and is a generalization of the Husimi function.

David Böhm, Basil J. Hiley

André Martin, Shasanka Mohan Roy

Yu. N. Orlov, V. Zh. Sakbaev

Abstract The paper uses an approach to the formal description of linear quantization as a statistical mixture of operators of the so-called tau quantization. This allows us to consider arbitrary Hermitian linear quantization as a linear integral operator acting on functions from the space of canonical variables and translating them into matrix elements of the quantum operator. In such a scheme, various quantization rules are uniformly studied: Weyl, Born, Jordan, and a number of others. The main result relates to the proof of the uniqueness of Weyl quantization as an operator, which is answered by the inverse operator generating a positive definite probability measure. For any other quantization, the inverse operator (dequantization) generates a measure that is not positive definite.

Ashok Das, L. Greenwood

Fan Hong‐Yi

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions' Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.