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Entangled Fourier Transformation and its Application in Weyl-Wigner Operator Ordering and Fractional Squeezing

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Abstract

In order to entangle the functions to be transformed, we proposed the entangled. Fourier integration transformation (EFIT) which has the property of keeping modulus-invariant for its inverse transformation. Then we then studied Wigner operator’s EFIT and found that a function’s EFIT is just related to its Weyl-corresponding operator’s matrix element, in so doing we also derived new operator re-ordering formulas \( \delta \left(x-P\right)\left(y-Q\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{-2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \);\( \delta \left(y-Q\right)\left(x-P\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \), where P, Q are momentum and coordinate operator respectively, the symbol \( {\displaystyle \begin{array}{c}:\\ {}:\end{array}}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \) denotes Weyl ordering. By virtue of EFIT we also found the operator which can generate fractional squeezing transformation.

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Acknowledgments

This work is supported by the Natural Science Foundation of the Anhui Higher Education Institutions of China [Grant No. KJ2014A236]; National Natural Science Foundation of China [Grant No. 11775208].

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Authors and Affiliations

  1. Key Laboratory of Atmospheric Optics, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei, 230031, China

    Ke Zhang & Chengyu Fan

  2. Science Island Branch of Graduate School, University of Science and Technology of China, Hefei, 230031, China

    Ke Zhang & Hongyi Fan

  3. Huainan Normal University, Huainan, 232038, China

    Ke Zhang

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  1. Ke Zhang
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Correspondence to Ke Zhang.

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Zhang, K., Fan, C. & Fan, H. Entangled Fourier Transformation and its Application in Weyl-Wigner Operator Ordering and Fractional Squeezing. Int J Theor Phys 58, 1687–1697 (2019). https://doi.org/10.1007/s10773-019-04066-y

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