Chapman & Hall/CRC eBooks

Wigner Transforms and Weyl Transforms

Man Wah Wong

Published Jun 28, 2022• 0 citations

Abstract

The Fourier–Wigner transform of two functions is first defined and is given an alternative expression. The Wigner transform is then defined to be the Fourier transform of the Fourier–Wigner transform. The Moyal identity, which is an analog of the Plancherel formula, is then proved for the Wigner transform. The Weyl transform with a given symbol is then defined in terms of the Wigner transform. Another formula for the Weyl transform with a given symbol is then proved and used to establish the fact that the product of two Weyl transforms is also a Weyl transform with symbol given by a twisted convolution of the two given symbols.

DOI: 10.1201/9781003206781-20

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Chapman & Hall/CRC eBooks

Wigner Transforms and Weyl Transforms

Man Wah Wong

Published Jun 28, 2022• 0 citations

Publisher

Abstract

DOI: 10.1201/9781003206781-20

Related papers

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We show that the cross Wigner function can be written in the form

W(\psi, \phi)= \hat S (\psi \otimes \overline{\hat\phi})

where

{\hat\phi}

is the Fourier transform of

\phi

and

\hat S

is a metaplectic operator that projects onto a linear symplectomorphism

S

consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl-Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schr\"odinger and the Bopp representations of quantum mechanics.