arXiv:2203.17210

A Few Almost Trivial Notes on the Symplectic Radon Transform and the Tomographic Picture of Quantum Mechanics

Maurice A. de Gosson

Published Mar 31, 2022• 0 citations

Abstract

We emphasize in these pedagogical notes the that the theory of the Radon transform and its applications is best understood using the theory of the metaplectic group and the quadratic Fourier transforms generating metaplectic operator.. Doing this we hope that these notes will be useful to a larger audience, including researchers in time-frequency analysis.

DOI: 10.48550/arxiv.2203.17210

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arXiv:2203.17210

A Few Almost Trivial Notes on the Symplectic Radon Transform and the Tomographic Picture of Quantum Mechanics

Maurice A. de Gosson

Published Mar 31, 2022• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2203.17210

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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.