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Metaplectic formulation of the Wigner transform and applications
- Source :
- Rev. Math. Phys. 25, No. 10 (2013) 1343010 (19 pages)
- Publication Year :
- 2014
-
Abstract
- 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.<br />Comment: 18 pages, latex file
- Subjects :
- Mathematical Physics
Mathematics - Functional Analysis
Quantum Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- Rev. Math. Phys. 25, No. 10 (2013) 1343010 (19 pages)
- Publication Type :
- Report
- Accession number :
- edsarx.1401.3388
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1142/S0129055X13430101