Wigner Functions, Fresnel Optics, and Symplectic Connections on Phase Space
نویسنده
چکیده
We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum–mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.
منابع مشابه
Fresnel tomography: a novel approach to the wave function reconstruction based on Fresnel representation of tomograms
New type of tomographic probability distribution which contains complete information on the density matrix (wave function) related to the Fresnel transform of the complex wave function is introduced. Relation to symplectic tomographic probability distribution is elucidated. Multimode generalization of the Fresnel tomography is presented. Examples of applications of the present approach are give...
متن کاملFractional Talbot effect in phase space: A compact summation formula.
A phase space description of the fractional Talbot effect, occurring in a one--dimensional Fresnel diffraction from a periodic grating, is presented. Using the phase space formalism a compact summation formula for the Wigner function at rational multiples of the Talbot distance is derived. The summation formula shows that the fractional Talbot image in the phase space is generated by a finite s...
متن کاملGeometry and dynamics in the Fresnel transforms of discrete systems.
Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally she...
متن کاملFe b 20 06 Abelian connection in Fedosov deformation quantization . I . The 2 – dimensional phase space
General properties of Abelian connection in Fedosov deformation quantization are investigated. Criteria of being a finite series for an Abelian connection are presented. Conditions for different symplectic connections to generate the same correction to the Abelian connection are given. A proof that in 2–dimensional case the Abelian connection is an ifinite series is done. A class of symplectic ...
متن کاملFocal shift, optical transfer function, and phase-space representations.
The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner di...
متن کامل