Quantum Mechanics on Phase Space

Schroeck, Franklin E.

doi:10.1007/978-94-017-2830-0

Cited by 136 publications

(149 citation statements)

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“…Let, say, X be a tight frame for H , let Y be a subsystem of X and M(Y ) = x∈Y ·, x x. Then the mapping M, defined on the family of all subsystems of X , is a simple example of a positive operator valued measure, a notion playing a major role in quantum information theory, describing generalized measurements [104]. In general, M(Y ) is not an orthogonal projection.…”

Section: Proposition 23mentioning

confidence: 99%

Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis J Fourier Anal Appl (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

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Section: Proposition 23mentioning

confidence: 99%

Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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“…Such applications have been well established for the continuum theory of phase space; see, for instance, [11,19,25,29,31,34]. A satisfactory discrete version of the theory would also be expected to have such applications.…”

Section: Introductionmentioning

confidence: 99%

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Barker

Candan

Hakioğlu

et al. 2000

J. Phys. A: Math. Gen.

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Abstract. Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

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“…With coworkers he has proposed a wide class of alternative wave equations which overcome these shortcomings [36,[40][41][42]. For the spinless non-relativistic case the phase-space wave function is written as [36] If for each p held fixed the functions ( , ; ) ψ t q p satisfy the regularity conditions of Theorem 1 then each these currents will all be non-radiating.…”

Section: Proofmentioning

confidence: 99%

Quantum wave equations and non-radiating electromagnetic sources

Davidson¹

2007

Annals of Physics

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A connection between classical non-radiating sources and free-particle wave equations in quantum mechanics is rigorously made. It is proven that free-particle wave equations for all spins have currents which can be defined which are non-radiating electromagnetic sources. It is also proven that the advanced and retarded fields are exactly equal for these sources. Implications of these results are discussed.

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Quantum Mechanics on Phase Space

Cited by 136 publications

References 0 publications

Duality for Frames

Duality for Frames

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Quantum wave equations and non-radiating electromagnetic sources

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