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International series of numerical mathematics

Radar Ambiguity Functions and the Linear Schrödinger Representation

Walter Schempp

Published Jan 1, 1984• 5 citations
Publisher

Abstract

In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. Quantum mechanics stands here for the quantum-mechanical description, at a given instant of time, of a non-relativistic particle. These uncertainty principles suggest that there should exist a common mathematical structure behind both quantum mechanics and the theory of signals. It is the aim of the present article to establish that the concept of real Heisenberg nilpotent group Ã(ℝ) lies at the foundations of both of these fields. The crucial point is to endow the time-frequency plane ℝ ⊕ ℝ with the structure of a two dimensional real symplectic vector space so that it gets symplectomorphic to the tangent plane to the “Schrodinger coadjoint orbit” at the point 1 in the Kirillov orbit picture for the unitary dual of Ã(ℝ). Various applications of this geometric relationship between signal theory and nil potent harmonic analysis are pointed out. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-5432-0_41

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We show that the paradigmatic quantum triad equation–Uncertainty principle–Heisenberg emerges mathematically from classical mechanics. In the case of the Schrodinger equation, this is done by extending the metaplectic representation of linear Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices to use tools from multivariate statistics together with the theory of John's minimum volume ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already exists in classical mechanics.
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
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In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase-space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.
the Heisenberg uncertainty Principle is a fundamental principle in quantum mechanics, which was developed by the German physicist Werner Heisenberg and was proposed by him in 1927. This principle states that for a pair of physical quantities that share phase space, such as position and momentum, it is impossible to accurately measure their values at the same time. There are several variants of it in harmonic analysis studies, and the article will introduce some of them in R^1 space and L^2 space. In the process of providing the Heisenberg inequality, the article proved the Plancherel identity and Schwartz inequality by using Fourier transform and inverse Fourier transform. Finally, author solved the equation of the wave function φ(x) . The famous physicists Heisenberg proposed one of the more novel ideas in quantum mechanics – the existence of unobservable orbits cannot be assumed, which did bring great influence in quantum mechanics. The article will introduce the conception of Heisenberg inequality and try to finish the proof.