16,198 research outputs found

    Uncertainty Principles and Vector Quantization

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    Given a frame in C^n which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashin's representation whose coefficients all have the smallest possible dynamic range O(1/\sqrt{n}). The information tends to spread evenly among these coefficients. As a consequence, Kashin's representations have a great power for reduction of errors in their coefficients, including coefficient losses and distortions.Comment: Final version, to appear in IEEE Trans. Information Theory. Introduction updated, minor inaccuracies corrected

    The Physical Principles of Quantum Mechanics. A critical review

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    The standard presentation of the principles of quantum mechanics is critically reviewed both from the experimental/operational point and with respect to the request of mathematical consistency and logical economy. A simpler and more physically motivated formulation is discussed. The existence of non commuting observables, which characterizes quantum mechanics with respect to classical mechanics, is related to operationally testable complementarity relations, rather than to uncertainty relations. The drawbacks of Dirac argument for canonical quantization are avoided by a more geometrical approach.Comment: Bibliography and section 2.1 slightly improve

    Quantization of Contact Manifolds and Thermodynamics

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    The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this `quantization' describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.Comment: Additional references; typos fixed; a clarifying remark adde

    Exact uncertainty principle and quantization: implications for the gravitational field

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    The quantization of the gravitational field is discussed within the exact uncertainty approach. The method may be described as a Hamilton-Jacobi quantization of gravity. It differs from previous approaches that take the classical Hamilton-Jacobi equation as their starting point in that it incorporates some new elements, in particular the use of a formalism of ensembles in configuration space and the postulate of an exact uncertainty relation. These provide the fundamental elements needed for the transition from the classical theory to the quantum theory.Comment: 6 pages; submitted to the proceedings of DICE2004, to appear in the Brazilian Journal of Physic
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