16,198 research outputs found
Uncertainty Principles and Vector Quantization
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
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
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
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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