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arXiv:quant-ph/0407129

On the goodness of "quantum blobs" in phase space quantization

Maurice A. de Gosson

Published Jul 16, 2004• 2 citations
arXivHTMLPDF

Abstract

We replace the usual notion of quantum cell from statistical mechanics by that of "quantum blob". A quantum blob is the transform, by a linaer symplectic transformation, of a phase space ball with radius equal to the square root of h-bar. The intersection of a quantum blob with any symplectic plane is an ellipse with area one half of h. This very special property, which is closed related to the principle of the symplectic camel, leads to a symplectic invariant statement of the uncertainty principle. We moreover prove that the average of a phase space Gaussian over a quantum blob is the Wigner transform of a minimum uncertainty Gaussian.

DOI
10.48550/arxiv.quant-ph/0407129

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We replace the usual notion of quantum cell from statistical mechanics by that of "quantum blob". A quantum blob is the transform, by a linaer symplectic transformation, of a phase space ball with radius equal to the square root of h-bar. The intersection of a quantum blob with any symplectic plane is an ellipse with area one half of h. This very special property, which is closed related to the principle of the symplectic camel, leads to a symplectic invariant statement of the uncertainty principle. We moreover prove that the average of a phase space Gaussian over a quantum blob is the Wigner transform of a minimum uncertainty Gaussian.
Abstract In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in con…guration space. To each of these states, we associate a class of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.
In earlier work, we introduced quantum blobs as minimum-uncertainty symplectic ellipsoids in phase space. These objects may be viewed as geometric monads in the Leibnizian sense, representing the elementary units of phase-space structure consistent with the uncertainty principle. We establish a one-to-one correspondence between such monads and generalized coherent states, represented by arbitrary non-degenerate Gaussian wave functions in configuration space. To each of these states, we associate a classs of Toeplitz operators that extends the standard anti-Wick quantization scheme. The mathematical and physical properties of these operators are analyzed, allowing for a generalized definition of density matrices within the phase-space formulation of quantum mechanics.
Quantum blobs are the smallest units of phase space that are compatible with the Robertson-Schrödinger indeterminacy relation and invariant under general symplectic transformations [1]. In this article, the distinguishability between pairs of quantum blobs, as measured by quantum fidelity, is defined on complex phase space. Fidelity is physically interpreted as the probability that the pair are mistaken for each other upon a measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of quantum blobs. The symplectic capacity for a given quantum blob, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalizes the Liouville theorem in classical mechanics, which states that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schrödinger equation is the mathematical representation for the conservation of fidelity.
We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson–Schrödinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Gaussian states in terms of partial information on the covariance ellipsoid, which allows us to formulate statements related to symplectic tomography.
In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schrödinger). In the present work we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of (small) ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to guide and control these motions as we like. In the classical case these ellipsoids correspond to a symplectic coarse graining of phase space, and in the quantum case they correspond to the "quantum blobs" we defined in previous work, and which can be viewed as minimum uncertainty phase space cells which are in a one-to-one correspondence with Gaussian pure states.
Quantum blobs are the smallest phase space units of phase space compatible with the uncertainty principle of quantum mechanics and having the symplectic group as group of symmetries. Quantum blobs are in a bijective correspondence with the squeezed coherent states from standard quantum mechanics, of which they are a phase space picture. This allows us to propose a substitute for phase space in quantum mechanics. We study the relationship between quantum blobs with a certain class of level sets defined by Fermi for the purpose of representing geometrically quantum states.