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"Gosson, Maurice de"
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Symplectic Radon Transform and the Metaplectic Representation
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a “demarginalization process” for the Wigner distribution.
Journal Article
The Symplectic Camel and Poincaré Superrecurrence: Open Problems
Poincaré’s Recurrence Theorem implies that any isolated Hamiltonian system evolving in a bounded Universe returns infinitely many times arbitrarily close to its initial phase space configuration. We discuss this and related recurrence properties from the point of view of recent advances in symplectic topology which have not yet reached the Physics community. These properties are closely related to Emergent Quantum Mechanics since they belong to a twilight zone between classical (Hamiltonian) mechanics and its quantization.
Journal Article
The Pauli Problem for Gaussian Quantum States: Geometric Interpretation
We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way.
Journal Article
Short-Time Propagators and the Born–Jordan Quantization Rule
We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.
Journal Article
On the Non-Uniqueness of Statistical Ensembles Defining a Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution
It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, i.e., that its integral is one and that the marginal properties are satisfied. However, this is generally not true. We introduced a class of quantum states for which this property is satisfied; these states are dubbed “Feichtinger states” because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980s by H. Feichtinger. The properties of these states were studied, giving us the opportunity to prove an extension to the general case of a result due to Jaynes on the non-uniqueness of the statistical ensemble, generating a density operator.
Journal Article
The principles of newtonian and quantum mechanics
This book deals with the foundations of classical physics from the \"symplectic\" point of view, and of quantum mechanics from the \"metaplectic\" point of view. The Bohmian interpretation of quantum mechanics is discussed. Phase space quantization is achieved using the \"principle of the symplectic camel\", which is a recently discovered deep topological property of Hamiltonian flows. The mathematical tools developed in this book are the theory of the metaplectic group, the Maslov index in a precise form, and the Leray index of a pair of Lagrangian planes. The concept of the \"metatron\" is introduced, in connection with the Bohmian theory of motion. A precise form of Feynman's integral is introduced in connection with the extended metaplectic representation.
eBook
A Pseudo-Quantum Triad: Schrödinger's Equation, the Uncertainty Principle, and the Heisenberg Group
We show that the paradigmatic quantum triad \"Schrödinger equation–Uncertainty principle–Heisenberg group\" emerges mathematically from classical mechanics. In the case of the Schrödinger 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.
Journal Article
Symplectic covariance properties for Shubin and Born–Jordan pseudo-differential operators
Among all classes of pseudo-differential operators only the Weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group. In this paper we show that there is, however, a weaker form of symplectic covariance for Shubin’s τ\\tau-dependent operators, in which the intertwiners are no longer metaplectic, but are still invertible non-unitary operators. We also study the case of Born–Jordan operators, which are obtained by averaging the τ\\tau-operators over the interval [0,1][0,1] (such operators have recently been studied by Boggiatto and his collaborators, and by Toft). We show that covariance still holds for these operators with respect to a subgroup of the metaplectic group.
Journal Article
Pointillisme à la Signac and Construction of a Quantum Fiber Bundle Over Convex Bodies
We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by Hilgevoord and Uffink.
Journal Article