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.

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.

Gaussian quantum states can be disentangled using symplectic rotations.

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a passive symplectic transformation, that is a metaplectic operator associated with a symplectic rotation. The main tools we use are the Werner-Wolf condition on covariance matrices and the symplectic covariance of Weyl quantization. Our result therefore complements a recent study by Lami, Serafini, and Adesso.

A Refinement of the Robertson-Schrödinger Uncertainty Principle and a Hirschman-Shannon Inequality for Wigner Distributions.

We propose a refinement of the Robertson-Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only.

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.

The Angular Momentum Dilemma and Born-Jordan Quantization.

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born-Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed " angular momentum dilemma" disappears if one uses Born-Jordan quantization. We argue that the latter is the only physically correct quantization procedure.

Hamiltonian deformations of Gabor frames: First steps.

Gabor frames can advantageously be redefined using the Heisenberg-Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies ( arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations.

Born-Jordan Quantization and the Equivalence of the Schrödinger and Heisenberg Pictures.

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using rule, and the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics.

Short-time quantum propagator and Bohmian trajectories.

We begin by giving correct expressions for the short-time action following the work Makri-Miller. We use these estimates to derive an accurate expression modulo [Formula: see text] for the quantum propagator and we show that the quantum potential is negligible modulo [Formula: see text] for a point source, thus justifying an unfortunately largely ignored observation of Holland made twenty years ago. We finally prove that this implies that the quantum motion is classical for very short times.

Quantum Blobs.

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.

Weak values of a quantum observable and the cross-Wigner distribution.

We study the weak values of a quantum observable from the point of view of the Wigner formalism. The main actor here is the cross-Wigner transform of two functions, which is in disguise the cross-ambiguity function familiar from radar theory and time-frequency analysis. It allows us to express weak values using a complex probability distribution.

A pseudo-differential calculus on non-standard symplectic space; Spectral and regularity results in modulation spaces.

The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on [Formula: see text]. In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on [Formula: see text] but rather on [Formula: see text]. These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of [Formula: see text] indexed by [Formula: see text].