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. 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.Comment: Prepublication. Dedicated to Basil Hile

The "Symplectic Camel Principle" and Semiclassical Mechanics

Gromov's nonsqueezing theorem, aka the property of the symplectic camel, leads to a very simple semiclassical quantiuzation scheme by imposing that the only "physically admissible" semiclassical phase space states are those whose symplectic capacity (in a sense to be precised) is nh + (1/2)h where h is Planck's constant. We the construct semiclassical waveforms on Lagrangian submanifolds using the properties of the Leray-Maslov index, which allows us to define the argument of the square root of a de Rham form.Comment: no figures. to appear in J. Phys. Math A. (2002

Short-time quantum propagator and Bohmian trajectorie

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 Δt2 for the quantum propagator and we show that the quantum potential is negligible modulo Δt2 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

On the Wigner Distribution of the Reduced Density Matrix

Consider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this Letter we show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to the subsystem B. Our proof makes use of the Weyl--Wigner--Moyal phase space formalism. Our result is applied to general Gaussian mixed states of which i gives a particularly simple and precise description. We also briefly discuss purification from the Wigner point of view

Extended Weyl Calculus and Application to the Phase-Space Schr\"{o}dinger Equation

We show that the Schr\"{o}dinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians.Comment: To appear in J. Phys. A: Math. and genera

The Reconstruction Problem and Weak Quantum Values

Quantum Mechanical weak values are an interference effect measured by the cross-Wigner transform W({\phi},{\psi}) of the post-and preselected states, leading to a complex quasi-distribution {\rho}_{{\phi},{\psi}}(x,p) on phase space. We show that the knowledge of {\rho}_{{\phi},{\psi}}(z) and of one of the two functions {\phi},{\psi} unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.Comment: To appear in J.Phys.: Math. Theo

On the notion of phase in mechanics

The notion of phase plays an esential role in both classical and quantum mechanics.But what is a phase? We show that if we define the notion of phase in phase (!) space one can very easily and naturally recover the Heisenberg-Weyl formalism; this is achieved using the properties of the Poincare-Cartan invariant, and without making any quantum assumption