Dynamical Equivalence, Commutation Relations and Noncommutative Geometry

И. В. Мингалев, О. В. Мингалев, Victor Valentinovich Vedenyapin

A. Horzela

The relation between canonical commutation rules and a shape of acting force is analysed. It is shown that canonical commutation relations and Newtonian equations of motion for a single particle dynamics do imply the force to be of the Lorentz type but the inverse is not true. The example of a single particle motion in a constant magnetic field shows that equations of motion allow an alternative to canonical commutation relations to exist and it is a Lie algebra. The algebra found is of the type of algebras found in studies leading towards noncommutative geometry approaches to physical problems.

Francesco D’Andrea

Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics.In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations.We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h-and q-derivatives, respectively.In these various examples, some points turn out to be at infinite distance.We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance.On the circle, this provides an explicit example of computation of the Wasserstein distance.

Mauro Spera

Mihail Vladimirovich Karasev

Quantum geometry, algebras with nonlinear commutation relations, representation theory, the coherent transform, and operator averaging are used to solve the perturbation problem for wave (quantum) systems near a resonance equilibrium point or a resonance equilibrium ray in two and three-dimensional spaces.

Gianfausto Dell’Antonio, Sergio Doplicher, David Pierre Ruelle