Quantum one-port networks and lobachevskian geometry.

R. Simon, Narasimhaiengar Mukunda

We bring out the connection between Berry's geometric phase and a gauge invariant expression introduced by Bargmann. We exploit this relationship to show that the G\"uoy effect, which referes to the phase jump experienced by a focused beam as it crosses the caustic, is the geometric phase associated with the underlying Lobachevskian hyperbolic geometry.

Monica Pate, Liang Fu

Andrei Yu. Khrennikov

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as the concrete example of the non-Kolmogorovian model of probability, similarly to the Lobachevskian model — the first example of the non-Euclidean model of geometry. This is the “probability model” interpretation of the violation of Bell’s inequality. We also criticize the standard interpretation—an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.

A. M. Perelomov

Antonín Hoskovec, Holger Frydrych, Igor Jex +1 more