Statistical interpretation of the p-adic quantum field theory

Andrei Yu. Khrennikov, J. S. Wood

Branko G Dragovich

Andrew Khrennikov

The development of p-adic quantum mechanics has made it necessary to construct a probability theory in which the probabilities of events are p-adic numbers. The foundations of this theory are developed here. The frequency definition of probability is used. A general principle of statistical stabilization of relative frequencies is formulated. By virtue of this principle, statistical stabilization of relative frequencies, which are, like all experimental data, rational numbers, can be considered not only in the real topology but also in p-adic topologies.

Paolo Aniello

Abstract In the framework of quantum mechanics constructed over a quadratic extension of the field of p-adic numbers, we consider an algebraic definition of physical states. Next, the corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a p-adic counterpart of the POVMs associated with a quantum system over the complex numbers. Differently from the standard complex setting, the space of all states of a p-adic quantum system has an affine — rather than convex — structure. Thus, a symmetry transformation may be defined, in a natural way, as a map preserving this affine structure. We argue that the group of all symmetry transformations of a p-adic quantum system has a richer structure wrt the case of standard quantum mechanics over the complex numbers.

C. Alacoque, Philippe Ruelle, E. Thiran +2 more