Mathematical and Physical Aspects of Stochastic Mechanics

Francesco Guerra

Michele Pavon

Simon Golin

G. Popov, V. Dzyuba

Abstract The discussion of stochasticity inherent in physical phenomena and processes occurring on our world is continued actively at present. For this purpose, various physical models and mathematical descriptions that are based most often on postulated are proposed. As a rule, stochasticity introduced in physical models phenomenologically, as well as its origin and mechanisms of emergence are not considered. However, the problem concerning the origin of stochasticity is far from being solved both in physical and mathematical aspects. In this article, the idea put forth by D.I. Blokhintsev and L.E. Ballentine concerning the emergence of stochasticity in classical and quantum field systems as a consequence of action of the surrounding stochastic background is developed. We are using the Lagrangian formalism and the categorically topological method of description for correct definition of the Lagrangian, action, the Euler–Lagrange equations, Noether currents, and generalized charges in stochastic surroundings. It is shown that Noether currents and conserved quantities become stochastic, and their probabilistic properties are determined by the background characteristics. The change in the action of the background on the observation interval determines the deviation of conserved quantities from their mean values, and the Heisenberg uncertainty relation can be treated as a manifestation of this stochasticity. As an illustration of the operability of a mathematical formulation of the stochasticity origin, we consider a classical scalar field. The proposed approach is naturally generalized to quantum fields. It can be used to obtain stochastic equations describing the behavior of material macroscopic systems in random fields and measurement processes.

Khavtgain Namsrai

Aubrey Truman, Ian M Davies