The Generalized “Position–Momentum” Uncertainty Relation in Quantum Mechanics and in the Theory of Brownian Motion

We show that the generalized Schrödinger uncertainty relations have the meaning of fundamental restrictions on the characteristics of the state space in any theory of a probabilistic type. Both quantum mechanics and the theory of Brownian motion for arbitrary time intervals are among these theories. We compare the “position-momentum” uncertainty relation in the theory of Brownian motion and a similar uncertainty relation for a microparticle in the Gaussian wave-packet state. We establish that the two theories are conceptually similar despite a serious distinction between their mathematical apparatus. This similarity manifests itself in alternative regimes such that small times in one theory correspond to large times in the other theory, and vice versa. In each of the theories, an uncontrollable effect of either quantum or thermal type is of crucial importance.