Quantum Mechanics is a Topic of the Theory of Real Pure-Jump Non-Markovian Stochastic Processes

摘要

From E. Madelung's work issue in 1926, it became clear that there is a one-to-one correspondence between the pair of adjoint Schrodinger equations and the pair of equations of the "hydrodynamic representation" for the probability density w(x,t) = |ψ|/(x,t)|~2 and mean momentum p(x,t). Both these equations can be deduced from the Quantum Transport Equation (QTE) for a probability density P(p,x,t), as two equations for the two first moments w(x,t) = fP(p,x,t)dp and > = (l/w)∫p(p,x,t)dp. In turn, the QTE can be deduced from the Quantum Kinetic Equation (QKE), that is from a non-Markovian stochastic Kolmogorov-Gikhman-Skorokhod equation for a real pure-jump process. At last, the QKE is a consequence of a non-Markovian analogue of the Kolmogorov-Chapman equation for a real transition probability F(q,y,s; p,x,t). Similarly, the Klein-Fock-Gordon equation follows from the relativistic QTE (RQTE). In turn, the RQTE can be deduced from the relativistic QKE, as well as from a non-Markovian analogue of the relativistic generalization of the functional Kolrnogorov-Chapman equation. Thus all quantum mechanics as a mathematical theory is a topic of theory of real pure-jump non-Markovian stochastic processes.