Physical equivalence based on the experimental uncertainties is shown to be described as the equivalence of eigenstates based on free ultra-filters on ∗H .The coarse-graining is realized by means of the equivalent treatment with respect to the equivalent eigenstates.Two different types of decoherence between quantum states are simultaneously derived in the coarse-graining process.One represents the decoherence for realizing microcanonical ensemble of statistical mechanics and the other is that required in quantum theory of measurements.
An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr¨odinger equation.
Abstract: An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schrödinger equation.