96 research outputs found
Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds
Let (S(t)) be a one-parameter family S = (S(t)) of positive integral
operators on a locally compact space L. For a possibly non-uniform partition of
[0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the
transition between cosecutive partition times of distance dt, and b) a suitable
continuous interpolation scheme (e.g. Brownian bridges or geodesics). If
necessary normalize to get a probability measure. We prove a version of
Chernoff's theorem of semigroup theory and tighness results which together
yield convergence in law of such measures as the partition gets finer. In
particular let L be a closed smooth submanifold of a Riemannian manifold M. We
prove convergence of Brownian motion on M, conditioned to visit L at all
partition times, to a process on L whose law has a Radon-Nikodym density with
repect to Brownian motion on L which contains scalar, mean and sectional
curvature terms. Various approximation schemes for Brownian motion are also
given. These results substantially extend earlier work by the authors and by
Andersson and Driver.Comment: 35 pages, revised version for publication, more detailed expositio
Noether theorems and quantum anomalies
In this communication, we show that both infinite-dimensional versions of
Noether's theorems, and the explanation of quantum anomalies can be obtained
using similar formulas for the derivatives of functions whose values are
measures (Smolyanov and von Weizsaecker, 1995) or pseudomeasures (Gough, Ratiu
and Smolyanov, 2015). In particular, we improve son these results.Comment: 8 pages, no figure
Feynman, Wigner, and Hamiltonian Structures Describing the Dynamics of Open Quantum Systems
This paper discusses several methods for describing the dynamics of open
quantum systems, where the environment of the open system is
infinite-dimensional. These are purifications, phase space forms, master
equation and liouville equation forms. The main contribution is in using
Feynman-Kac formalisms to describe the infinite-demsional components
Wigner Measures and Coherent Quantum Control
We introduce Wigner measures for infinite-dimensional open quantum systems; important examples of such systems are encountered in quantum control theory. In addition, we propose an axiomatic definition of coherent quantum feedback
The Bell Theorem as a Special Case of a Theorem of Bass
The theorem of Bell states that certain results of quantum mechanics violate
inequalities that are valid for objective local random variables. We show that
the inequalities of Bell are special cases of theorems found ten years earlier
by Bass and stated in full generality by Vorob'ev. This fact implies precise
necessary and sufficient mathematical conditions for the validity of the Bell
inequalities. We show that these precise conditions differ significantly from
the definition of objective local variable spaces and as an application that
the Bell inequalities may be violated even for objective local random
variables.Comment: 15 pages, 2 figure
Generalized probabilities taking values in non-Archimedean fields and topological groups
We develop an analogue of probability theory for probabilities taking values
in topological groups. We generalize Kolmogorov's method of axiomatization of
probability theory: main distinguishing features of frequency probabilities are
taken as axioms in the measure-theoretic approach. We also present a review of
non-Kolmogorovian probabilistic models including models with negative, complex,
and -adic valued probabilities. The latter model is discussed in details.
The introduction of -adic (as well as more general non-Archimedean)
probabilities is one of the main motivations for consideration of generalized
probabilities taking values in topological groups which are distinct from the
field of real numbers. We discuss applications of non-Kolmogorovian models in
physics and cognitive sciences. An important part of this paper is devoted to
statistical interpretation of probabilities taking values in topological groups
(and in particular in non-Archimedean fields)
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