122 research outputs found
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in ℝ d is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of ℤ d ,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)N(d,β), anyx ∈ ℤ d , and anyA ⊂ {1, ,n} d •{y∈ℤ whereν A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ →
Time and Ensemble Averages in Bohmian Mechanics
We show that in the framework of one-dimensional Bohmian Quantum
Mechanics[1], for a particle subject to a potential undergoing a weak adiabatic
change, the time averages of the particle's positions typically differ markedly
from the ensemble averages. We Apply this result to the case where the weak
perturbing potential is the back-action of a measuring device (i.e. a
protective measurement). It is shown that under these conditions, most
trajectories never cross the position measured (as already shown for a
particular example in [3]).Comment: 6 page
Time of Arrival from Bohmian Flow
We develop a new conception for the quantum mechanical arrival time
distribution from the perspective of Bohmian mechanics. A detection probability
for detectors sensitive to quite arbitrary spacetime domains is formulated.
Basic positivity and monotonicity properties are established. We show that our
detection probability improves and generalises earlier proposals by Leavens and
McKinnon. The difference between the two notions is illustrated through
application to a free wave packet.Comment: 18 pages, 8 figures, to appear in Journ. Phys. A; representation of
ref. 5 improved (thanks to Rick Leavens
Feynman's Path Integrals and Bohm's Particle Paths
Both Bohmian mechanics, a version of quantum mechanics with trajectories, and
Feynman's path integral formalism have something to do with particle paths in
space and time. The question thus arises how the two ideas relate to each
other. In short, the answer is, path integrals provide a re-formulation of
Schroedinger's equation, which is half of the defining equations of Bohmian
mechanics. I try to give a clear and concise description of the various aspects
of the situation.Comment: 4 pages LaTeX, no figures; v2 shortened a bi
Hypersurface Bohm-Dirac models
We define a class of Lorentz invariant Bohmian quantum models for N entangled
but noninteracting Dirac particles. Lorentz invariance is achieved for these
models through the incorporation of an additional dynamical space-time
structure provided by a foliation of space-time. These models can be regarded
as the extension of Bohm's model for N Dirac particles, corresponding to the
foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to
more general foliations. As with Bohm's model, there exists for these models an
equivariant measure on the leaves of the foliation. This makes possible a
simple statistical analysis of position correlations analogous to the
equilibrium analysis for (the nonrelativistic) Bohmian mechanics.Comment: 17 pages, 3 figures, RevTex. Completely revised versio
Classical and Non-Relativistic Limits of a Lorentz-Invariant Bohmian Model for a System of Spinless Particles
A completely Lorentz-invariant Bohmian model has been proposed recently for
the case of a system of non-interacting spinless particles, obeying
Klein-Gordon equations. It is based on a multi-temporal formalism and on the
idea of treating the squared norm of the wave function as a space-time
probability density. The particle's configurations evolve in space-time in
terms of a parameter {\sigma}, with dimensions of time. In this work this model
is further analyzed and extended to the case of an interaction with an external
electromagnetic field. The physical meaning of {\sigma} is explored. Two
special situations are studied in depth: (1) the classical limit, where the
Einsteinian Mechanics of Special Relativity is recovered and the parameter
{\sigma} is shown to tend to the particle's proper time; and (2) the
non-relativistic limit, where it is obtained a model very similar to the usual
non-relativistic Bohmian Mechanics but with the time of the frame of reference
replaced by {\sigma} as the dynamical temporal parameter
Non-Locality and Theories of Causation
The aim of the paper is to investigate the characterization of an unambiguous
notion of causation linking single space-llike separated events in EPR-Bell
frameworks. This issue is investigated in ordinary quantum mechanics, with some
hints to no collapse formulations of the theory such as Bohmian mechanics.Comment: Presented at the NATO Advanced Research Workshop on Modality,
Probability and Bell's Theorems, Cracow, Poland, August 19-23, 200
Willmore minimizers with prescribed isoperimetric ratio
Motivated by a simple model for elastic cell membranes, we minimize the
Willmore functional among two-dimensional spheres embedded in R^3 with
prescribed isoperimetric ratio
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
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