535 research outputs found
Dirac Equation with External Potential and Initial Data on Cauchy Surfaces
With this paper we provide a mathematical review on the initial-value problem
of the one-particle Dirac equation on space-like Cauchy hypersurfaces for
compactly supported external potentials. We, first, discuss the physically
relevant spaces of solutions and initial values in position and mass shell
representation; second, review the action of the Poincar\'e group as well as
gauge transformations on those spaces; third, introduce generalized Fourier
transforms between those spaces and prove convenient Paley-Wiener- and
Sobolev-type estimates. These generalized Fourier transforms immediately allow
the construction of a unitary evolution operator for the free Dirac equation
between the Hilbert spaces of square-integrable wave functions of two
respective Cauchy surfaces. With a Picard-Lindel\"of argument this evolution
map is generalized to the Dirac evolution including the external potential. For
the latter we introduce a convenient interaction picture on Cauchy surfaces.
These tools immediately provide another proof of the well-known existence and
uniqueness of classical solutions and their causal structure
Linearly edge-reinforced random walks
We review results on linearly edge-reinforced random walks. On finite graphs,
the process has the same distribution as a mixture of reversible Markov chains.
This has applications in Bayesian statistics and it has been used in studying
the random walk on infinite graphs. On trees, one has a representation as a
random walk in an independent random environment. We review recent results for
the random walk on ladders: recurrence, a representation as a random walk in a
random environment, and estimates for the position of the random walker.Comment: Published at http://dx.doi.org/10.1214/074921706000000103 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Moderate deviations for longest increasing subsequences : the lower tail
We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity
ATLAS Monitored Drift Tube Chambers in E = 11 MeV Neutron Background
The influence of fast neutrons on the occupancy and the single tube
resolution of ATLAS muon drift detectors was investigated by exposing a chamber
built out of 3 layers of 3 short standard drift tubes to neutron flux-densities
of up to 16 kHz/cm2 at a neutron energy of E=11 MeV. Pulse shape capable NE213
scintillaton detectors and a calibrated BF3 neutron detector provided
monitoring of the neutron flux-density and energy. The sensitivity of the drift
chamber to the neutrons was measured to be 4*10-4 by comparing data sets with
and without neutron background. For the investigation of tracks of cosmic muons
two silicon-strip detectors above and underneath the chamber allow to compare
measured drift-radii with reference tracks. Alternatively, the single tube
resolution was determined using the triple-sum method. The comparison between
data with and without neutron irradiation shows only a marginal effect on the
resolution and little influence on the muon track reconstruction.Comment: 4 pages, 11 figures, conferenc
Atomic Zitterbewegung
Ultra-cold atoms which are subject to ultra-relativistic dynamics are
investigated. By using optically induced gauge potentials we show that the
dynamics of the atoms is governed by a Dirac type equation. To illustrate this
we study the trembling motion of the centre of mass for an effective two level
system, historically called Zitterbewegung. Its origin is described in detail,
where in particular the role of the finite width of the atomic wave packets is
seen to induce a damping of both the centre of mass dynamics and the dynamics
of the populations of the two levels.Comment: 6 pages, 4 figure
- …