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