The Fermionic Projector, Entanglement, and the Collapse of the Wave Function

After a brief introduction to the fermionic projector approach, we review how entanglement and second quantized bosonic and fermionic fields can be described in this framework. The constructions are discussed with regard to decoherence phenomena and the measurement problem. We propose a mechanism leading to the collapse of the wave function in the quantum mechanical measurement process.Comment: 17 pages, LaTeX, 2 figures, minor changes (published version

On the Regularized Fermionic Projector of the Vacuum

We construct families of fermionic projectors with spherically symmetric regularization, which satisfy the condition of a distributional ${\mathcal{M}} P$-product. The method is to analyze regularization tails with a power-law or logarithmic scaling in composite expressions in the fermionic projector. The resulting regularizations break the Lorentz symmetry and give rise to a multi-layer structure of the fermionic projector near the light cone. The remaining freedom for the regularization parameters and the consequences for the normalization of the fermionic states are discussed.Comment: 66 pages, LaTeX, 8 figures, minor improvements (published version

Non-Existence of Time-Periodic Solutions of the Dirac Equation in a Reissner-Nordstrom Black Hole Background

It is shown analytically that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordstrom black hole background; in particular, there are no static solutions of the Dirac equation in such a background field. The physical interpretation is that Dirac particles can either disappear into the black hole or escape to infinity, but they cannot stay on a periodic orbit around the black hole.Comment: 24 pages, 2 figures (published version

Fermion Systems in Discrete Space-Time Exemplifying the Spontaneous Generation of a Causal Structure

As toy models for space-time on the Planck scale, we consider examples of fermion systems in discrete space-time which are composed of one or two particles defined on two up to nine space-time points. We study the self-organization of the particles as described by a variational principle both analytically and numerically. We find an effect of spontaneous symmetry breaking which leads to the emergence of a discrete causal structure.Comment: 37 pages, LaTeX, 12 figures, minor changes (published version

A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.Comment: 19 pages, LaTeX, proof of Propositions 2.3 and 3.1 given in more detai

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.Comment: 41 pages, 4 figures, minor correction

Spatial Patterns of Soil Development, Methane Oxidation, and Methanotrophic Diversity along a Receding Glacier Forefield, Southeast Greenland

Increasing global annual temperature leads to massive loss of ice cover worldwide. Consequently, glaciers retreat and ice-covered areas become exposed. We report on a study from the Mittivakkat Gletscher forefield in Southeast Greenland with special focus on methanotrophy in relation to exposure time to the atmosphere. The Mittivakkat Gletscher has receded since the end of the Little Ice Age (LIA; about AD 1850) and has left behind a series of deposits of decreasing age concurrently with its recession. Soil samples from this chronosequence were examined in order to elucidate main soil variables, as well as the activity and community structure of methanotrophs, a group of microorganisms involved in regulation of atmospheric methane. Soil variables revealed poor soil development, and incubation experiments showed methane consumption rates of 2.14 nmol CH4 day−1 gsoil −1 at 22 °C and 1.24 nmol CH4 day−1 gsoil −1 at 10 °C in the LIA terminal moraine. Methane consumption was not detected in younger samples, despite the presence of high-affinity methanotrophs in all samples. This was indicated by successful amplification of partial pmoA genes, which code for a subunit of a key enzyme involved in methane oxidation. In addition, the results of the diversity study show that the diversity of the methanotrophic community at the younger, recently deglaciated site P5 is poorer than the diversity of the community retrieved from the LIA moraine. We put forward the hypothesis that aerobic methanotrophs were at very low abundance and diversity during glaciation probably due to anoxia at the ice-sediment interface and that colonization after deglaciation is not completed yet. More detailed studies are required to explain the causes of discrepancy between activity and presence of high-affinity methanotrophs and its relation to the transit from ice-covered probably anoxic to ice-free oxi

The Dirac Equation and the Normalization of its Solutions in a Closed Friedmann-Robertson-Walker Universe

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form.Comment: 22 pages, LaTeX, sign error in equation (3.7) correcte

Particle-Like Solutions of the Einstein-Dirac Equations

The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of soliton-like solutions of these equations. The stability of the solutions is analyzed. For weak coupling (i.e., small rest mass of the fermions), all the solutions are linearly stable (with respect to spherically symmetric perturbations), whereas for stronger coupling, both stable and unstable solutions exist. For the physical interpretation, we discuss how the energy of the fermions and the (ADM) mass behave as functions of the rest mass of the fermions. Although gravitation is not renormalizable, our solutions of the Einstein-Dirac equations are regular and well-behaved even for strong coupling.Comment: 31 pages, LaTeX, 21 PostScript figures, some references adde