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Felix Finster

Publications and source records attributed to Felix Finster.

159 records · Page 9Linked to original sources
arXiv2000

The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Black Holes

We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU(2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.

gr-qc↗
arXiv2000

Light-Cone Expansion of the Dirac Sea in the Presence of Chiral and Scalar Potentials

We study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green's functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The light-cone expansion expresses the Green's functions as an infinite sum of line integrals over the external potential and its partial derivatives. The Dirac sea is decomposed into a causal and a non-causal contribution. The causal contribution has a light-cone expansion which is closely related to the light-cone expansion of the Green's functions; it describes the singular behavior of the Dirac sea in terms of nested line integrals along the light cone. The non-causal contribution, on the other hand, is, to every order in perturbation theory, a smooth function in position space.

hep-th↗
arXiv2000

Curvature Estimates and the Positive Mass Theorem

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to (R^3,delta_{ij}), in the sense that there is an upper bound for the L^2 norm of the Riemannian curvature tensor over the manifold except for a set of small measure. This curvature estimate allows us to extend the case of equality of the Positive Mass Theorem to include non-smooth manifolds with generalized non-negative scalar curvature, which we define.

math.DG↗
arXiv2000

Non-Existence of Time-Periodic Solutions of the Dirac Equation in an Axisymmetric Black Hole Geometry

We prove that, in the non-extreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chandrasekhar's separation of the Dirac equation in this geometry. A similar non-existence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast with the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity.

gr-qc↗
arXiv2000

Some Recent Progress in Classical General Relativity

In this short survey paper, we discuss certain recent results in classical gravity. Our main attention is restricted to two topics: the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).

gr-qc↗
arXiv2000

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.

gr-qc↗
arXiv1999

Particle-Like Solutions of the Einstein-Dirac-Maxwell Equations

We consider the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions are discussed for different values of the electromagnetic coupling constant. We find solutions even when the electromagnetic coupling is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that the combined electromagnetic and gravitational self-interaction of the Dirac particles is finite.

gr-qc↗
arXiv1999

The Coupling of Gravity to Spin and Electromagnetism

The coupled Einstein-Dirac-Maxwell equations are considered for a static, spherically symmetric system of two fermions in a singlet spinor state. Stable soliton-like solutions are shown to exist, and we discuss the regularizing effect of gravity from a Feynman diagram point of view.

gr-qc↗
arXiv1999

Derivation of Local Gauge Freedom from a Measurement Principle

We define operator manifolds as manifolds on which a spectral measure on a Hilbert space is given as additional structure. The spectral measure mathematically describes space as a quantum mechanical observable. We show that the vectors of the Hilbert space can be represented as functions on the manifold. The arbitrariness of this representation is interpreted as local gauge freedom. In this way, the physical gauge principle is linked with quantum mechanical measurements of the position variable. We derive the restriction for the local gauge group to be U(m), where m is the number of components of the wave functions.

funct-an↗
arXiv1999

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.

gr-qc↗
arXiv1997

Light-Cone Expansion of the Dirac Sea with Light Cone Integrals

The Dirac sea is calculated in an expansion around the light cone. The method is to analyze the perturbation expansion for the Dirac sea in position space. This leads to integrals over expressions containing distributions which are singular on the light cone. We derive asymptotic formulas for these "light cone integrals" in terms of line integrals over the external potential and its partial derivatives. The calculations are based on the perturbation expansion for the Dirac sea in the preprint gr-qc/9606040 and yield the formulas listed in the appendix of this preprint. The results can be obtained easier with a combination of calculations in position and momentum space (see the corresponding preprints on the hep-th server). Therefore the calculations are preliminary and will remain unpublished; they are intended as reference for people who encounter similar mathematical problems.

funct-an↗
arXiv1997

Node Theorem for Matrix Schroedinger Operators

In this paper we study the ground states of a matrix Schroedinger operator, that is an operator of the type (-Laplace) + V acting on m-component wave functions in R^n. We prove in generalization of the classical node theorem that the ground states of this operator have no zeros, if the potential V satisfies the following conditions: 1) V is bounded from below. 2) V is strictly positive at infinity with respect to the ground state energy. 3) The partial derivatives of V do not grow too fast in the order of the derivative. Furthermore we show that the degeneracy of the ground states is at most m. For the proof we reformulate the problem with path integrals and perform a cluster expansion with large/small field conditions.

funct-an↗
arXiv1996

Derivation of Field Equations from the Principle of the Fermionic Projector

With the concept of "discrete space-time" the space-time continuum is resolved into discrete points at the scale of the Planck length. We postulate with the "principle of the fermionic projector" that physical equations must be formulated intrinsically in discrete space-time with the projector on fermionic states. This principle combines the Pauli principle, a local gauge principle and the principle of relativity, but does not include a locality or causality principle. It is shown that a well-established limit yields the structure of a Lorentzian manifold. The equations of discrete space-time reduce to classical field equations. This preliminary text is intended as introduction. Detailed calculations can be obtained from the author on request.

gr-qc↗