Isotropic Kähler structures on Engel 4-manifolds

Examples of isotropic Kähler manifolds (i.e., J²=0) which are neither complex nor symplectic, and therefore not indefinite Kähler, are constructed.     

Kähler-Dirac ghosts for self-dual fields

We present the generalization to spacetime dimension D=4n+2 of the Lorentz covariant quadratic lagrangian for pairs of (anti)self-dual fields previously obtained by the authors in D=2. In the process BRST quantizing this lagrangian a first-order quadratic lagrangian for ghost (anti)self-dual fields is found which, after gauge fixing, can be written in terms of bispinors and it turns out to be a Kähler-Dirac lagrangian. The coupling to gravity is straightforward and the gravitational anomaly due to (anti)self-dual fields is obtained directly from an action principle.     

Integrabilität,Irreversibilität und Kosmogonie

Integrability, Irreversibility, and Cosmogony Leading the famous discussion with Boltzmann on the foundation of statistical thermodynamics Zermelo was backed by Planck himself. Zermelo's objections to Boltzmann's atomism and Boltzmann's answers are again vital today. After all Boltzmann founded statistical thermodynamics cosmogonically.     

The first eigenvalue of the Dirac operator on Kähler manifolds

If M^2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r (r {1, …, [(m + 1)/2]}) of the Dirac operator D satisfies the inequality λ² ≥ rR₀/4r − 2, where R₀ is the minimum of R on M^2m. Hence, if the complex dimension m is odd (even) we have the estimation

Full-size image

for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2r − 1 the manifold M^2m must be Einstein. The manifolds S², S² × S², S² × T², P³( ), F( ), P³( ) × T² and F( ³) × T², where F( ³) denotes the flag manifold and T² the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M^2m is an example of odd complex dimension m, then M^2m × T² is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D² which are Kählerian twistor-spinors.     

Hermitizität und Eichinvarianz

Hermiticity and Gauge Invariance In the Theory of Hermitian Relativity (HRT) the postulates of hermiticity and gauge invariance are formulated in different ways, due to a different understanding of the idea of hermiticity. However all hermitian systems of equations have to satisfy Einstein's weak system of equations being equivalent to Einstein-Schrödinger equations.