K. Scharnhorst

Angles in Complex Vector Spaces

The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and Hermitian angles, (Kasners) pseudo-angle, the Kähler angle (synonyms for the latter used in the literature are angle of inclination, characteristic deviation, holomorphic deviation, holomorphy angle, Wirtinger angle, slant angle).

On Propagation of light in the vacuum between plates

QED is considered in the presence of two parallel plates (Casimir effect type configuration) imposing boundary conditions on the photon vacuum fluctuations. Two-loop corrections arising from the boundary conditions for the photon vacuum fluctuations to the QED effective action are calculated in a physically reasonable approximation. From this effective action we find that for light propagating perpendicular to the plates in the vacuum between them, their impact phrased in the simplest terms consists in causing a change in the velocity of light.

QED between parallel mirrors : light signals faster than c, or amplified by the vacuum

Because it is scattered by the zero-point oscillations of the quantized fields, light of frequency omega travelling normally to two parallel mirrors experiences the vacuum between them as a dispersive medium with refractive index n( omega ). An earlier low-frequency result that n(0)<1 is combined with the Kramers-Kronig dispersion relation for n and with the classic Sommerfeld-Brillouin argument to show (under certain physically reasonable assumptions) that either n( infinity )<1, in which case the signal velocity c/n( infinity ) exceeds c; or that the imaginary part of n is negative at least for some ranges of frequency, in which case the vacuum between the mirrors fails to respond to a light probe like a normal passive medium. Further, the optical theorem suggests that n exhibits no dispersion to order e4, i.e. that n( infinity )=n(0) up to corrections of order e6 at most.

The velocities of light in modified QED vacua

QED vacua under the influence of external conditions (background fields, finite temperature, boundary conditions) can be considered as dispersive media whose complex behaviour can no longer be described in terms of a single universal vacuum velocity of light c. Beginning in the early 1950s (J.S. Toll), quantum field theoretic investigations have led to considerable insight into the relation between the vacuum structure and the propagation of light. Recent years have witnessed a significant growth of activity in this field of research. After a short overview, two characteristic situations are discussed: the propagation of light in a constant homogeneous magnetic field and in a Casimir vacuum. The latter appears to be particularly interesting because the Casimir vacuum has been found to exhibit modes of the propagation of light with phase and group velocities larger than c in the low frequency domain ω ≪ m where m is the electron mass. The impact of this result on the front velocity of light in a Casimir vacuum is discussed by means of the Kramers-Kronig relations.

Radiative Corrections to the Casimir Pressure Under the Influence of Temperature and External Fields

Generalizing the quantum field theory (QFT) with boundary conditions in covariant gauge to the case of finite temperature, we develop the quantum electrodynamics (QED) with boundary conditions in the Matsubara approach as well as in the thermofield formulation. We rederive the known results of the free-field theory for the pressure and the free energy of the Casimir problem. For infinitely thin plates we calculate the radiative corrections in second-order perturbation theory at finite temperature. Thereby it turns out that the calculation in of the vacuum energy at the vanishing temperature via the Z functional is much simplier than the calculation via the energy momentum tensor. This observation allows determination of the influence of static electromagnetic fields on the Casimir problem. copyright 1987 Academic Press, Inc.

O(alpha) radiative correction to the Casimir energy for penetrable mirrors

The leading radiative correction to the Casimir energy for two parallel penetrable mirrors (realized by

A Grassmann integral equation

\ensuremath{\delta}

The exact equivalence of the one-flavour lattice Thirring model with Wilson fermions to a two-colour loop model

-function potentials) is calculated within QED perturbation theory. It is found to be of order

Nonlinear Bogolyubov-Valatin transformations: Two modes

\ensuremath{\alpha}

Results for the strong coupling lattice Schwinger model with Wilson fermions from a study of the equivalent loop model

like the known radiative correction for ideally reflecting mirrors from which it differs, for a mirror distance much larger than the electron Compton wavelength, only by a monotonic, powerlike function of the frequency at which the mirrors become transparent. This shows that the