152 research outputs found
The classical point-electron in Colombeau's theory of nonlinear generalized functions
The electric and magnetic fields of a pole-dipole singularity attributed to a
point-electron-singularity in the Maxwell field are expressed in a Colombeau
algebra of generalized functions. This enables one to calculate dynamical
quantities quadratic in the fields which are otherwise mathematically
ill-defined: The self-energy (i.e., `mass'), the self-angular momentum (i.e.,
`spin'), the self-momentum (i.e., `hidden momentum'), and the self-force. While
the total self-force and self-momentum are zero, therefore insuring that the
electron-singularity is stable, the mass and the spin are diverging integrals
of delta-squared-functions. Yet, after renormalization according to standard
prescriptions, the expressions for mass and spin are consistent with quantum
theory, including the requirement of a gyromagnetic ratio greater than one. The
most striking result, however, is that the electric and magnetic fields differ
from the classical monopolar and dipolar fields by delta-function terms which
are usually considered as insignificant, while in a Colombeau algebra these
terms are precisely the sources of the mechanical mass and spin of the
electron-singularity.Comment: 30 pages. Final published version with a few minor correction
Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities
In a mathematical context in which one can multiply distributions the
"`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field
Theory makes sense mathematically, which can be understood a priori from the
fact the so called "`infinite quantities"' make sense unambiguously (but are
not classical real numbers). The perturbation series does not make sense. A
novelty appears when one starts to compute the transition probabilities. The
transition probabilities have to be computed in a nonperturbative way which, at
least in simplified mathematical examples (even those looking like
nonrenormalizable series), gives real values between 0 and 1 capable to
represent probabilities. However these calculations should be done numerically
and we have only been able to compute simplified mathematical examples due to
the fact these calculations appear very demanding in the physically significant
situation with an infinite dimensional Fock space and the QFT operators
Expanding impulsive gravitational waves
We explicitly demonstrate that the known solutions for expanding impulsive
spherical gravitational waves that have been obtained by a "cut and paste"
method may be considered to be impulsive limits of the Robinson-Trautman vacuum
type N solutions. We extend these results to all the generically distinct
subclasses of these solutions in Minkowski, de Sitter and anti-de Sitter
backgrounds. For these we express the solutions in terms of a continuous
metric. Finally, we also extend the class of spherical shock gravitational
waves to include a non-zero cosmological constant.Comment: 11 pages, LaTeX, To appear in Class. Quantum Gra
An algebraic approach to manifold-valued generalized functions
We discuss the nature of structure-preserving maps of varies function
algebras. In particular, we identify isomorphisms between special Colombeau
algebras on manifolds with invertible manifold-valued generalized functions in
the case of smooth parametrization. As a consequence, and to underline the
consistency and validity of this approach, we see that this generalized version
on algebra isomorphisms in turn implies the classical result on algebras of
smooth functions.Comment: 7 page
On the completeness of impulsive gravitational wave space-times
We consider a class of impulsive gravitational wave space-times, which
generalize impulsive pp-waves. They are of the form ,
where is a Riemannian manifold of arbitrary dimension and carries
the line element with the line
element of and the Dirac measure. We prove a completeness result
for such space-times with complete Riemannian part .Comment: 13 pages, minor changes suggested by the referee
On the Geroch-Traschen class of metrics
We compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea
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