256 research outputs found
Unification of Gravity and Yang-Mills-Higgs Gauge Theories
In this letter we show how the action functional of the standard model and of
gravity can be derived from a specific Dirac operator. Far from being exotic
this particular Dirac operator turns out to be structurally determined by the
Yukawa coupling term. The main feature of our approach is that it naturally
unifies the action of the standard model with gravity.Comment: 8 pages, late
The generalized Lichnerowicz formula and analysis of Dirac operators
We study Dirac operators acting on sections of a Clifford module \
over a Riemannian manifold . We prove the intrinsic decomposition formula
for their square, which is the generalisation of the well-known formula due to
Lichnerowicz [L]. This formula enables us to distinguish Dirac operators of
simple type. For each Dirac operator of this natural class the local
Atiyah-Singer index theorem holds. Furthermore, if \ is compact and
{{\petit \rm dim}\;M=2n\ge 4}, we derive an expression for the Wodzicki
function , which is defined via the non-commutative residue on the
space of all Dirac operators . We calculate this function
for certain Dirac operators explicitly. From a physical point of view this
provides a method to derive gravity, resp. combined gravity/Yang-Mills actions
from the Dirac operators in question.Comment: 25 pages, plain te
Z_2-Bi-Gradings, Majorana Modules and the Standard Model Action
The action functional of the Standard Model of particle physics is intimately
related to a specific class of first order differential operators called Dirac
operators of Pauli type ("Pauli-Dirac operators"). The aim of this article is
to carefully analyze the geometrical structure of this class of Dirac operators
on the basis of real Dirac operators of simple type. On the basis of simple
type Dirac operators, it is shown how the Standard Model action (STM action)
may be viewed as generalizing the Einstein-Hilbert action in a similar way the
Einstein-Hilbert action is generalized by a cosmological constant. Furthermore,
we demonstrate how the geometrical scheme presented allows to naturally
incorporate also Majorana mass terms within the Standard Model. For reasons of
consistency these Majorana mass terms are shown to dynamically contribute to
the Einstein-Hilbert action by a "true" cosmological constant. Due to its
specific form, this cosmological constant can be very small. Nonetheless, this
cosmological constant may provide a significant contribution to dark
matter/energy. In the geometrical description presented this possibility arises
from a subtle interplay between Dirac and Majorana masses
Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians
We consider the problem of minimising the th eigenvalue, , of
the (-)Laplacian with Robin boundary conditions with respect to all domains
in of given volume . When , we prove that the second
eigenvalue of the -Laplacian is minimised by the domain consisting of the
disjoint union of two balls of equal volume, and that this is the unique domain
with this property. For and , we prove that in many cases a
minimiser cannot be independent of the value of the constant in the
boundary condition, or equivalently of the volume . We obtain similar
results for the Laplacian with generalised Wentzell boundary conditions .Comment: 16 page
A remark on an overdetermined problem in Riemannian Geometry
Let be a Riemannian manifold with a distinguished point and
assume that the geodesic distance from is an isoparametric function.
Let be a bounded domain, with , and consider
the problem in with on ,
where is the -Laplacian of . We prove that if the normal
derivative of along the boundary of is a
function of satisfying suitable conditions, then must be a
geodesic ball. In particular, our result applies to open balls of
equipped with a rotationally symmetric metric of the form
, where is the standard metric of the sphere.Comment: 8 pages. This paper has been written for possible publication in a
special volume dedicated to the conference "Geometric Properties for
Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in
Palinuro in May 201
On the regularity up to the boundary for certain nonlinear elliptic systems
We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems
Monochromatization of femtosecond XUV light pulses with the use of reflection zone plates
We report on a newly built laser based tabletop setup which enables generation of femtosecond light pulses in the XUV range via employing the process of high order harmonic generation HHG in a gas medium. The spatial, spectral, and temporal characteristics of the XUV beam are presented. Monochromatization of XUV light with minimum temporal pulse distortion is the central issue of this work. Off center reflection zone plates are shown to be superior to gratings when selection of a desired harmonic is carried out with the use of a single optical element. A cross correlation technique was applied to characterize the performance of zone plates in the time domain. By using laser pulses of 25 fs length to pump the HHG process, a pulse duration of 45 fs for monochromatized harmonics was achieved in the present setu
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