3,145 research outputs found
Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space
We construct the Continuous Wavelet Transform (CWT) on the homogeneous space
(Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2)
(locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be
mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski
space through a Cayley transformation, where other kind of (electromagnetic)
wavelets have already been proposed in the literature. We study the unitary
irreducible representations of the conformal group on the Hilbert spaces
L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square
integrable holomorphic functions with scale dimension \lambda and continuous
mass spectrum, prove the isomorphism (equivariance) between both Hilbert
spaces, admissibility and tight-frame conditions, provide reconstruction
formulas and orthonormal basis of homogeneous polynomials and discuss symmetry
properties and the Euclidean limit of the proposed conformal wavelets. For that
purpose, we firstly state and prove a \lambda-extension of Schwinger's Master
Theorem (SMT), which turns out to be a useful mathematical tool for us,
particularly as a generating function for the unitary-representation functions
of the conformal group and for the derivation of the reproducing (Bergman)
kernel of L^2_h(D_4,d\nu_\lambda). SMT is related to MacMahon's Master Theorem
(MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is
also provided for completeness. Convergence conditions are also studied.Comment: LaTeX, 40 pages, three new Sections and six new references added. To
appear in ACH
Algebraic Structures in the Coupling of Gravity to Gauge Theories
This article is an extension of the author's second master thesis [1]. It
aims to introduce to the theory of perturbatively quantized General Relativity
coupled to Spinor Electrodynamics, provide the results thereof and set the
notation to serve as a starting point for further research in this direction.
It includes the differential geometric and Hopf algebraic background, as well
as the corresponding Lagrange density and some renormalization theory. Then, a
particular problem in the renormalization of Quantum General Relativity coupled
to Quantum Electrodynamics is addressed and solved by a generalization of
Furry's Theorem. Next, the restricted combinatorial Green's functions for all
two-loop propagators and all one-loop divergent subgraphs thereof are
presented. Finally, relations between these one-loop restricted combinatorial
Green's functions necessary for multiplicative renormalization are discussed.
Keywords: Quantum Field Theory; Quantum Gravity; Quantum General Relativity;
Quantum Electrodynamics; Perturbative Quantization; Hopf Algebraic
RenormalizationComment: 57 pages, 259 Feynman diagrams, article; minor revisions; version to
appear in Annals of Physic
A remark on the Alexandrov-Fenchel inequality
In this article, we give a complex-geometric proof of the Alexandrov-Fenchel
inequality without using toric compactifications. The idea is to use the
Legendre transform and develop the Brascamp-Lieb proof of the Pr\'ekopa
theorem. New ingredients in our proof include an integration of Timorin's mixed
Hodge-Riemann bilinear relation and a mixed norm version of H\"ormander's
-estimate, which also implies a non-compact version of the
Khovanski\u{i}-Teissier inequality.Comment: New version, "on line first" in Journal of Functional Analysis:
https://doi.org/10.1016/j.jfa.2018.01.01
A curvature formula associated to a family of pseudoconvex domains
We shall give a definition of the curvature operator for a family of weighted
Bergman spaces associated to a smooth family of smoothly
bounded strongly pseudoconvex domains . In order to study the boundary
term in the curvature operator, we shall introduce the notion of geodesic
curvature for the associated family of boundaries . As an
application, we get a variation formula for the norms of Bergman projections of
currents with compact support. A flatness criterion for and
its applications to triviality of fibrations are also given in this paper.Comment: 35 pages, to appear in Annales de l'Institut Fourie
Temporal Lorentzian Spectral Triples
We present the notion of temporal Lorentzian spectral triple which is an
extension of the notion of pseudo-Riemannian spectral triple with a way to
ensure that the signature of the metric is Lorentzian. A temporal Lorentzian
spectral triple corresponds to a specific 3+1 decomposition of a possibly
noncommutative Lorentzian space. This structure introduces a notion of global
time in noncommutative geometry. As an example, we construct a temporal
Lorentzian spectral triple over a Moyal--Minkowski spacetime. We show that,
when time is commutative, the algebra can be extended to unbounded elements.
Using such an extension, it is possible to define a Lorentzian distance formula
between pure states with a well-defined noncommutative formulation.Comment: 25 pages, a proposition has been added (Prop. 11) concerning the
recovering of the Lorentzian signature, final versio
The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein-Gordon fields
We study the initial value problem for two fundamental theories of gravity,
that is, Einstein's field equations of general relativity and the
(fourth-order) field equations of f(R) modified gravity. For both of these
physical theories, we investigate the global dynamics of a self-gravitating
massive matter field when an initial data set is prescribed on an
asymptotically flat and spacelike hypersurface, provided these data are
sufficiently close to data in Minkowski spacetime. Under such conditions, we
thus establish the global nonlinear stability of Minkowski spacetime in
presence of massive matter. In addition, we provide a rigorous mathematical
validation of the f(R) theory based on analyzing a singular limit problem, when
the function f(R) arising in the generalized Hilbert-Einstein functional
approaches the scalar curvature function R of the standard Hilbert-Einstein
functional. In this limit we prove that f(R) Cauchy developments converge to
Einstein's Cauchy developments in the regime close to Minkowski space. Our
proofs rely on a new strategy, introduced here and referred to as the
Euclidian-Hyperboloidal Foliation Method (EHFM). This is a major extension of
the Hyperboloidal Foliation Method (HFM) which we used earlier for the
Einstein-massive field system but for a restricted class of initial data. Here,
the data are solely assumed to satisfy an asymptotic flatness condition and be
small in a weighted energy norm. These results for matter spacetimes provide a
significant extension to the existing stability theory for vacuum spacetimes,
developed by Christodoulou and Klainerman and revisited by Lindblad and
Rodnianski.Comment: 127 pages. Selected chapters from a boo
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