3,935 research outputs found
Generalized Hadamard Product and the Derivatives of Spectral Functions
In this work we propose a generalization of the Hadamard product between two
matrices to a tensor-valued, multi-linear product between k matrices for any . A multi-linear dual operator to the generalized Hadamard product is
presented. It is a natural generalization of the Diag x operator, that maps a
vector into the diagonal matrix with x on its main diagonal.
Defining an action of the orthogonal matrices on the space of
k-dimensional tensors, we investigate its interactions with the generalized
Hadamard product and its dual. The research is motivated, as illustrated
throughout the paper, by the apparent suitability of this language to describe
the higher-order derivatives of spectral functions and the tools needed to
compute them. For more on the later we refer the reader to [14] and [15], where
we use the language and properties developed here to study the higher-order
derivatives of spectral functions.Comment: 24 page
Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime
We establish the existence of local, covariant time ordered products of local
Wick polynomials for a free scalar field in curved spacetime. Our time ordered
products satisfy all of the hypotheses of our previous uniqueness theorem, so
our construction essentially completes the analysis of the existence,
uniqueness and renormalizability of the perturbative expansion for nonlinear
quantum field theories in curved spacetime. As a byproduct of our analysis, we
derive a scaling expansion of the time ordered products about the total
diagonal that expresses them as a sum of products of polynomials in the
curvature times Lorentz invariant distributions, plus a remainder term of
arbitrary low scaling degree.Comment: Minor revisions, 2 references added. 43 pages, Latex format, no
figure
Supersymmetric Field-Theoretic Models on a Supermanifold
We propose the extension of some structural aspects that have successfully
been applied in the development of the theory of quantum fields propagating on
a general spacetime manifold so as to include superfield models on a
supermanifold. We only deal with the limited class of supermanifolds which
admit the existence of a smooth body manifold structure. Our considerations are
based on the Catenacci-Reina-Teofillatto-Bryant approach to supermanifolds. In
particular, we show that the class of supermanifolds constructed by
Bonora-Pasti-Tonin satisfies the criterions which guarantee that a
supermanifold admits a Hausdorff body manifold. This construction is the
closest to the physicist's intuitive view of superspace as a manifold with some
anticommuting coordinates, where the odd sector is topologically trivial. The
paper also contains a new construction of superdistributions and useful results
on the wavefront set of such objects. Moreover, a generalization of the
spectral condition is formulated using the notion of the wavefront set of
superdistributions, which is equivalent to the requirement that all of the
component fields satisfy, on the body manifold, a microlocal spectral condition
proposed by Brunetti-Fredenhagen-K\"ohler.Comment: Final version to appear in J.Math.Phy
On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist
A parametric spectral estimation problem in the style of Byrnes, Georgiou,
and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was
only proved in a special case. Based on their results, we show that a solution
indeed exists given an arbitrary matrix-valued prior density. The main tool in
our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA
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