2,192 research outputs found
A new class of Fermionic Projectors: M{\o}ller operators and mass oscillation properties
Recently, a new functional analytic construction of quasi-free states for a
self-dual CAR algebra has been presented in \cite{Felix2}. This method relies
on the so-called strong mass oscillation property. We provide an example where
this requirement is not satisfied, due to the nonvanishing trace of the
solutions of the Dirac equation on the horizon of Rindler space, and we propose
a modification of the construction in order to weaken this condition. Finally,
a connection between the two approaches is built.Comment: 21 pages, accepted for publication in Letters in Mathematical Physics
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Models of free quantum field theories on curved backgrounds
Free quantum field theories on curved backgrounds are discussed via three
explicit examples: the real scalar field, the Dirac field and the Proca field.
The first step consists of outlining the main properties of globally hyperbolic
spacetimes, that is the class of manifolds on which the classical dynamics of
all physically relevant free fields can be written in terms of a Cauchy
problem. The set of all smooth solutions of the latter encompasses the
dynamically allowed configurations which are used to identify via a suitable
pairing a collection of classical observables. As a last step we use such
collection to construct a -algebra which encodes the information on the
dynamics and on the canonical commutation or anti-commutation relations
depending whether the underlying field is a Fermion or a Boson.Comment: 41 page
Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
An integrable self-adjoint 7-point scheme on the triangular lattice and an
integrable self-adjoint scheme on the honeycomb lattice are studied using the
sublattice approach. The star-triangle relation between these systems is
introduced, and the Darboux transformations for both linear problems from the
Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A
geometric interpretation of the Laplace transformations of the self-adjoint
7-point scheme is given and the corresponding novel integrable discrete 3D
system is constructed.Comment: 15 pages, 6 figures; references added, some typos correcte
The Cauchy Problem for the Wave Equation in the Schwarzschild Geometry
The Cauchy problem is considered for the scalar wave equation in the
Schwarzschild geometry. We derive an integral spectral representation for the
solution and prove pointwise decay in time.Comment: 33 page
On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat
equation with inverse singular potential, is proved for 2mth-order linear
parabolic equations with Hardy-supercritical singular potentials. Extensions to
other linear and nonlinear singular PDEs are discussed.Comment: 22 page
Block-diagonalisation of matrices and operators
In this short note we deal with a constructive scheme to decompose a
continuous family of matrices asymptotically as into
blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also
discuss the extension of the scheme to matrix families depending upon
additional parameters and operators on Hilbert spaces.Comment: 8 page
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