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Convolution of n-dimensional Tempered Ultradistributions and Field Theory
In this work, a general definition of convolution between two arbitrary
Tempered Ultradistributions is given. When one of the Tempered
Ultradistributions is rapidly decreasing this definition coincides with the
definition of J. Sebastiao e Silva. In the four-dimensional case, when the
Tempered Ultradistributions are even in the variables and (see
Section 5) we obtain an expression for the convolution, which is more suitable
for practical applications. The product of two arbitrary even (in the variables
and ) four dimensional distributions of exponential type is defined
via the convolution of its corresponding Fourier Transforms. With this
definition of convolution, we treat the problem of singular products of Green
Functions in Quantum Field Theory. (For Renormalizable as well as for
Nonrenormalizable Theories). Several examples of convolution of two Tempered
Ultradistributions are given. In particular we calculate the convolution of two
massless Wheeeler's propagators and the convolution of two complex mass
Wheeler's propagators.Comment: 28 page
Convolution of Lorentz Invariant Ultradistributions and Field Theory
In this work, a general definition of convolution between two arbitrary four
dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in
both: Minkowskian and Euclidean Space (Spherically symmetric tempered
ultradistributions). The product of two arbitrary fdLi distributions of
exponential type is defined via the convolution of its corresponding Fourier
Transforms. Several examples of convolution of two fdLi Tempered
Ultradistributions are given. In particular we calculate exactly the
convolution of two Feynman's massless propagators. An expression for the
Fourier Transform of a Lorentz invariant Tempered Ultradistribution in terms of
modified Bessel distributions is obtained in this work (Generalization of
Bochner's formula to Minkowskian space). At the same time, and in a previous
step used for the deduction of the convolution formula, we obtain the
generalization to the Minkowskian space, of the dimensional regularization of
the perturbation theory of Green Functions in the Euclidean configuration space
given in ref.[12]. As an example we evaluate the convolution of two
n-dimensional complex-mass Wheeler's propagators.Comment: LaTeX, 52 pages, no figure
Possible Divergences in Tsallis' Thermostatistics
Trying to compute the nonextensive q-partition function for the Harmonic
Oscillator in more than two dimensions, one encounters that it diverges, which
poses a serious threat to the whole of Tsallis' thermostatistics. Appeal to the
so called q-Laplace Transform, where the q-exponential function plays the role
of the ordinary exponential, is seen to save the day.Comment: Text has change
Classical and \textcolor{blue}{Quantum} Field-Theoretical approach to the non-linear q-Klein-Gordon Equation
\color{blue}{In the wake of efforts made in [EPL {\bf 97}, 41001 (2012)] and
[J. Math. Phys. {\bf 54}, 103302 (2913)], we extend them here by developing the
conventional Lagrangian treatment of a classical field theory (FT) to the
q-Klein-Gordon equation advanced in [Phys. Rev. Lett. {\bf 106}, 140601 (2011)]
and [J. Math. Phys. {\bf 54}, 103302 (2913)], and the quantum theory
corresponding to . This makes it possible to generate a
putative conjecture regarding black matter. Our theory reduces to the usual FT
for .}Comment: Title has changed. Text has change
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