Convolution of n-Dimensional Tempered Ultradistributions and Field Theory

Bollini, C. G.; Rocca, Mario Carlos

doi:10.1023/b:ijtp.0000028850.35090.24

Cited by 32 publications

(29 citation statements)

References 7 publications

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“…9) with (6.12) of [3] confirms the validity of the results obtained in section 6 of this paper. We emphasize that the present results are obtained in a manner considerably simpler to that of [3]. Resorting again to [66] we have:…”

Section: The Euclidean Space Casesupporting

confidence: 89%

“…For the first convolution of (6. This result was obtained in the [3], formula (6.12) using the convolution of even Tempered Ultradistributions. The coincidence of (6.1.…”

Section: The Euclidean Space Casementioning

confidence: 67%

“…In [1][2][3][4] it was demonstrated that it is possible to define a general convolution between the ultradistributions of JSS [5] (Ultrahyperfunctions). This convolution yields another Ultrahyperfunction.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum field theory, Feynman-, Wheeler propagators, dimensional regularization in configuration space and convolution of Lorentz Invariant Tempered Distributions

Plastino

Rocca

2018

J. Phys. Commun.

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The Dimensional Regularization (DR) of Bollini and Giambiagi (BG) can not be defined for all Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI) L  ¢ . In this paper we overcome such limitation and show that it can be generalized to all aforementioned STDELI and obtain a product in a ring with zero divisors. For this purpose, we resort to a formula obtained by Bollini and Rocca and demonstrate the existence of the convolution (in Minkowskian space) of such distributions. This is done by following a procedure similar to that used so as to define a general convolution between the Ultradistributions of J Sebastiao e Silva (JSS), also known as Ultrahyperfunctions, obtained by Bollini et al. Using the Inverse Fourier Transform we get the ring with zero divisors LA  ¢ , defined asdenotes the Inverse Fourier Transform. In this manner we effect a dimensional regularization in momentum space (the ring L  ¢ ) via convolution, and a product of distributions in the corresponding configuration space (the ring LA  ¢ ). This generalizes the results obtained by BG for Euclidean space. We provide several examples of the application of our new results in Quantum Field Theory (QFT). In particular, the convolution of n massless Feynman's propagators and the convolution of n massless Wheeler's propagators in Minkowskian space. The results obtained in this work have already allowed us to calculate the classical partition function of Newtonian gravity, for the first time ever, in the Gibbs' formulation and in the Tsallis' one. It is our hope that this convolution will allow one to quantize non-renormalizable Quantum Field Theories (QFT's).

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Section: The Euclidean Space Casesupporting

confidence: 89%

“…For the first convolution of (6. This result was obtained in the [3], formula (6.12) using the convolution of even Tempered Ultradistributions. The coincidence of (6.1.…”

Section: The Euclidean Space Casementioning

confidence: 67%

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Quantum field theory, Feynman-, Wheeler propagators, dimensional regularization in configuration space and convolution of Lorentz Invariant Tempered Distributions

Plastino

Rocca

2018

J. Phys. Commun.

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“…In other paper ref. [4] we have extended these procedure to n-dimensional space. In four-dimensional space we have given an expression for the convolution of two tempered ultradistributions even in the variables k 0 and ρ.…”

Section: Discussionmentioning

confidence: 99%

Convolution of Lorentz Invariant Ultradistributions and Field Theory

Bollini

Rocca

2004

International Journal of Theoretical Physics

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“…Further, aside from the mathematical interest of the results presented in Refs. [1][2][3][4][5], Brüning and Nagamachi have conjectured that the properties of tempered ultrahyperfunctions are well adapted for their use in quantum field theory with a fundamental length, while Bollini and Rocca [6] have given a general definition of convolution between two arbitrary tempered ultrahyperfunctions in order to treat the problem of singular products of functions Green also in quantum field theory. In another interesting recent work [7], Schmidt has given an insight in the operations of duality and Fourier transform on the space of test and generalized functions belonging to new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, which is very similar to the studies by Sebastião e Silva [1] and Hasumi [3] about tempered ultrahyperfunctions, and eventually suggests applications to quantum field theory.…”

Section: Introductionmentioning

confidence: 99%

A note on Fourier–Laplace transform and analytic wave front set in theory of tempered ultrahyperfunctions

Franco¹,

Renoldi

2007

Journal of Mathematical Analysis and Applications

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In this paper we study the Fourier-Laplace transform of tempered ultrahyperfunctions introduced by Sebastião e Silva and Hasumi. We establish a generalization of Paley-Wiener-Schwartz theorem for this setting. This theorem is interesting in connection with the microlocal analysis. For this reason, the paper also contains a description of the singularity structure of tempered ultrahyperfunctions in terms of the concept of analytic wave front set.

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Convolution of n-Dimensional Tempered Ultradistributions and Field Theory

Cited by 32 publications

References 7 publications

Quantum field theory, Feynman-, Wheeler propagators, dimensional regularization in configuration space and convolution of Lorentz Invariant Tempered Distributions

Quantum field theory, Feynman-, Wheeler propagators, dimensional regularization in configuration space and convolution of Lorentz Invariant Tempered Distributions

Convolution of Lorentz Invariant Ultradistributions and Field Theory

A note on Fourier–Laplace transform and analytic wave front set in theory of tempered ultrahyperfunctions

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