Abstract
The Dimensional Regularization technique of Bollini and Giambiagi (BG) [Phys. Lett. B 40, 566 (1972); Il Nuovo Cim. B 12, 20 (1972); Phys. Rev. D 53, 5761 (1996)] cannot be employed for all Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI) S′L. We lifted such limitation in [J. Phys. Comm. 2 115029 (2018)], which opens new QFT possibilities, centering in the use of STDELI that allows one to obtain a product in a ring with zero divisors. This in turn, overcomes all problems regrading QFT infinities. We provide here three examples of the application of our STDELI-extension to quantum field theory (A) the exact evaluation of an electron’s self energy to one loop, (B) the exact evaluation of QED’s vacuum polarization, and C) the theory for six dimensions, that is non-renormalizable.
Highlights
Brief Summary of the Mathematical Results to Be EmployedThis subsection may be omitted at a first reading. Our work revolves around the problem of using a workable version of the product of two distributions (a product in a ring with divisors of zero)
We provide here three examples of the application of our Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI)-extension to quantum field theory (A) the exact evaluation of an electron’s self energy to one loop, (B) the exact evaluation of QED’s vacuum polarization, and the λ φ4 4!
Praesent eget The current paradigm in quantum field theory (QFT) asserts that non sem vel leo ultrices bibendum
Summary
This subsection may be omitted at a first reading. Our work revolves around the problem of using a workable version of the product of two distributions (a product in a ring with divisors of zero). In quantum field theory (QFT), the problem of evaluating the product of distributions with coincident point singularities is related to the asymptotic behaviour of loop integrals of propagators, becoming an obstacle that is to the essence to overcome We did it in references [1,2,3,4,5]. If one adds to it our DRgeneralization [1], the above referred to convolution happens to be both one of STDELI in momentum space and a product in a ring with divisors of zero in configuration space. That the result obtained for finite convolutions will coincide with such a term This fact translates to configuration space the product-operation on a ring with divisors of zero
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