Noncommutative Momentum and Torsional Regularization
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Noncommutative Momentum and Torsional Regularization Nikodem Popławski1 Received: 25 November 2019 / Accepted: 3 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We show that in the presence of the torsion tensor Sijk , the quantum commutation relation for the four-momentum, traced over spinor indices, is given by [pi , pj ] = 2iℏSijk pk . In the Einstein–Cartan theory of gravity, in which torsion is coupled to spin of fermions, this relation in a coordinate frame reduces to a commutation relation of noncommutative momentum space, [pi , pj ] = i𝜖ijk Up3 pk , where U is a constant on the order of the squared inverse of the Planck mass. We propose that this relation replaces the integration in the momentum space in Feynman diagrams with the summation over the discrete momentum eigenvalues. We derive a prescription for this summation that agrees with √ convergent integrals: 4p ∑∞ 𝜋∕2 sin4 𝜙 ns−3 ∫ (p2d+𝛥) → 4𝜋U s−2 l=1 ∫0 d𝜙 [sin , where n = l(l + 1) and 𝛥 does not s 𝜙+U𝛥n]s depend on p. We show that this prescription regularizes ultraviolet-divergent integrals in loop diagrams. We extend this prescription to tensor integrals. We derive a finite, gauge-invariant vacuum polarization tensor and a finite running coupling. Including loops from all charged fermions, we find a finite value for the bare electric charge of an electron: ≈ −1.22 e . This torsional regularization may therefore provide a realistic, physical mechanism for eliminating infinities in quantum field theory and making renormalization finite. Keywords Torsion · Einstein–Cartan theory · Noncommutative momentum · Regularization · Finite renormalization · Vacuum polarization
1 Introduction In quantum electrodynamics (QED) [1–8], a calculation of the amplitude for a physical process must include perturbative corrections involving Feynman diagrams [9] with closed loops of virtual particles (radiative corrections). The integration in the
* Nikodem Popławski [email protected] 1
Department of Mathematics and Physics, University of New Haven, 300 Boston Post Road, West Haven, CT 06516, USA
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resulting integrals is taken in the four-momentum space, with the magnitudes of the energy and momentum running to infinity and not restricted to the relativistic energy-momentum relation (off-shell particles). Many integrals that appear in radiative corrections are divergent, which is referred to as the ultraviolet divergence. The ultraviolet divergence results from the asymptotic, high-energy behavior of the Feynman propagators [9]. Physically, the divergence is a consequence of an incompleteness of our understanding of the physics at large energies and momenta [10]. There are three one-loop divergent diagrams in QED: vacuum polarization (a photon creating a virtual electron-positron pair which then annihilates), self-energy (an electron emits and reabsorbs a virtual photon), and vertex (an electron emits a photon, emits a second photon, and then reabsorbs the first) [11
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