Noncommutative Momentum and Torsional Regularization
Abstract
We show that in the presence of the torsion tensor S_{ij}^{k}, the quantum commutation relation for the fourmomentum, traced over spinor indices, is given by [p_{i},p_{j}] =2 i ħ S_{ij}^{k}p_{k} . In the EinsteinCartan 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, [p_{i},p_{j}] =i ɛ_{ijk}U p^{3}p_{k} , 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: ∫d/^{4}p (p^{2}+Δ) s→4 π U^{s 2∑l=1∞∫0π/2d ϕ sin4/ϕns 3 [sinϕ+U Δ n ] s, where n =√{l (l +1 ) } and Δ does not depend on p. We show that this prescription regularizes ultravioletdivergent integrals in loop diagrams. We extend this prescription to tensor integrals. We derive a finite, gaugeinvariant 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. }
 Publication:

Foundations of Physics
 Pub Date:
 July 2020
 DOI:
 10.1007/s10701020003571
 arXiv:
 arXiv:1712.09997
 Bibcode:
 2020FoPh...50..900P
 Keywords:

 Torsion;
 EinsteinCartan theory;
 Noncommutative momentum;
 Regularization;
 Finite renormalization;
 Vacuum polarization;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 14 pages