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### Description

We show that in the presence of the torsion tensor $S^k_{\phantom{k}ij}$, whose existence is required by the consistency of the conservation law for the total angular momentum of a Dirac particle in curved spacetime with relativistic quantum mechanics, the quantum commutation relation for the four-momentum is given by $[p_i,p_j]=2i\hbar S^k_{\phantom{k}ij}p_k$.

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:

$\int\frac{d^4p}{(p^2+\Delta)^s}\rightarrow 4\pi U^{s-2}\sum_{l=1}^\infty \int_0^{\pi/2} d\phi \frac{\sin^4\phi\,n^{s-3}}{[\sin\phi+U\Delta n]^s}$,

where $n=\sqrt{l(l+1)}$ and $1/\sqrt{U}$ is a constant on the order of the Planck mass, determined by the Einstein-Cartan theory of gravity.

We show that this prescription regularizes ultraviolet-divergent integrals in loop diagrams.

We extend this prescription to tensor integrals and apply it to vacuum polarization.

We derive a finite, gauge-invariant vacuum polarization tensor and a finite running coupling that agrees with the low-energy limit of the standard quantum electrodynamics.

Including loops from all charged fermions, we find a finite value for the bare electric charge of an electron: $\approx -1.22\,e$.

Torsional regularization, originating from the noncommutativity of the momentum and spin-torsion coupling, therefore provides a realistic, physical mechanism for eliminating infinities in quantum field theory: quantum electrodynamics with torsion is ultraviolet complete.

arXiv | 1712.09997 |
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