  We show that in the presence of the torsion tensor $S^k_{\phantom{k}ij}$, the
quantum commutation relation for the four-momentum, traced over spinor indices,
is given by $[p_i,p_j]=2i\hbar S^k_{\phantom{k}ij}p_k$. 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, $[p_i,p_j]=i\epsilon_{ijk}Up^3 p_k$, 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:
\[ \int\frac{d^4p}{(p^2+\Delta)^s}\rightarrow 4\pi U^{s-2}\sum_{l=1}^\infty
\int_0^{\pi/2} d\phi \frac{\sin^4\phi\,n^{s-3}}{[\sin\phi+U\Delta n]^s}, \]
where $n=\sqrt{l(l+1)}$ and $\Delta$ does not 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: $\approx -1.22\,e$. This torsional
regularization may therefore provide a realistic, physical mechanism for
eliminating infinities in quantum field theory and making renormalization
finite.
