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Noncommutative Momentum and Torsional Regularization

Abstract

We show that in the presence of the torsion tensor \(S^k_{ij}\), the quantum commutation relation for the four-momentum, traced over spinor indices, is given by \([p_i,p_j]=2i\hbar S^k_{ij}p_k\). 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: \(\int \frac{d^4p}{(p^2+\varDelta )^s}\rightarrow 4\pi U^{s-2}\sum _{l=1}^\infty \int _0^{\pi /2} d\phi \frac{\sin ^4\phi \,n^{s-3}}{[\sin \phi +U\varDelta n]^s}\), where \(n=\sqrt{l(l+1)}\) and \(\varDelta \) 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.

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Notes

  1. The contraction of the curvature tensor in its first two indices vanishes if spacetime satisfies the metric compatibility of the affine connection: \(\nabla _i g_{jk}=0\). This condition holds in GR and EC.

  2. We assume that the Feynman propagator for a Dirac particle with four-momentum \(p_\mu \) and mass m is \(\tilde{S}_\text {F}(p)=i(\gamma ^\mu p_\mu +m)/(p^2-m^2+i\epsilon )\) and for a photon (in the Feynman gauge) is \(\tilde{D}^{\mu \nu }_\text {F}(p)=-ig^{\mu \nu }/(p^2+i\epsilon )\), where \(\gamma ^\mu \) are the Dirac matrices in flat spacetime (with \(g_{\mu \nu }\) being the Minkowski metric tensor): \(\gamma ^{(\mu }\gamma ^{\nu )}=g^{\mu \nu }I_4\), \(p^2=p_\mu p^\nu \), and \(\epsilon \rightarrow 0^{+}\) [1,2,3,4,5,6,7,8]. The Fourier transform of a propagator in the four-momentum representation, with the exponential factor \(e^{-ip_\mu (x-y)^\mu }\), gives the position representation of the propagator (\(S_\text {F}(x-y)\) or \(D_\text {F}(x-y)\)) describing a particle moving from one point x in spacetime to another point y that is infinitesimally close to x. In the Fourier transform, the integration over the four-momentum must be replaced with the summation over the four-momentum eigenvalues, according to the presented prescription. A propagator describing the motion between two points separated by a finite distance can be constructed as a sequence of infinitesimal propagators using the Riemann normal coordinates, which depend on the curvature and torsion tensors [100].

  3. Our regularization prescription does not introduce an explicit momentum space cutoff \(\varLambda \) and thus does not have ultraviolet-infrared divergence mixing that occurs for field theories with noncommutative coordinate space that do have such a cutoff [30, 31].

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Acknowledgements

I am grateful to my parents Bożenna Popławska and Janusz Popławski for their support, and to Gabe Unger for inspiring my research. This work was funded by the University Research Scholar program at the University of New Haven.

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Popławski, N. Noncommutative Momentum and Torsional Regularization. Found Phys 50, 900–923 (2020). https://doi.org/10.1007/s10701-020-00357-1

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Keywords

  • Torsion
  • Einstein–Cartan theory
  • Noncommutative momentum
  • Regularization
  • Finite renormalization
  • Vacuum polarization