[Paper Review] Divergence free quantum field theory using a spectral calculus of Lorentz invariant measures

John Mashford|arXiv (Cornell University)|Mar 13, 2018

Noncommutative and Quantum Gravity Theories11 references3 citations

TL;DR

This paper introduces a spectral calculus for causal Lorentz-invariant Borel measures on Minkowski space, enabling the rigorous computation of densities and spectra for quantum field theory (QFT) measures. It demonstrates that divergent integrals in QFT can be treated as well-defined Lorentz-covariant measures, yielding a finite, convergent running coupling constant that agrees with experiment and avoids the Landau pole, unlike dimensional regularization approaches.

ABSTRACT

This paper presents a spectral calculus for computing the spectrum of a causal Lorentz invariant Borel complex measure on Minkowski space, thereby enabling one to compute the density for such a measure with respect to Lebesque measure. It is proved that the convolution of arbitrary causal Lorentz invariant Borel measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation are presented. Divergent integrals in quantum field theory (QFT) are shown to have a well defined existence as Lorentz covariant measures. The case of vacuum polarization is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function the exact Uehling potential function is derived. The spectral running coupling constant is computed and is shown to converge for all energies while the integral defining the running coupling constant obtained using dimensional regularization / renormalization is shown to diverge for all non-zero energies.

Motivation & Objective

• To provide a mathematically rigorous framework for divergent integrals in QFT by treating them as Lorentz-covariant measures.
• To eliminate reliance on regularization and renormalization by defining well-posed initial equations using spectral representations of causal measures.
• To compute the vacuum polarization function and running coupling constant without divergences, offering a physically consistent alternative to dimensional regularization.
• To derive the exact Uehling potential and Lamb shift contribution using the spectral vacuum polarization function.
• To establish a foundation for reformulating QED and QCD using spectral calculus, with implications for asymptotic freedom and renormalization.

Proposed method

• Develops a spectral calculus to compute the spectrum of causal Lorentz-invariant Borel complex measures on Minkowski space.
• Uses spectral representations to derive densities with respect to Lebesgue measure, enabling physical computation.
• Applies the spectral calculus to the vacuum polarization tensor, computing its contracted spectrum and resulting vacuum polarization function.
• Establishes that convolutions and products of causal Lorentz-invariant measures exist and can be computed via spectral methods.
• Derives the Uehling potential via the Born approximation using the spectral vacuum polarization function.
• Computes the running coupling constant using the spectral vacuum polarization function, showing convergence for all energies.

Experimental results

Research questions

• RQ1 Can divergent integrals in QFT be rigorously defined as Lorentz-covariant measures without regularization?
• RQ2 Does the spectral vacuum polarization function derived via spectral calculus agree with the renormalized function obtained via dimensional regularization in the timelike domain?
• RQ3 Can the exact Uehling potential and Lamb shift be derived from a well-defined spectral vacuum polarization function?
• RQ4 Does the running coupling constant computed using the spectral method remain finite and convergent at all energies, unlike the dimensional regularization approach?
• RQ5 Can the spectral calculus be extended to self-energy and other renormalization issues in QED and QCD?

Key findings

• The spectral vacuum polarization function shows very close agreement with the renormalized vacuum polarization function obtained via dimensional regularization in the timelike domain.
• The exact Uehling potential is derived from the spectral vacuum polarization function, enabling exact computation of the Uehling contribution to the Lamb shift.
• The spectral running coupling constant is finite and convergent for all non-zero energies, in contrast to the divergent integral obtained using dimensional regularization.
• The spectral running coupling function achieves a value of approximately 1/127 at high energies, consistent with experimental observations.
• The spectral method avoids the Landau pole, providing a physically consistent running coupling without divergences.
• The spectral calculus framework provides a mathematically well-defined foundation for QFT, replacing regularization and renormalization with rigorous measure-theoretic constructions.