Gauge invariance, gauge field quantization, and renormalization in autoregularization

Autoregularization, a new divergence-free framework for calculating scattering amplitudes, uses a Lorentz-invariant scale harvested from the kinematics of a scattering process to regularize the amplitude of the process [N. Prabhu, J. Phys. Commun. 7, 115002 (2023).]. Preliminary validation studies show that autoregularization's predictions are in good agreement with experimental data—across several scattering processes and a wide range of energy scales. Further, ab initio tree-level calculation of the vacuum energy density of the free fields in the Standard Model, using autoregularization, is shown to yield a value that is smaller than the current estimate of the cosmic critical density. In this paper, we prove that the scattering amplitudes in QED, calculated using autoregularization, are gauge invariant. Our proof, which is valid both for autoregularization and current theory, is stronger in that it shows the amplitude of every Feynman diagram is gauge invariant in contrast to previous proofs, which establish gauge invariance only for sum of amplitudes of Feynman diagrams of a process. Next, we show that—unlike in the standard quantization framework, which requires modification of both the quantization framework itself as well as the Lagrangian in order to quantize gauge fields in covariant gauge—in autoregularization the gauge field in QED can be quantized, in covariant gauge, without modifying the standard quantization procedure or the Lagrangian and without introducing the ghost field. Finally, we illustrate renormalization based on autoregularization up to 1-loop in ${\phi }^4$ theory. Since perturbative corrections are finite in autoregularization, the counterterms are not designed to remove divergences but to implement renormalization prescriptions at every order of perturbation. We also derive the renormalization group equation (RGE). Unlike in some regularization schemes (such as dimensional regularization), in which the physical meaning of the fictitious scale introduced by regularization is unclear, in autoregularization the scale in RGE has a transparent physical meaning—it is the Lorentz-invariant kinematic scale of the scattering process of interest. The increasing simplifications resulting within autoregularization and the agreement between its predictions and experimental data, together with the underlying thermodynamic argument, which shows that the framework is essential for a complete description of quantum fields, all converge to suggest that autoregularization provides the proper framework for the description of quantum fields.