The mathematical justification for the renormalization of infinite series in Quantum Field Theory (QFT) lies in the concept of divergent series and the need to extract meaningful and finite physical results from these series. Renormalization is a technique used to remove infinities that arise in certain calculations of QFT and provide well-defined, finite predictions.
In QFT, various physical observables, such as particle masses and interaction strengths, are calculated using perturbation theory. Perturbation theory involves expanding quantities in a series called a Feynman diagram series, which represents the contributions of different particle interactions.
However, in many cases, these Feynman diagram series contain divergent terms that result in infinite values. Divergences arise due to the presence of ultraviolet (UV) divergences and infrared (IR) divergences.
UV divergences arise from the integration over high-energy virtual particles and can lead to infinities in the calculations. IR divergences arise from the integration over low-energy or long-wavelength virtual particles and can also produce divergent results.
To deal with these divergences and obtain meaningful physical results, renormalization is applied. Renormalization involves introducing counterterms into the theory to cancel out the infinities. These counterterms are chosen in such a way that the infinities in the calculations are absorbed into them, resulting in finite, observable quantities.
The renormalization process typically involves three steps:
Regularization: A regularization scheme is employed to regulate the divergent integrals by introducing a parameter, often denoted as ε, that controls the behavior of the integral. Common regularization methods include dimensional regularization or cutoff regularization.
Calculation of divergent terms: The divergent terms in the series are isolated and identified. These divergences can be expressed in terms of the counterterms, which are additional parameters or terms that are introduced into the theory.
Absorption of divergences: The counterterms are adjusted to cancel out the divergences and make the final result finite. The values of the counterterms are usually determined by matching calculated observables with experimental measurements.
The mathematical justification for renormalization comes from the fact that the counterterms can be interpreted as the effective parameters of the theory that take into account the effects of all possible interactions, including those at higher energy scales. By choosing appropriate counterterms, the divergences are absorbed into them, resulting in finite and physically meaningful predictions.
It's worth noting that renormalization is a highly nontrivial and intricate procedure, and its justification and mathematical foundations have been the subject of extensive study and development in the field of QFT.