Studying Quantum Field Theory

TL;DR

This paper reviews various aspects of quantum field theory, including the properties of Feynman amplitudes, the role of conformal group representations, and recent advances in renormalization techniques, highlighting interdisciplinary collaborations.

Contribution

It synthesizes old and new insights on quantum fields, emphasizing the Epstein-Glaser renormalization approach and the significance of conformal group representations in gauge theories.

Findings

Enhanced understanding of Feynman amplitude properties

Application of Epstein-Glaser method to divergent graphs

Interdisciplinary outlook on perturbative QFT developments

Abstract

The paper puts together some loosely connected observations, old and new, on the concept of a quantum field and on the properties of Feynman amplitudes. We recall, in particular, the role of (exceptional) elementary induced representations of the quantum mechanical conformal group SU(2,2) in the study of gauge fields and their higher spin generalization. A recent revival of the (Bogolubov-)Epstein-Glaser approach to position space renormalization is reviewed including an application to the calculation of residues of primitively divergent graphs. We end up with an optimistic outlook of current developments of analytic methods in perturbative QFT which combine the efforts of theoretical physicists, algebraic geometers and number theorists.