Abstract
A concise discussion of a \(3+1\)-dimensional derivative coupling model, in which a massive Dirac field couples to the four-gradient of a massless scalar field, is given in order to elucidate the role of different concepts in quantum field theory like the regularization of quantum fields as operator-valued distributions, correlation distributions, locality, causality, and field operator gauge transformations.
Highlights
Quantum field theory (QFT) is plagued by many conceptual problems
Whereas ultraviolet divergences are rather related to the short distance behavior of a QFT, integrals over infinite space-time result in some sort of infrared difficulties when massless fields are involved, depending on the approach that was chosen to formulate the theory
One may conclude that even a physically trivial interaction may enforce a formalism which goes beyond the well-behaved setting of Schwartz distributions, which lies at the basis of perturbatively renormalizable QFTs
Summary
It has hitherto been impossible to prove the existence of a non-trivial QFT in four space-time dimensions. It is notoriously difficult for perturbative QFTs to establish convergence of expansions of the S-matrix and related observable quantities. Despite this fact, perturbative QFT has been very successful in predicting measurable quantities in elementary particle physics. Whereas ultraviolet divergences are rather related to the short distance behavior of a QFT, integrals over infinite space-time result in some sort of infrared difficulties when massless fields are involved, depending on the approach that was chosen to formulate the theory. The derivative coupling model, which serves thereby as a trivial, but stunning example for this fact, will be discussed in two different versions
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