UltravioletDivergenzen

UltravioletDivergenzen, also known as ultraviolet divergences, refer to infinities that arise in quantum field theory and statistical physics when integrating over high-energy, short-distance degrees of freedom. In perturbative quantum field theory, loop diagrams generate integrals over internal momenta that diverge as the momentum goes to infinity. These divergences indicate that the theory’s predictions depend on physics at arbitrarily high energies and are not well-defined without a regulator.

Regularization provides a temporary mathematical device to tame the divergences by introducing a finite scale or modification, such as a momentum cutoff, Pauli–Villars fields, or dimensional regularization, in which the number of spacetime dimensions is continued away from four. Renormalization follows, absorbing the divergent parts into redefined, physical parameters like masses and coupling constants, so that observable quantities remain finite. In renormalizable theories, only a finite set of parameters requires redefinition; in nonrenormalizable theories, an infinite set of counterterms appears, though effective field theory methods allow a controlled expansion by energy scale.

The Wilsonian perspective frames ultraviolet divergences as reflections of sensitivity to high-energy details, with low-energy physics described by an effective theory that encapsulates high-energy effects in a finite number of parameters. This viewpoint emphasizes the separation of scales and the idea that UV completions influence but do not typically determine low-energy observables within a given accuracy.

Common contexts include quantum electrodynamics, quantum chromodynamics, and scalar field theories, where consistent renormalization yields precise, finite predictions across a range of energies.