NichtRenormierbarkeit

Nichtrenormierbarkeit, commonly written Nichtrenormierbarkeit in German, denotes the property of a quantum field theory that cannot be renormalized within the standard perturbative framework. In such theories, the divergences that arise at intermediate steps of perturbation theory cannot be absorbed into a finite number of redefined parameters, requiring an infinite tower of counterterms. Consequently, if treated as a fundamental theory, its predictive power would be lost at high energies. In modern usage, Nichtrenormierbarkeit is often discussed in contrast to renormalizable theories, where a finite set of couplings suffices to absorb divergences.

Power counting and operator dimensions provide practical criteria: in four spacetime dimensions, the renormalizability of a local interaction is determined by the mass dimension of its coupling. Operators with dimension greater than four are non-renormalizable in perturbation theory because their associated couplings have negative mass dimension. Classic examples include the Fermi four-fermion interaction, a dimension-6 operator with coupling ~ 1/Λ^2, and Einstein gravity, for which Newton’s constant has negative mass dimension and perturbative loops generate infinities that would require an infinite series of counterterms.

Modern perspective treats non-renormalizable theories as effective field theories valid up to a cutoff scale. The Wilsonian view regards renormalizability as a statement about low-energy behavior, while the underlying ultraviolet (UV) completion—such as a quantum theory of gravity or string theory—provides a finite description at high energies. Concepts like asymptotic safety explore whether a nontrivial UV fixed point can render a non-renormalizable theory predictive at all scales.