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Wave function renormalization

Process to make sure wave functions can induce probability distributions

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated July 28, 2025

In quantum field theory, wave function renormalization is a rescaling (or renormalization) of quantum fields to take into account the effects of interactions.

For a noninteracting or free field, the field operator creates or annihilates a single particle with probability 1. Once interactions are included, however, this probability is modified in general to Z \neq 1.

This appears when one calculates the propagator beyond leading order; e.g. for a scalar field,

{\frac {i}{p^{2}-m_{0}^{2}+i\varepsilon }}\rightarrow {\frac {iZ}{p^{2}-m^{2}+i\varepsilon }}

(The shift of the mass from m0 to m constitutes the mass renormalization.)

One possible wave function renormalization, which happens to be scale independent, is to rescale the fields so that the Lehmann weight (Z in the formula above) of their quanta is 1. For the purposes of studying renormalization group flows, if the coefficient of the kinetic term in the action at the scale Λ is Z, then the field is rescaled by {\sqrt {Z}}. A scale dependent wave function renormalization for a field means that that field has an anomalous scaling dimension.

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This article is adapted from the Wikipedia article “Wave function renormalization”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Renormalization

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. Even if no infinities arose in loop diagrams in quantum field theory, it can be shown that it is necessary to renormalize the mass and fields appearing in the original Lagrangian.