# Wave function renormalization

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Wave_function_renormalization
> Markdown URL: https://mediated.wiki/source/Wave_function_renormalization.md
> Source: https://en.wikipedia.org/wiki/Wave_function_renormalization
> Source revision: 1303020961
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Process to make sure wave functions can induce probability distributions}}
{{Use American English|date=January 2019}}
{{refimprove |date=June 2025}}
In [quantum field theory](/source/quantum_field_theory), '''wave function renormalization''' is a rescaling (or [renormalization](/source/renormalization)) of quantum fields to take into account the effects of interactions.<ref>{{cite book|url=https://books.google.com/books?id=cGa5Q9BeMNUC&dq=%22Wave+function+renormalization%22+-wikipedia&pg=PA115|pages=113–117|title=Introduction to the Functional Renormalization Group|first1=Peter |last1=Kopietz|first2=Lorenz |last2=Bartosch|first3=Florian |last3=Schütz |year=2010|isbn=9783642050930|publisher=Springer|access-date=June 9, 2025}}</ref><ref>{{cite book|pages=384–385|url=https://books.google.com/books?id=rB_wdTnfTmoC&dq=%22Wave+function+renormalization%22+-wikipedia&pg=PA385|title=An Introduction to Quantum Field Theory|first=George |last=Sterman|year=1993|isbn=9780521311328|publisher=Cambridge University Press |access-date=June 9, 2025}}</ref>

For a noninteracting or [free field](/source/free_field), the [field operator](/source/field_operator) creates or annihilates a single particle with [probability](/source/probability) 1. Once interactions are included, however, this probability is modified in general to ''Z'' <math>\neq</math> 1.<ref>{{cite book|url=https://books.google.com/books?id=yykTDAAAQBAJ&dq=%22Wave+function+renormalization%22+-wikipedia&pg=PA137|pages=137–138|title=A Modern Introduction to Quantum Field Theory|first=Michele |last=Maggiore|year= 2005|isbn=9780198520733|publisher=Oxford University Press |access-date=June 9, 2025}}</ref>

This appears when one calculates the [propagator](/source/propagator) beyond [leading order](/source/leading-order); e.g. for a scalar field,

:<math>\frac{i}{p^2 - m_0^2 + i \varepsilon} \rightarrow \frac{i Z}{p^2 - m^2 + i \varepsilon}</math>

(The shift of the mass from ''m''<sub>0</sub> to m constitutes the [mass renormalization](/source/mass_renormalization).)

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

==See also==
*[Renormalization](/source/Renormalization)

==References==
{{reflist}}

{{DEFAULTSORT:Wave Function Renormalization}}
Category:Renormalization group

{{Quantum-stub}}

---
Adapted from the Wikipedia article [Wave function renormalization](https://en.wikipedia.org/wiki/Wave_function_renormalization) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Wave_function_renormalization?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.