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{{Short description|Procedure in quantum field theory that removes infinities by redefining physical parameters such as mass and charge}}

{{Quantum book backlink|Quantum field theory}}
'''Renormalization in quantum field theory''' is the systematic procedure used to handle divergences that arise in perturbative calculations by absorbing them into redefined physical parameters such as mass, charge, and field normalization.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref> It allows quantum field theories to produce finite, physically meaningful predictions.

<div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Renormalization_flow_diagram.jpg|400px]]
<div style="font-size:90%;">Renormalization: scale-dependent behavior of physical parameters and absorption of divergences into redefined quantities</div>
</div>

== Origin of divergences ==
In quantum field theory, higher-order corrections involve integrals over all possible momenta. These integrals often diverge at high energies (ultraviolet divergences).<ref name="weinberg">Weinberg, S. ''The Quantum Theory of Fields'' (1995).</ref>

For example, loop diagrams in perturbation theory can produce expressions such as:
<math>
\int^\infty \frac{d^4p}{p^2 - m^2}
</math>

which are not finite without additional procedures.

== Regularization ==
The first step in renormalization is regularization, where divergences are controlled by introducing a parameter that makes the integrals finite.

Common methods include:

* momentum cutoff
* dimensional regularization
* Pauli–Villars regularization

For instance, a momentum cutoff replaces divergent integrals with:
<math>
\int^{\Lambda} d^4p
</math>

where <math>\Lambda</math> is a finite cutoff scale.<ref name="schwartz">Schwartz, M. D. ''Quantum Field Theory and the Standard Model'' (2014).</ref>

== Renormalization procedure ==
After regularization, divergences are absorbed into redefined parameters:

* bare mass <math>m_0</math> → physical mass <math>m</math>
* bare charge <math>e_0</math> → physical charge <math>e</math>

The Lagrangian is rewritten in terms of renormalized quantities plus counterterms:
<math>
\mathcal{L} = \mathcal{L}_{\text{ren}} + \mathcal{L}_{\text{counter}}
</math>

These counterterms cancel the divergences arising in loop calculations.<ref name="peskin"/>

== Running coupling constants ==
Renormalization introduces a dependence of physical parameters on the energy scale. This is described by the renormalization group.

For example, the coupling constant becomes scale-dependent:
<math>
\alpha(\mu)
</math>

where <math>\mu</math> is the renormalization scale.

The evolution of parameters with scale is governed by equations such as:
<math>
\mu \frac{d g}{d\mu} = \beta(g)
</math>

where <math>\beta(g)</math> is the beta function.<ref name="zee">Zee, A. ''Quantum Field Theory in a Nutshell'' (2010).</ref>

== Renormalizable theories ==
A theory is called renormalizable if all divergences can be absorbed into a finite number of parameters.

Examples include:

* quantum electrodynamics (QED)
* quantum chromodynamics (QCD)

Non-renormalizable theories can still be useful as effective field theories valid at a limited energy scale.<ref name="weinberg"/>

== Physical interpretation ==
Renormalization reflects the fact that physical measurements depend on the energy scale at which they are performed.

Quantum fluctuations at different scales modify the effective values of parameters, leading to observable effects such as charge screening in QED.

== Conceptual importance ==
Renormalization is one of the central concepts of modern quantum field theory. It explains how:

* infinities are handled consistently
* physical predictions remain finite
* interactions depend on scale

It also provides the foundation for the renormalization group and modern effective field theory approaches.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum field theory (QFT) core|1}}

Physics:Quantum Renormalization in field theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template