Beta function (physics): Difference between revisions

Revision as of 21:01, 7 June 2025

Latest revision as of 18:22, 9 June 2025

(4 intermediate revisions by one other user not shown)

Line 4:

{{quantum field theory}}

In [[theoretical physics]], specifically [[quantum field theory]], a '''beta function''' or '''Gell-Mann–Low function''', ''β(g)'', encodes the dependence of a [[Coupling constant|coupling parameter]], ''g'', on the [[energy scale]], ''μ'', of a given physical process described by [[quantum field theory]].

It is defined by the '''Gell-Mann–Low equation'''<ref>{{Cite book |last=Tsvelik |first=Alexei M. |url=https://www.google.fr/books/edition/Quantum_Field_Theory_in_Condensed_Matter/78t7iDTth2YC?hl=en&gbpv=1&dq=%22Gell-Mann-Low+equation%22+renormalization&pg=PA61&printsec=frontcover |title=Quantum Field Theory in Condensed Matter Physics |date=2007-01-18 |publisher=Cambridge University Press |isbn=978-0-521-52980-8 |language=en}}</ref> or '''renormalization group equation,''' given by

It is defined by the '''Gell-Mann–Low equation''' or '''renormalization flow equation,''' given by

:: <math>\beta(g) = \mu \frac{\partial g}{\partial \mu} = \frac{\partial g}{\partial \ln(\mu)} ~,</math>

and, because of the underlying [[renormalization group]], it has no explicit dependence on ''μ'', so it only depends on ''μ'' implicitly through ''g''.

This dependence on the energy scale thus specified is known as the [[Coupling constant#Running coupling|running]] of the coupling parameter, a fundamental

feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of beta function was first introduced by [[Ernst Stueckelberg]] and [[André Petermann]] in 1953,<ref>{{cite journal |author1-link=Ernst Stueckelberg |last1=Stueckelberg |first1=E.C.G. |author2-link=André Petermann |first2=A. |last2=Petermann |year=1953 |url=https://www.e-periodica.ch/cntmng?pid=hpa-001:1953:26::894 |title=La renormalisation des constants dans la théorie de quanta |journal=Helv. Phys. Acta |volume=26 |pages=499–520 |language=FR}}</ref> and independently postulated by [[Murray Gell-Mann]] and [[Francis E. Low]] in 1954.<ref>{{Cite journal |last=Fraser |first=James D. |date=2021-10-01 |title=The twin origins of renormalization group concepts |url=https://www.sciencedirect.com/science/article/pii/S0039368121001126 |journal=Studies in History and Philosophy of Science Part A |volume=89 |pages=114–128 |doi=10.1016/j.shpsa.2021.08.002 |issn=0039-3681}}</ref>

== History ==

{{Excerpt|Renormalization group#Beginnings}}

==Scale invariance==

If the beta functions of a [[quantum field theory]] (QFT) vanish, usually at particular values of the coupling parameters, then the theory is said to be [[Scale invariance|scale-invariant]]. Almost all scale-invariant QFTs are also [[Conformal symmetry|conformally invariant]]. The study of such theories is [[conformal field theory]].

The coupling parameters of a quantum field theory can run even if the corresponding [[classical field theory]] is scale-invariant. In this case, the non-zero beta function tells us that the classical scale invariance is [[Conformal anomaly|anomalous]].

Line 107 ⟶ 110:

==References==

{{notelist}}

{{Reflist}}