Editing Quantum field theory (section)

====Renormalization group====
{{Main|Renormalization group}}

The [[renormalization group]], developed by [[Kenneth G. Wilson|Kenneth Wilson]], is a mathematical apparatus used to study the changes in physical parameters (coefficients in the Lagrangian) as the system is viewed at different scales.{{r|peskin|page1=393}} The way in which each parameter changes with scale is described by its [[beta function (physics)|''β'' function]].{{r|peskin|page1=417}} Correlation functions, which underlie quantitative physical predictions, change with scale according to the [[Callan–Symanzik equation]].{{r|peskin|page1=410–411}}

As an example, the coupling constant in QED, namely the [[elementary charge]] {{math|''e''}}, has the following ''β'' function:
:<math>\beta(e) \equiv \frac{1}{\Lambda}\frac{de}{d\Lambda} = \frac{e^3}{12\pi^2} + O\mathord\left(e^5\right),</math>
where {{math|Λ}} is the energy scale under which the measurement of {{math|''e''}} is performed. This [[differential equation]] implies that the observed elementary charge increases as the scale increases.<ref>{{cite arXiv |last=Fujita |first=Takehisa |eprint=hep-th/0606101 |title=Physics of Renormalization Group Equation in QED |date=2008-02-01 }}</ref> The renormalized coupling constant, which changes with the energy scale, is also called the running coupling constant.{{r|peskin|page1=420}}

The coupling constant {{math|''g''}} in [[quantum chromodynamics]], a non-Abelian gauge theory based on the symmetry group {{math|[[special unitary group|SU(3)]]}}, has the following ''β'' function:
:<math>\beta(g) \equiv \frac{1}{\Lambda}\frac{dg}{d\Lambda} = \frac{g^3}{16\pi^2}\left(-11 + \frac 23 N_f\right) + O\mathord\left(g^5\right),</math>
where {{math|''N<sub>f</sub>''}} is the number of [[quark]] [[flavour (particle physics)|flavours]]. In the case where {{math|''N<sub>f</sub>'' ≤ 16}} (the Standard Model has {{math|''N<sub>f</sub>'' {{=}} 6}}), the coupling constant {{math|''g''}} decreases as the energy scale increases. Hence, while the strong interaction is strong at low energies, it becomes very weak in high-energy interactions, a phenomenon known as [[asymptotic freedom]].{{r|peskin|page1=531}}

[[Conformal field theories]] (CFTs) are special QFTs that admit [[conformal symmetry]]. They are insensitive to changes in the scale, as all their coupling constants have vanishing ''β'' function. (The converse is not true, however — the vanishing of all ''β'' functions does not imply conformal symmetry of the theory.)<ref>{{Cite journal |last1=Aharony |first1=Ofer |last2=Gur-Ari |first2=Guy |last3=Klinghoffer |first3=Nizan |arxiv=1501.06664 |title=The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants |journal=Journal of High Energy Physics |volume=2015 |issue=5 |pages=31 |date=2015-05-19 |bibcode=2015JHEP...05..031A |doi=10.1007/JHEP05(2015)031 |s2cid=115167208 }}</ref> Examples include [[string theory]]<ref name="polchinski1">{{cite book |last=Polchinski |first=Joseph |date=2005 |title=String Theory |volume=1 |publisher=Cambridge University Press |isbn=978-0-521-67227-6 |author-link=Joseph Polchinski }}</ref> and [[N = 4 supersymmetric Yang–Mills theory|{{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory]].<ref>{{cite arXiv |last=Kovacs |first=Stefano |eprint=hep-th/9908171 |title={{math|''N'' {{=}} 4}} supersymmetric Yang–Mills theory and the AdS/SCFT correspondence |date=1999-08-26 }}</ref>

According to Wilson's picture, every QFT is fundamentally accompanied by its energy cut-off {{math|Λ}}, ''i.e.'' that the theory is no longer valid at energies higher than {{math|Λ}}, and all degrees of freedom above the scale {{math|Λ}} are to be omitted. For example, the cut-off could be the inverse of the atomic spacing in a condensed matter system, and in elementary particle physics it could be associated with the fundamental "graininess" of spacetime caused by quantum fluctuations in gravity. The cut-off scale of theories of particle interactions lies far beyond current experiments. Even if the theory were very complicated at that scale, as long as its couplings are sufficiently weak, it must be described at low energies by a renormalizable [[effective field theory]].{{r|peskin|page1=402–403}} The difference between renormalizable and non-renormalizable theories is that the former are insensitive to details at high energies, whereas the latter do depend on them.{{r|shifman|page1=2}} According to this view, non-renormalizable theories are to be seen as low-energy effective theories of a more fundamental theory. The failure to remove the cut-off {{math|Λ}} from calculations in such a theory merely indicates that new physical phenomena appear at scales above {{math|Λ}}, where a new theory is necessary.{{r|zee|page1=156}}