Editing Scale invariance (section)

==Scale invariance in quantum field theory==
The scale-dependence of a [[quantum field theory]] (QFT) is characterised by the way its [[coupling constant|coupling parameters]] depend on the energy-scale of a given physical process. This energy dependence is described by the [[renormalization group]], and is encoded in the [[Beta function (physics)|beta-function]]s of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as [[Renormalization group|fixed points]] of the corresponding renormalization group flow.<ref>[[Jean Zinn-Justin|J. Zinn-Justin]] (2010)  Scholarpedia article [http://www.scholarpedia.org/article/Critical_Phenomena:_field_theoretical_approach "Critical Phenomena: field theoretical approach"].</ref>

===Quantum electrodynamics===
A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since [[photon]]s are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

However, in nature the electromagnetic field is coupled to charged particles, such as [[electron]]s. The QFT describing the interactions of photons and charged particles is [[quantum electrodynamics]] (QED), and this theory is not scale-invariant. We can see this from the [[beta function (physics)#Quantum electrodynamics|QED beta-function]]. This tells us that the [[electric charge]] (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles '''is''' scale-invariant, QED is '''not''' scale-invariant.

===Massless scalar field theory===
Free, massless [[scalar field (quantum field theory)|quantized scalar field theory]] has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the [[Gaussian fixed point]].

However, even though the classical massless ''φ''<sup>4</sup> theory is scale-invariant in ''D''&nbsp;=&nbsp;4, the quantized version is '''not''' scale-invariant. We can see this from the [[Beta function (physics)|beta-function]] for the coupling parameter, ''g''.

Even though the quantized massless ''φ''<sup>4</sup> is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the '''Wilson–Fisher fixed point''', below.

===Conformal field theory===
Scale-invariant QFTs are almost always invariant under the full [[conformal symmetry]], and the study of such QFTs is [[conformal field theory]] (CFT). [[operator (physics)|Operators]] in a CFT have a well-defined [[scaling dimension]], analogous to the [[scaling dimension]], ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as [[anomalous scaling dimension]]s.

===Scale and conformal anomalies===
The φ<sup>4</sup> theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be [[conformal anomaly|anomalous]]. A classically scale-invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called [[Inflation (cosmology)|cosmic inflation]], as long as the theory can be studied through [[perturbation theory]].<ref>{{cite journal|last=Salvio, Strumia|title=Agravity|journal=JHEP  |volume=2014 |issue=6|pages=080|date=2014-03-17|url=http://inspirehep.net/record/1286134|arxiv = 1403.4226|bibcode = 2014JHEP...06..080S|doi=10.1007/JHEP06(2014)080|s2cid=256010671 }}</ref>