Editing Conformal field theory (section)

== Scale invariance vs conformal invariance ==
In [[quantum field theory]], [[scale invariance]] is a common and natural symmetry, because any fixed point of the [[renormalization group]] is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions<ref name="Polchinski88"/> to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have their currents given by <math>T_{\mu \nu} \xi^\nu</math> where <math>\xi^\nu</math> is a [[Killing vector]] and <math>T_{\mu \nu}</math> is a conserved operator (the stress-tensor) of dimension exactly {{tmath|1= d }}. For the associated symmetries to include scale but not conformal transformations, the trace <math>T_\mu{}^\mu</math> has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly {{tmath|1= d - 1 }}.

Under some assumptions it is possible to completely rule out this type of non-renormalization and hence prove that scale invariance implies conformal invariance in a quantum field theory, for example in [[unitarity (physics)|unitary]] compact conformal field theories in two dimensions.

While it is possible for a [[quantum field theory]] to be [[scale invariance|scale invariant]]  but not conformally invariant, examples are rare.<ref>One physical example is the theory of elasticity in two and three dimensions (also known as the theory of a vector field without gauge invariance). See 
{{cite journal|author=Riva V, Cardy J|title=Scale and conformal invariance in field theory: a physical counterexample|journal=	Phys. Lett. B|volume=622|issue=3–4|pages=339–342|year=2005|doi=10.1016/j.physletb.2005.07.010|arxiv=hep-th/0504197|bibcode = 2005PhLB..622..339R |s2cid=119175109}}</ref> For this reason, the terms are often used interchangeably in the context of quantum field theory.