Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.
Conformal symmetry is stronger than scale invariance, and one needs additional assumptions[2] to argue that it should appear in nature.
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 unitary compact conformal field theories in two dimensions.
The development of conformal field theory has been earlier and deeper in the two-dimensional case, in particular after the 1983 article by Belavin, Polyakov and Zamolodchikov.
Strictly speaking, it is possible for a two-dimensional conformal field theory to be local (in the sense of possessing a stress-tensor) while still only exhibiting invariance under the global
Unless the central charge vanishes, there cannot exist a state that leaves the entire infinite dimensional conformal symmetry unbroken.
As shown by Lüscher and Mack, it is possible to restore the invariance under the conformal group by extending the flat Minkowski space into a Lorentzian cylinder.
[8] The original Minkowski space is conformally equivalent to a region of the cylinder called a Poincaré patch.
In the cylinder, global conformal transformations do not violate causality: instead, they can move points outside the Poincaré patch.
behaves as a highest-weight vector in an induced representation of the conformal group from the subgroup generated by dilations and rotations.
The invariance of correlation functions under conformal transformations severely constrain their dependence on field positions.
In the case of two- and three-point functions, that dependence is determined up to finitely many constant coefficients.
If the dilation operator is diagonalizable (i.e. if the theory is not logarithmic), there exists a basis of primary fields such that two-point functions are diagonal, i.e.
Similarly, two-point functions of non-scalar primary fields are determined up to a coefficient, which can be set to one.
This is because in conformal field theory, the operator product expansion's radius of convergence is finite (i.e. it is not zero).
Furthermore, if the space is Euclidean, the OPE must be commutative, because correlation functions do not depend on the order of the fields, i.e.
, the same four-point function is written in terms of t-channel conformal blocks or u-channel conformal blocks, The equality of the s-, t- and u-channel decompositions is called crossing symmetry: a constraint on the spectrum of primary fields, and on the three-point structure constants.
In Euclidean space, conformal blocks are single-valued real-analytic functions of the positions except when the four points
lie on a circle but in a singly-transposed cyclic order [1324], and only in these exceptional cases does the decomposition into conformal blocks not converge.
From the spectrum and OPE coefficients (collectively referred to as the CFT data), correlation functions of arbitrary order can be computed.
A conformal field theory is unitary if its space of states has a positive definite scalar product such that the dilation operator is self-adjoint.
In Euclidean conformal field theories, unitarity is equivalent to reflection positivity of correlation functions: one of the Osterwalder-Schrader axioms.
For scalar fields, the unitarity bound is[11] In a unitary theory, three-point structure constants must be real, which in turn implies that four-point functions obey certain inequalities.
The name comes from the fact that if a 2D conformal field theory is also a sigma model, it will satisfy these conditions if and only if its target space is compact.
The corresponding mean field theory is then non-local (e.g. it does not have a conserved stress tensor operator).
Continuous phase transitions (critical points) of classical statistical physics systems with D spatial dimensions are often described by Euclidean conformal field theories.
If the classical statistical physics system is reflection positive, the corresponding Euclidean CFT describing its critical point will be unitary.
Apart from translation and rotation invariance, an additional necessary condition for this to happen is that the dynamical critical exponent z should be equal to 1.
CFTs describing such quantum phase transitions (in absence of quenched disorder) are always unitary.
Coordinates of the spacetime in which string theory lives correspond to bosonic fields of this CFT.