Critical exponents and conformal transformations

2 Critical exponents and conformal transformations 4.2.1 Conformal transformations and field theory 

As we saw previously, the correlation functions of a magnetic system remain invariant for translations and rotations. When the system becomes critical a new invariance appears for dilatations. Thus, the simplest correlation function associated with a field component q p(f) reads 

This invariance is trivial, but as was shown by Polyakov in 1970,51 a critical system is also invariant for transformations belonging to the conformal group. 

The conformal group consists of a set of point transformations preserving angles. In other words, if a conformal transformation transforms a point M into M, the neighbourhood of M can be deduced from the neighbourhood of M by local rotation and dilatation. 

The special conformal group is a subgroup of the conformal group and it is generated by the following transformations (r - r ) 

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