Editing Conformal field theory (section)

=== Conformal group ===

The [[conformal group]] of Euclidean space is locally isomorphic to {{tmath|1= \mathrm{SO}(1, d + 1) }}, and of Minkowski space is {{tmath|1= \mathrm{SO}(2,d) }}. This includes translations, rotations (Euclidean) or Lorentz transformations (Minkowski), and dilations i.e. scale transformations 
: <math>
x^\mu \to \lambda x^\mu.
</math>

This also includes special conformal transformations. For any translation {{tmath|1= T_a(x) = x + a }}, there is a '''special conformal transformation'''
: <math>
S_a = I \circ T_a \circ I,
</math>
where <math> I </math> is the '''inversion''' such that 
: <math>
I\left(x^\mu\right) = \frac{x^\mu}{x^2}.
</math>

In the sphere {{tmath|1= S^d = \mathbb{R}^d \cup \{\infty\} }}, the inversion exchanges <math>0</math> with {{tmath|1= \infty }}. Translations leave <math>\infty</math> fixed, while special conformal transformations leave <math>0</math> fixed.