Editing Conformal field theory (section)

== Conformal symmetry ==
{{main|Conformal symmetry}}

=== Definition and Jacobian ===
For a given spacetime and metric, a conformal transformation is a transformation that preserves angles. We will focus on conformal transformations of the flat <math>d</math>-dimensional Euclidean space <math>\mathbb{R}^d</math> or  of the Minkowski space {{tmath|1= \mathbb{R}^{1,d-1} }}.

If <math>x\to f(x)</math> is a conformal transformation, the Jacobian <math>J^\mu_\nu(x) = \frac{\partial f^\mu(x)}{\partial x^\nu} </math> is of the form 
: <math>
J^\mu_\nu(x) = \Omega(x) R^\mu_\nu(x),
</math>
where <math>\Omega(x)</math> is the scale factor, and <math>R^\mu_\nu(x)</math> is a rotation (i.e. an orthogonal matrix) or Lorentz transformation.

=== 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.

=== Conformal algebra ===
The commutation relations of the corresponding Lie algebra are
: <math>\begin{align}[]
  [P_\mu, P_\nu] &=  0, \\[]
      [D, K_\mu] &= -K_\mu, \\[]
      [D, P_\mu] &=  P_\mu, \\[]
  [K_\mu, K_\nu] &=  0, \\[]
  [K_\mu, P_\nu] &=  \eta_{\mu\nu}D - iM_{\mu\nu},
\end{align}</math>
where <math>P</math> generate [[translation (physics)|translation]]s, <math>D</math> generates dilations, <math>K_\mu</math> generate special conformal transformations, and <math>M_{\mu\nu}</math> generate rotations or Lorentz transformations. The tensor <math>\eta_{\mu\nu}</math> is the flat metric.

=== Global issues in Minkowski space ===
In Minkowski space, the conformal group does not preserve [[causality (physics)|causality]]. Observables such as correlation functions are invariant under the conformal algebra, but not under the conformal group. 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.<ref name="lm75"/> 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.