Conformal field theory

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Conformal field theories ( English Conformal Field Theory , abbreviation CFT ) are quantum field theories or statistical field theories that are invariant under any conformal transformations . Most renormalizable field theories fall into this category at their critical points , because the system has scale invariance there (described by the renormalization group ), see also Figure 1.

Fig. 1: A conformal transformation can be described locally by a translation, a rotation and a change in the scale. For a physical system with short-range interaction (think of a crystal, for example), translation and rotation are insignificant. This illustrates why scale-invariant (critical) systems with short-range interaction usually also have conformal invariance.

The group of conformal transformations of the 2-dimensional Euclidean space is generated by an infinite-dimensional algebra of generators . This high degree of symmetry enables a classification of 2-dimensional field theories and sometimes an exact solution. For this reason, the critical exponents of 2-dimensional systems are often rational numbers (examples: Ising model , isotropic percolation ).

Other applications are found in the string theory , as a string in spacetime spans a two-dimensional surface.

For d- dimensional Euclidean spaces with d> 2, however , the algebra of the generators is only (d + 1) (d + 2) / 2 -dimensional, and the conformal invariance is less useful here.

See also

literature

  • Malte Henkel: Conformal invariance and critical phenomena. Springer, Berlin a. a. 1999, ISBN 3-540-65321-X ( Texts and Monographs in Physics ).
  • John Cardy : Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge u. a. 1996, ISBN 0-521-49959-3 ( Cambridge Lecture Notes in Physics 5).

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