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Résumé
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.
Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.
In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that local scale invariant theories have their currents given by where is a Killing vector and is a conserved operator (the stress-tensor) of dimension exactly . For the associated symmetries to include scale but not conformal transformations, the trace has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly .
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.
While it is possible for a quantum field theory to be scale invariant but not conformally invariant, examples are rare. For this reason, the terms are often used interchangeably in the context of quantum field theory.
The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraining in two dimensions. All conformal field theories share the ideas and techniques of the conformal bootstrap.
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