In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.
other commutators vanish. Here is the Minkowski metric tensor.
Additionally, is a scalar and is a covariant vector under the Lorentz transformations.
The special conformal transformations are given by
where is a parameter describing the transformation. This special conformal transformation can also be written as , where
which shows that it consists of an inversion, followed by a translation, followed by a second inversion.
In more than two dimensions, Euclidean conformal transformations map circles to circles, and hyperspheres to hyperspheres with a straight line considered a degenerate circle and a hyperplane a degenerate hypercircle.
Conformal field theory
The largest possible[why?] global symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. Such theories are known as conformal field theories.
Second-order phase transitions
One particular application is to critical phenomena in systems with local interactions. Fluctuations[clarification needed] in such systems are conformally invariant at the critical point. That allows for classification of universality classes of phase transitions in terms of conformal field theories
Conformal invariance is also present in two-dimensional turbulence at high Reynolds number.
Many theories studied in high-energy physics admit the conformal symmetry[why?]. A famous[why?] example is the N=4 supersymmetric Yang-Mills theory. Also, the worldsheet in string theory is described by a two-dimensional conformal field theory coupled to the two-dimensional gravity.
- Conformal group
- Coleman–Mandula theorem
- Renormalization group
- Scale invariance
- Superconformal algebra
- Conformal Killing equation
- Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. p. 98. ISBN 978-0-387-94785-3.
- Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. p. 97. ISBN 978-0-387-94785-3.
- Juan Maldacena; Alexander Zhiboedov (2013). "Constraining conformal field theories with a higher spin symmetry". Journal of Physics A: Mathematical and Theoretical. 46 (21): 214011. arXiv:1112.1016. Bibcode:2013JPhA...46u4011M. doi:10.1088/1751-8113/46/21/214011.