Compliant group

The conformal group of a semiriemannian manifold is the (component of the one of the) Lie group of conformal mappings of the manifold in itself. It is thus a subgroup of the diffeomorphism group and contains the isometric group of the manifold.

For physics, the conformal groups of manifolds with flat metrics are particularly important. For the Euclidean space of dimension d the conformal group is isomorphic to the group SO (d + 1,1) . Maxwell's electrodynamics is not only invariant under the Lorentz group , but also under a conformal 15-parameter group of spherical wave transformations . In solid state physics and string theory , systems occur that are scale-invariant , at least to a good approximation . These systems are quantum-physically described with conformal quantum field theories that are invariant under the conformal group.

For string theory, the two-dimensional case is particularly interesting, where space then represents the world surface of a string. In the two-dimensional plane case with the Minkowski metric, the Lie algebra for the conformal group contains the infinite-dimensional Witt algebra of the polynomial vector fields on the unit circle line (see conformal figure ).

literature

• R. Penrose , W. Rindler : Spinors and space-time. Volume 2: Spinor and Twistor Methods in Space Time Geometry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, ISBN 0-521-34786-6
• J. Dieudonne : Linear Algebra and Geometry , Boston, MA Houghton Mifflin, 1969, ISBN 0-901665-01-0
• F. Scheck: Theoretical Physics 1. , Springer, 2007, ISBN 3-540-71377-8
• Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory , Springer-Verlag , Berlin Heidelberg , 1997 . ISBN 3-540-61753-1 , 2nd edition 2008, ISBN 978-3-540-68625-5 .