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History Related: Original Paper
On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics †
Article first published online: 3 SEP 2008
Copyright © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Annalen der Physik
Volume 17, Issue 9-10, pages 631–690, September-October 2008
How to Cite
Kastrup, H.A. (2008), On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics . Ann. Phys., 17: 631–690. doi: 10.1002/andp.200810324
Dedicated to the memory of Julius Wess (1934–2007), colleague and friend for many years.
- Issue published online: 3 SEP 2008
- Article first published online: 3 SEP 2008
- Manuscript Revised: 12 AUG 2008
- Manuscript Accepted: 12 AUG 2008
- Manuscript Received: 11 JUL 2008
- Stereographic projection;
- transformation by reciprocal radii;
- conformally invariant quantum field theories;
- superconformal quantum systems.
The historical developments of conformal transformations and symmetries are sketched: Their origin from stereographic projections of the globe, their blossoming in two dimensions within the field of analytic complex functions, the generic role of transformations by reciprocal radii in dimensions higher than two and their linearization in terms of polyspherical coordinates by Darboux, Weyl's attempt to extend General Relativity, the slow rise of finite dimensional conformal transformations in classical field theories and the problem of their interpretation, then since about 1970 the rapid spread of their acceptance for asymptotic and structural problems in quantum field theories and beyond, up to the current AdS/CFT conjecture. The occasion for the present article: hundred years ago Bateman and Cunningham discovered the form invariance of Maxwell's equations for electromagnetism with respect to conformal space-time transformations.