Bulletin of the Belgian Mathematical Society - Simon Stevin

On the Geometry of the Conformal Group in Spacetime

N. G. Gresnigt and P. F. Renaud

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Abstract

The study of the conformal group in $R^{p,q}$ usually involves the conformal compactification of $R^{p,q}$. This allows the transformations to be represented by linear transformations in $R^{p+1,q+1}$. So, for example, the conformal group of Minkowski space, $R^{1,3}$ leads to its isomorphism with $SO(2,4)$. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. This is particularly a problem in $R^{1,3}$ where we might well prefer to keep the geometric nature of the various types of transformations in sight. In this note, we show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra. Although we are mostly concerned with the conformal group of Minkowski space (where the geometry is clearest), generalization to the general case is straightforward.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2 (2010), 193-200.

Dates
First available in Project Euclid: 26 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1274896198

Digital Object Identifier
doi:10.36045/bbms/1274896198

Mathematical Reviews number (MathSciNet)
MR2667386

Zentralblatt MATH identifier
1192.22011

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 17B15: Representations, analytic theory 22E70: Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10]

Keywords
Clifford algebra conformal group Minkowski space

Citation

Gresnigt, N. G.; Renaud, P. F. On the Geometry of the Conformal Group in Spacetime. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 193--200. doi:10.36045/bbms/1274896198. https://projecteuclid.org/euclid.bbms/1274896198


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