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april 2010 On the Geometry of the Conformal Group in Spacetime
N. G. Gresnigt, P. F. Renaud
Bull. Belg. Math. Soc. Simon Stevin 17(2): 193-200 (april 2010). DOI: 10.36045/bbms/1274896198


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.


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N. G. Gresnigt. P. F. Renaud. "On the Geometry of the Conformal Group in Spacetime." Bull. Belg. Math. Soc. Simon Stevin 17 (2) 193 - 200, april 2010.


Published: april 2010
First available in Project Euclid: 26 May 2010

zbMATH: 1192.22011
MathSciNet: MR2667386
Digital Object Identifier: 10.36045/bbms/1274896198

Primary: 22E46
Secondary: 17B15 , 22E70

Keywords: Clifford algebra , conformal group , Minkowski space

Rights: Copyright © 2010 The Belgian Mathematical Society


Vol.17 • No. 2 • april 2010
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