On the geometry of the conformal group in spacetime

N. G. Gresnigt, P. F. Renaud

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2 Citations (Scopus)


The study of the conformal group in Rp,q usually involves the conformal compactification of Rp,q. This allows the transformations to be represented by linear transformations in Rp+1,q+1 So, for example, the conformal group of Minkowski space, R1,3 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 R1,3 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.

Original languageEnglish
Pages (from-to)193-200
Number of pages8
JournalBulletin of the Belgian Mathematical Society - Simon Stevin
Issue number2
Publication statusPublished - 2010
Externally publishedYes


  • Clifford algebra
  • Conformal group
  • Minkowski space

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