Advances in Applied Clifford Algebras

, Volume 21, Issue 4, pp 721–756 | Cite as

On Conformal Infinity and Compactifications of the Minkowski Space

  • Arkadiusz JadczykEmail author


Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the “cone at infinity” but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge \({\star}\) operator twistors (i.e. vectors of the pseudo-Hermitian space H 2,2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4, 2)/Z 2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H p,q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.


Compactification Minkowski space-time Conformal spinors Twistors SU(2) SU(2,2) U(2) SO(4,2) Null geodesics Penrose diagram Conformal infinity Conformal group Conformal inversion Hodge star bivectors Grassmann manifold Clifford algebra Cℓ(4, 2) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Armin Uhlmann, The Closure of Minkowski Space. Acta Physica Polonica, Vol. XXIV, Fasc. 2(8) (1963), pp. 295–296.Google Scholar
  2. 2.
    R. Penrose and W. Rindler, Spinors and Space-Time, Vol. 2 – Spinor and Twistor Methods in Space-Time Geometry. Cambridge University Press, Cambridge, England, 1984.Google Scholar
  3. 3.
    S. A. Huggett and K. P. Tod, An Introduction to Twistor Theory. Cambridge University Press (1994).Google Scholar
  4. 4.
    Beem John K., Ehrlich Paul E., Easley Kevin L.: Global Lorentzian Geometry, Second Edition. Marcel Dekker Inc., New York (1996)Google Scholar
  5. 5.
    Flores José L.: The Causal Boundary of Spacetimes Revisited. Commun. Math. Phys. 276, 611–643 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Lerner David E.: Global Properties of Massless Free Fields. Commun. Math. Phys. 55, 179–182 (1977)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Claude Chevalley, Theory of Lie Groups I. Princeton Mathematical Series, no. 8, Princeton University Press, (1946), Chapter I, §X, Proposition 6.Google Scholar
  8. 8.
    A. V. Levichev, Parallelizations of Chronometric Bundles Based on the Sub-group U(2). Izvestia RAEN, ser.MMMIU, 10 (2006), n.1-2, pp. 51–61, (in Russian).Google Scholar
  9. 9.
    W. Kopczyński and L. S. Woronowicz, A geometrical approach to the twistor formalism. Rep. Math. Phys. Vol 2 (1971), pp. 35–51.Google Scholar
  10. 10.
    Stolarczyk Leszek Z.: The Hodge Operator in Fermionic Fock Space. Collect. Czech. Chem. Commun. 70, 979–1016 (2005)CrossRefGoogle Scholar
  11. 11.
    René Deheuvels, Formes quadratiques et groupes classiques. Presse Universitaires de France, Paris (1981).Google Scholar
  12. 12.
    Aubert Daigneault, Irving Segal’s axiomatization of spacetime and its cosmological consequences. Preprint
  13. 13.
    Schmidt B.G.: A New Definition of Conformal and Projective Infinity of Space-Times. Commun. math. Phys. 36, 73–90 (1974)ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Akivis Maks A., Goldberg Vladislav V.: Conformal Differential Geometry and its Generalizations. A Wiley Interscience Publications, New York (1996)zbMATHCrossRefGoogle Scholar
  15. 15.
    N. M. Todorov, I. T. Todorov, Conformal Quantum Field Theory in Two and Four Dimensions. Vienna, Preprint ESI 1155 (2002),
  16. 16.
    Nikolay M. Nikolov, Rationality of Conformally Invariant Local Correlation Functions on Compactified Minkowski Space. Commun. Math. Phys. 218 (2001), pp. 417–436.Google Scholar
  17. 17.
    Nikolov Nikolay M.: Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory. Commun. Math. Phys. 253, 283–322 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Roger Penrose, Conformal traetment of Infinity. In “Relativity, groups and topology”, Lectures delivered at Les Houches during the 1963 session of the Summer School of Theoretical Physics, University of Grenoble, ed. DeWitt, Gordon and Brach, New York (1964), pp. 563–584.Google Scholar
  19. 19.
    Misner C.W., Thorne K.S., Wheeler J.A.: Gravitation. Freeman and Co., New York (1973)Google Scholar
  20. 20.
    Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1976)Google Scholar
  21. 21.
    Roger Penrose, The Light Cone at Infinity. In Relativistic Theories of Gravitation, ed. L. Infeld, Pergamon Press, Oxford (1964), pp. 369–373.Google Scholar
  22. 22.
    Rühl W.: Distributions on Minkowski Space and their Connection with Analytic Representations of the Conformal Group. Commun.math. Phys. 27, 53–86 (1972)ADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Pierre Anglès, Conformal Groups in Geometry and Spin Structures. Birkhäuser, Progress in Mathematical Physics, Vol. 50 (2008).Google Scholar
  24. 24.
    Roger Penrose, Structure of space-time. In Battelle Rencontres, ed. by C. M. DeWitt, J. A. Wheeler, Benjamin, New York (1969), pp. 121–235.Google Scholar
  25. 25.
    Penrose Roger: Twistor Algebra. J. Math. Phys. 8, 345–366 (1967)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Kuiper N.H.: On conformally flat spaces in the large. Ann. of Math. 50, 916–924 (1949)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Penrose R.: Zero Rest-Mass Fields Including Gravitation: Asymptotic Behaviour. Proc. R. Soc. London 284, 159–203 (1965)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Anıl Zenginoğlu, Hyperboloidal foliations and scri-fixing.
  29. 29.
    Robert Geroch, Asymptotic Structure of Space-Time. In Asymptotic Structure of Space-Time, ed. F. Esposito and L. Witten, Plenum Press (1977), pp. 1–105.Google Scholar
  30. 30.
    Albert Crumeyrolle, Orthogonal and Symplectic Clifford Algebras. Spinor structures. Kluwer Academic Publishers (1990).Google Scholar
  31. 31.
    Chisholm M.: Such Silver Currents. The Story of William and Lucy Clifford 1845–1929. The Lutterworth Press, Cambridge (2002)Google Scholar
  32. 32.
    Penrose Roger: The Road to Reality. Jonathan Cape, London (2004)Google Scholar
  33. 33.
    Werner Greub, Multilinear Algebra. Springer (1978).Google Scholar
  34. 34.
    John. C. Baez, Irving E. Segal, Zhengfang Zhou, Introduction to Algebraic and Constructive Quantum Field Theory. Princeton University Press (1992).Google Scholar
  35. 35.
    Curtis W.D., Miller F.R.: Differential Manifolds and Theoretical Physics. Academic Press, New York (1985)zbMATHGoogle Scholar
  36. 36.
    Fecko M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  37. 37.
    Lerner David E.: Twistors and induced representations of SU(2,2). J. Math. Phys. 18, 1812–1817 (1977)ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Shoshichi Kobayashi, Transformation Groups in Differential Geometry. Springer (1972).Google Scholar
  39. 39.
    A. Jadczyk, Born’s Reciprocity in the Conformal Domain. In Z. Oziewicz et al (eds.); Spinors, Twistors, Clifford Algebras and Quantum Deformations. Kluwer Academic Publishers (1993), pp. 129–140.Google Scholar
  40. 40.
    Felipe Leitner, Twistor Spinors and Normal Cartan Connections in Conformal Geometries.
  41. 41.
    J. Mickelsson, J. Niederle, Conformally invariant field equations. Ann. de l’I.H.P., section A, tome 23, no 3 (1975), p. 277–295.Google Scholar
  42. 42.
    Bourbaki Nicolas, Éléments de Mathématique. Algèbre, Chapitre 9. Springer (2007). First edition: Hermann, Paris (1959).Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Center CAIROS, Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France

Personalised recommendations