Foundations of Physics

, Volume 35, Issue 9, pp 1617–1642 | Cite as

Clifford Space as a Generalization of Spacetime: Prospects for QFT of Point Particles and Strings

  • Matej Pavšič

The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacuum state. As an introduction into the subject, a toy model of the harmonic oscillator in pseudo-Euclidean space is studied.


quantum field theory strings Clifford algebra 


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  1. 1.
    P. A. M. Dirac, talk presented at International School of Subnuclear Physics; 19th Course: The Unity of the Fundamental Interactions, 31 July–11 August (Erice, Sicily, Italy, 1981).Google Scholar
  2. 2.
    Feynman, R.P. 1951Phys. Rev.84108CrossRefMATHMathSciNetADSGoogle Scholar
  3. 3.
    Schweber, S.S. 1986Rev. Mod. Phys.58449CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    V. Fock, Phys. Z. Sowj. 12, 404 (1937); E. C. G. Stueckelberg, Helv. Phys. Acta. 14, 322 (1941); 14, 588 (1941); 15, 23 (1942).Google Scholar
  5. 5.
    L. P. Horwitz and C. Piron, Helv. Phys. Acta 46, 316 (1973); L. P. Horwitz and F. Rohrlich, Phys. Rev. D. 24, 1528 (1981); 26, 3452 (1982); L. P. Horwitz, R. I. Arshansky and A. C. Elitzur, Found. Phys. 18, 1159 (1988); R. Arshansky, L. P. Horwitz and Y. Lavie, Found. Phys. 13, 1167 (1983); L. P. Horwitz, In Old and New Questions in Physics, Cosmology, Philosophy and Theoretical Biology, Alwyn van der Merwe ed. Plenum, New York, 1983). L. P. Horwitz and Y. Lavie, Phys. Rev. D 26, 819 (1982); L. Burakovsky, L. P. Horwitz, and W. C. Schieve, Phys. Rev. D 54, 4029 (1996); L. P. Horwitz, and W. C. Schieve, Ann. Phys. 137, 306 (1981); J. R. Fanchi, Phys. Rev. D 20, 3108 (1979); see also the review J. R. Fanchi, Found. Phys. 23, 287 (1993), and many references therein; J. R. Fanchi Parametrized Relativistic Quantum Theory (Kluwer Academic, Dordrecht, 1993); M. Pavšič, Found. Phys. 21, 1005 (1991); M. Pavšič,Nuovo Cim. A104, 1337 (1991).Google Scholar
  6. 6.
    Pavšič, M. 2001The Landscape of Theoretical Physics: A Global View; From Point Particle to the Brane World and Beyond, in Search of Unifying PrincipleKluwer AcademicDordrechtGoogle Scholar
  7. 7.
    W. Pezzaglia, “Physical Applications of a Generalized Geometric Calculus” [arXiv: gr-qc/9710027]; “Dimensionally Democratic calculus and Principles of Polydimensional Physics” [arXiv: gr-qc/9912025]; “Classification of Multivector Theories and Modifications of the Postulates of Physics” [arXiv: gr-qc/9306006]; “Physical Applications of Generalized Clifford Calculus: Papatetrou equations and Metamorphic Curvature” [arXiv: gr-qc/9710027]; “Classification of Multivector theories and modification of the postulates of Physics” [arXiv: gr-qc/9306006].Google Scholar
  8. 8.
    C. Castro, Chaos, Solitons Fractals 10, 295 (1999); Chaos, Solitons Fractals 12, 1585 (2001); “The Search for the Origins of M Theory: Loop Quantum Mechanics, Loops/Strings and Bulk/Boundary Dualities” [arXiv: hep-th/9809102]; C. Castro, Chaos, Solitons Fractals 11, 1663 (2000); Found. Phys. 30, 1301 (2000).Google Scholar
  9. 9.
    Pavšič, M. 2001Found. Phys.311185[arXiv:hep-th/0011216]MathSciNetGoogle Scholar
  10. 10.
    Pavšič, M. 2003Found. Phys.331277[arXiv:gr-qc/0211085]MathSciNetGoogle Scholar
  11. 11.
    Castro, C., Pavšič, M. 2002Phys. Lett. B539133[arXiv:hep-th/0110079]MathSciNetADSGoogle Scholar
  12. 12.
    Aurilia, A., Ansoldi, S., Spallucci, E. 2002Class. Quant. Grav.193207[arXiv:hep-th/0205028]MathSciNetADSGoogle Scholar
  13. 13.
    Castro, C., Pavšič, M. 2003Int. J. Theor. Phys.421693[arXiv:hep-th/0203194]CrossRefGoogle Scholar
  14. 14.
    D. Cangemi, R. Jackiw, and B. Zwiebach, Ann. of Phys. 245, 408 (1996); E. Benedict, R. Jackiw, and H.-J. Lee, Phys. Rev. D 54, 6213 (1996)Google Scholar
  15. 15.
    S. Teitler, Supplemento al Nuovo Cimento III, 1 (1965) and references therein; Supplemento al Nuovo Cimento III, 15 (1965); J. Math.Phys. 7, 1730 (1966); J. Math. Phys. 7, 1739 (1966); L. P. Horwitz, J. Math. Phys. 20, 269 (1979); H. H. Goldstine and L. P. Horwitz, Mathe. Ann. 164, 291 (1966).Google Scholar
  16. 16.
    D. Hestenes, Space-Time Algebra (Gordon & Breach, New York, 1966); D. Hestenes and G. Sobcyk, Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).Google Scholar
  17. 17.
    Lounesto, P. 2001Clifford Algebras and SpinorsCambridge University PressCambridgeGoogle Scholar
  18. 18.
    N. S. Mankoč Borštnik and H. B. Nielsen, J. Math. Phys. 43 5782 (2002) [arXiv:hep-th/0111257]; J. Math. Phys. 44 4817 (2003) [arXiv:hep-th/0303224]Google Scholar
  19. 19.
    Weinberg, S. 1989Rev. Mod. Phys.611See, e.g.CrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Baylis, W. 1999Electrodynamics, A Modern Geometric ApproachBirkhäuserBostonGoogle Scholar
  21. 21.
    G. Trayling and W. Baylis, J. Phys. A 34, 3309 (2001); Int. J. Mod. Phys. A 16, Suppl. 1C, 909 (2001)Google Scholar
  22. 22.
    Jancewicz, B. 1989Multivectors and Clifford Algebra in ElectrodynamicsWorld ScientificSingaporeGoogle Scholar
  23. 23.
    Clifford Algebras and their Applications in Mathematical Physics, Vol. 1: Algebras and Physics, R. Ablamowicz and B. Fauser, eds. Vol. 2: Clifford Analysis, J. Ryan and W. Sprosig, eds. (Birkhäuser, Boston, 2000)Google Scholar
  24. 24.
    Lasenby, A., Doran, C. 2002Geometric Algebra for PhysicistsCambridge University. PressCambridgeGoogle Scholar
  25. 25.
    A. M. Moya, V. V Fernandez and W. A. Rodrigues, Int.J.Theor.Phys. 40 (2001) 2347–2378 [arXiv: math-ph/0302007]; Multivector Functions of a Multivector Variable [arXiv: math.GM/0212223]; Multivector Functionals [arXiv: math.GM/0212224]; W.A. Rodrigues, Jr, J. Vaz, Jr. [Adv. Appl. Clifford Algebras 7 457–466 (1997); E. C de Oliveira and W. A. Rodrigues, Jr. Ann. Phys. 7, 654–659 (1998); Phys. Lett. A 291 367 (2001); W.A. Rodrigues, Jr, J. Y. Lu, Found. Phys. 27 (1997) 435–508Google Scholar
  26. 26.
    S. Perlmutter et al., Astrophys. J. 517, 565 (1999); A. G. Riess et al., Astron. J. 116, 1009 (1998); D. N. Spergel, et al., Astrophys. J.Suppl 148, 175 (2003); L. Page et al., Astrophys. J.Suppl. 148, 233 (2003)Google Scholar
  27. 27.
    Pavšič, M. 1999Phys. Lett. A254119[arXiv:hep-th/9812123]MathSciNetADSGoogle Scholar
  28. 28.
    Y. S. Kim and M. E. Noz, Phys. Rev. D 8, 3521 (1973); 12, 122 (1975); 15,335 (1977); Phys. Rev. Lett. 63, 348 (1989); Y. S. Kim and M. E. Noz, Theory Applications of the Poincaré Group (Reidel, Dordrecht, 1986)Google Scholar
  29. 29.
    See, e.g., M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge University Press, Cambridge, 1987); M. Kaku, Introduction to Superstrings (Springer, New York, N.Y., 1988)Google Scholar
  30. 30.
    M. Faux and S. J. Gates, “Adinkras: A Graphical Technology for Supersymmetric Representation Theory” [arXiv:hep-th/040800]4; S. J. J. Gates, W. D. Linch, and J. Phillips, “When Superspace is not Enough” [arXiv:hep-th/0211034]; S. J. J. Gates and L. Rana, Phys. Lett. B 369, 262 (1996) [arXiv:hep-th/9510151]; S. J. Gates and L. Rana, Phys. Lett. B 352, 50 (1995) [arXiv:hep-th/9504025]Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Jožef Stefan InstituteLjubljanaSlovenia

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