Clifford Space as the Arena for Physics


A new theory is considered according to which extended objects in n-dimensional space are described in terms of multivector coordinates which are interpreted as generalizing the concept of center of mass coordinates. While the usual center of mass is a point, by generalizing the latter concept, we associate with every extended object a set of r-loops, r=0,1,...,n−1, enclosing oriented (r+1)-dimensional surfaces represented by Clifford numbers called (r+1)-vectors or multivectors. Superpositions of multivectors are called polyvectors or Clifford aggregates and they are elements of Clifford algebra. The set of all possible polyvectors forms a manifold, called C-space. We assume that the arena in which physics takes place is in fact not Minkowski space, but C-space. This has many far reaching physical implications, some of which are discussed in this paper. The most notable is the finding that although we start from the constrained relativity in C-space we arrive at the unconstrained Stueckelberg relativistic dynamics in Minkowski space which is a subspace of C-space.

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  1. 1.

    D. Hestenes, Space-Time Algebra (Gordon & Breach, New York, 1966). D. Hestenes, Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).

    Google Scholar 

  2. 2.

    W. M. Pezzaglia, A Clifford Algebra Multivector Reformulation of Field Theory, Dissertation, University of California, Davis, 1983. W. M. Pezzaglia, “Classification of multivector theories and modification of the postulates of physics” [arXiv:gr-qc/9306006]; W. M. Pezzaglia“Polydimensional relativity, a classical generalization of the automorphism invariance principle” [arXiv:gr-qc/9608052]; W. M. Pezzaglia“Physical applications of a generalized Clifford calculus: Papapetrou equations and metamorphic curvature” [arXiv:gr-qc/9710027]. W. M. Pezzaglia, Jr. and J. J. Adams, “Should metric signature matter in Clifford algebra formulation of physical theories?” [arXiv:gr-qc/9704048]. W. M. Pezzaglia, Jr. and A. W. Differ, “A Clifford dyadic superfield from bilateral interactions of geometric multispin Dirac theory” [arXiv:gr-qc/9311015]. W. M. Pezzaglia, Jr., “Dimensionally democratic calculus and principles of polydimensional physics” [arXiv:gr-qc/9912025].

    Google Scholar 

  3. 3.

    C. Castro, J. Chaos, Solitons, and Fractals 11, 1721(2000) [arXiv:hep-th/9912113]; C. Castro J. Chaos, Solitons, and Fractals 12, 1585 (2001) [arXiv:physics/0011040].

    Google Scholar 

  4. 4.

    M. Pavšič, Found. Phys. 31, 1185(2001) [arXiv:hep-th/0011216]; M. Pavšič, The Landscape of Theoretical Physics: A Global View. From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle (Kluwer Academic, Dordrecht, 2001).

    Google Scholar 

  5. 5.

    V. Fock, Phys. Z. Sowj. 12, 404(1937). E. C. G. Stueckelberg, Helv. Phys. Acta 14, 322 (1941); 14, 588 (1941); 15, 23 (1942). R. P. Feynman, Phys. Rev. 84, 108 (1951).

    Google Scholar 

  6. 6.

    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, 1683). 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); Nuovo Cim. A 104, 1337 (1991).

    Google Scholar 

  7. 7.

    M. Pavšič, Nuovo Cim. B 41, 397(1977); J. Phys. A 13, 1367 (1980).

    Google Scholar 

  8. 8.

    A. Aurilla, C. Castro, and M. Pavšič, in preparation.

  9. 9.

    J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater, and J. D. Barrow, Phys. Rev. Lett. 82, 884(1999). J. K. Webb, M. T. Murphy, V. V. Flambaum, J. D. Dzuba, J. D. Barrow, C. W. Churchill, J. X. Prochaska and A. M. Wolfe, Phys. Rev. Lett. 87, 091391 (2001). M. T. Murphy, J. K. Webb, V. V. Flambaum, V. A. Dzuba, C. W. Churchill, J. X. Prochaska, J. D. Barrow, and A. M. Wolfe, Mon. Not. R. Astron. Soc. 327, 1208 (2001).

    Google Scholar 

  10. 10.

    M. Pavšič, Found. Phys. 26, 159(1996) [arXiv:gr-qc/9506057].

    Google Scholar 

  11. 11.

    J. Greensite, Class. Grav. 13, 1339(1996); Phys. Rev. D 49, 930 (1994). A. Carlini and J. Greensite, Phys. Rev. D 52, 936 (1995); 52, 6947 (1995); A. Carlini and J. Greensite, Phys. Rev. D 55, 3514 (1997).

    Google Scholar 

  12. 12.

    P. S. Howe and R. W. Tucker, J. Phys. A: Math. Gen. 10, L155(1977). A. Sugamoto, Nucl. Phys. B 215, 381 (1983). E. Bergshoeff, E. Sezgin, and P. K. Townsend, Phys. Lett. B 189, 75 (1987). A. Achucarro, J. M. Evans, P. K. Townsend, and D. L. Wiltshire, Phys. Lett. B 198, 441 (1987). M. Pavšič, Class. Quant. Grav. 5, 247 (1988); M. Pavšič Phys. Lett. B 197, 327 (1987).

    Google Scholar 

  13. 13.

    D. Colladay and P. McDonald, J. Math. Phys. 43, 3554(2002), and references therein.

    Google Scholar 

  14. 14.

    A. Connes, “C*-Algèbres et Géometrie Différentielle, ” C. R. Acad. Sci. Paris 290, 599(1980); For a recent review see, e.g., J. Madore, An Introduction to Noncommutative Differential Geometry and Physical Applications (Cambridge University Press, 2000).

  15. 15.

    M. Pavšič, Nuovo Cim. A 108, 221(1995) [arXiv:gr-qc/9501036]; Found. Phys. 25, 819 (1995); Nuovo Cim. A 110, 369 (1997) [arXiv:hep-th/9704154]; Found. Phys. 28, 1443 (1998); Found. Phys. 28, 1453 (1998).

    Google Scholar 

  16. 16.

    M. Pavšič, Found. Phys. 21, 1005(1991).

    Google Scholar 

  17. 17.

    C. Castro and M. Pavšič, Phys. Lett. B 539, 133(2002) [arXiv:hep-th/0110079]; “Clifford Algebra of Spacetime and the conformal Group, ” [arXiv:hep-th/0203194].

    Google Scholar 

  18. 18.

    S. Teitler, Suppl. Nuovo Cimento III, 1(1965); Suppl. Nuovo Cimento III, 15 (1965); J. Math. Phys. 7, 730 (1966); J. Math. Phys. 7, 1739 (1966). L. P. Horwitz, J. Math. Phys. 20, 269 (1979). H. H. Goldstine and L. P. Horwitz, Math. Ann. 164, 291 (1966).

    Google Scholar 

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Pavšsič, M. Clifford Space as the Arena for Physics. Foundations of Physics 33, 1277–1306 (2003).

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  • relativistic dynamics
  • Clifford space
  • branes
  • geometric calculus