Foundations of Physics

, Volume 37, Issue 8, pp 1197–1242 | Cite as

On a Unified Theory of Generalized Branes Coupled to Gauge Fields, Including the Gravitational and Kalb–Ramond Fields


We investigate a theory in which fundamental objects are branes described in terms of higher grade coordinates \(X^{\mu{_1}\ldots \mu{_n}}\) encoding both the motion of a brane as a whole, and its volume evolution. We thus formulate a dynamics which generalizes the dynamics of the usual branes. Geometrically, coordinates \(X^{\mu{_1} \ldots \mu{_n}}\) and associated coordinate frame fields {\({\gamma_{\mu{_1}\ldots\mu{_n}}}\)} extend the notion of geometry from spacetime to that of an enlarged space, called Clifford space or C-space. If we start from four-dimensional spacetime, then the dimension of C-space is 16. The fact that C-space has more than four dimensions suggests that it could serve as a realization of Kaluza-Klein idea. The “extra dimensions” are not just the ordinary extra dimensions, they are related to the volume degrees of freedom, therefore they are physical, and need not be compactified. Gauge fields are due to the metric of Clifford space. It turns out that amongst the latter gauge fields there also exist higher grade, antisymmetric fields of the Kalb–Ramond type, and their non-Abelian generalization. All those fields are naturally coupled to the generalized branes, whose dynamics is given by a generalized Howe–Tucker action in curved C-space.


Strings branes Clifford algebra unification Kaluza-Klein theories gauge fields 


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  1. 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-Verlag, New York, 1988); U. Danielsson, Rep. Progr. Phys. 64, 51 (2001).Google Scholar
  2. J. Polchinski, “Lectures on D-branes,” [arXiv:hep-th/9611050]; W. I. Taylor, “Lectures on D-branes, gauge theory and M(atrices),” arXiv:hep-th/9801182. H. Nicolai and R. Helling, “Supermembranes and M(atrix) theory,” arXiv:hep-th/9809103; J.H. Schwarz, Phys. Rep. 315, 107 (1999).Google Scholar
  3. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999) [arXiv:hep-th/9906064]. M. J. Duff, “Benchmarks on the brane,” arXiv:hep-th/0407175.Google Scholar
  4. M. J. Duff, R. R. Khuri, and J. X. Lu, Phys. Rept. 259, 213 (1995) [arXiv:hep-th/9412184]. M. J. Duff and J. X. Lu, Class. Quant. Grav. 9, 1 (1992). M. J. Duff and J. X. Lu, Phys. Rev. Lett. 66, 1402 (1991).Google Scholar
  5. Schild A. (1977). Phys. Rev. D 16: 1722CrossRefADSGoogle Scholar
  6. Eguchi T. (1980). Phys. Rev. Lett. 44: 126CrossRefADSGoogle Scholar
  7. A. Aurilia, A. Smailagic, and E. Spallucci, Phys. Rev. D 47, 2536 (1993) [arXiv:hep-th/9301019]; A. Aurilia and E. Spallucci, Class. Quant. Grav. 10, 1217 (1993) [arXiv:hep-th/9305020]; S. Ansoldi, A. Aurilia, and E. Spallucci, Phys. Rev. D 53, 870 (1996) [arXiv:hep-th/9510133].Google Scholar
  8. D. Hestenes, Space-Time Algebra (Gordon and Breach, New York, 1966); D. Hestenes and G. Sobcyk, Clifford Algebra to Geometric Calculus (D. Reidel, Dordrecht, 1984).Google Scholar
  9. P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2001); B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, Singapore, 1988); R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995); W. Baylis, Electrodynamics, A Modern Geometric Approach (Boston, Birkhauser, 1999); A. Lasenby and C. Doran, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2002); Clifford Algebras and their applications in Mathematical Physics, Vol 1: Algebras and Physics, eds. R. Ablamowicz and B. Fauser; Vol 2: Clifford analysis. eds. J. Ryan and W. Sprosig (Birkhauser, Boston, 2000); A. M. Moya, V. V Fernandez, and W. A. Rodrigues, Int. J. Theor. Phys. 40, 2347 (2001) [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 (1997); E. C de Oliveira and W. A. Rodrigues, Jr., Ann. der Physik 7, 654 (1998). Phys. Lett A 291, 367 (2001). W. A. Rodrigues, Jr., J. Y. Lu, Found. Phys. 27, 435 (1997); S. Vacaru, P. Stavrinos, E. Gaburov, and D. Gonta, “Clifford and Riemann-Finsler structures in geometric mechanics and gravity,” [arXiv:gr-qc/0508023].Google Scholar
  10. M. Pavšič, Found. Phys. 33, 1277 (2003) [arXiv:gr-qc/0211085].Google Scholar
  11. M. Pavšič, The Landscape of Theoretical Physics: A Global View; From Point Particle to the Brane World and Beyond, in Search of Unifying Principle (Kluwer Academic, Dordrecht, 2001).Google Scholar
  12. S. Ansoldi, A. Aurilia, C. Castro, and E. Spallucci, Phys. Rev. D 64, 026003 (2001) [arXiv:hep-th/0105027].Google Scholar
  13. A. Aurilia, S. Ansoldi, and E. Spallucci, Class. Quant. Grav. 19, 3207 (2002) [arXiv:hep-th/0205028].Google Scholar
  14. C. Castro and M. Pavšič, Progr. Phys. 1, 31 (2005).Google Scholar
  15. M. Pavšič, Found. Phys. 35, 1617 (2005) [arXiv:hep-th/0501222].Google Scholar
  16. E. C. G. Stueckelberg, Helv. Phys. Acta 14, 322 (1941); 14, 588 (1941); 15, 23 (1942).Google Scholar
  17. Feynman R.P. (1951). Phys. Rev. 84: 108MATHCrossRefADSGoogle Scholar
  18. Schwinger J. (1951). Phys. Rev. 82: 664MATHCrossRefADSGoogle Scholar
  19. W. C. Davidon, Phys. Rev. 97, 1131 (1955); 97, 1139 (1955).Google Scholar
  20. 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, A. 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).Google Scholar
  21. 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, Dordrecht, 1993).Google Scholar
  22. H. Enatsu, Progr. Theor. Phys. 30, 236 (1963); Nuovo Cimento A 95, 269 (1986); F. Reuse, Found. Phys. 9, 865 (1979); A. Kyprianidis Phys. Rep. 155, 1 (1987); R. Kubo, Nuovo Cim. A, 293 (1985); M. B. Mensky and H. von Borzeszkowski, Phys. Lett. A 208, 269 (1995); J. P. Aparicio, F. H. Gaioli, and E. T. Garcia-Alvarez, Phys. Rev. A 51, 96 (1995); Phys. Lett. A 200, 233 (1995); L. Hannibal, Int. J. Theoret. Phys. 30, 1445 (1991); F. H. Gaioli and E. T. Garcia-Alvarez, Gen. Relat. Grav. 26, 1267 (1994).Google Scholar
  23. M. Pavšič, Found. Phys. 21, 1005 (1991); M. Pavšič, Nuovo Cim. A 104, 1337 (1991); Doga, Turkish J. Phys. 17, 768 (1993).Google Scholar
  24. Pavšič M. (1996). Found. Phys. 26: 159 [arXiv:gr-qc/9506057].CrossRefGoogle Scholar
  25. M. Pavšič, Found. Phys. 31, 1185 (2001) [arXiv:hep-th/0011216].Google Scholar
  26. M. Riesz, “Sur certaines notions fondamentales en théorie quantiques relativiste’, in Dixième Congrès Math. des Pays Scandinaves, Copenhagen, 1946 (Jul. Gjellerups Forlag, Copenhagen, 1947), pp. 123–148; M. Riesz, Clifford Numbers and Spinors, E. Bolinder and P. Lounesto, eds. (Kluwer, 1993); 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); 7, 1739 (1966); W. A. Rodrigues, Jr., J. Math. Phys. 45, 2908 (2004).Google Scholar
  27. Pavšič M. (2005) . Phys. Lett. B 614: 85 [arXiv:hep-th/0412255].CrossRefADSGoogle Scholar
  28. M. Pavšič, “Spin gauge theory of gravity in Clifford space: A realization of Kaluza-Klein theory in 4-dimensional spacetime,” Int. J. Mod. Phys. A 21, 5905 (2006). arXiv:gr-qc/0507053.Google Scholar
  29. L. P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1926).Google Scholar
  30. H. Rund, Invariant Theory of Variational Problems on Subspaces of a Riemannian Manifold (Van Den Hoeck & Rupert, Göottingen, 1971).Google Scholar
  31. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman and Company, San Francisco, 1973).Google Scholar
  32. M. Pavšič, Class. Quant. Grav. 20, 2697 (2003) [arXiv:gr-qc/0111092].Google Scholar
  33. P. S. Howe and R. W. Tucker, J. Phys. A: Math. Gen. 10, L155 (1977); A. Sugamoto, Nucl. Phys. B 215, 381 (1983); M. Pavšič, Class. Quant. Grav. 5, 247 (1988).Google Scholar
  34. W. Pezzaglia, “Physical applications of a generalized geometric calculus,” in Dirac Operators in Analysis (Pitman Research Notes in Mathematics, Number 394), J. Ryan and D. Struppa, eds. (Longmann, 1997) pp. 191–202 [arXiv: gr-qc/9710027]; “Dimensionally democratic calculus and principles of polydimensional physics,” in Clifford Algebras and their Applications in Mathematical Physics, R. Ablamowicz and B. Fauser, eds. (Birkhauser, 2000), pp. 101–123,[arXiv: gr-qc/9912025]; “Classification of Multivector Theories and Modifications of the Postulates of Physics”, in Clifford Algebras and their Applications in Mathematical Physics, Brackx, Delanghe & Serras eds. (Kluwer, 1993) pp. 317–323, [arXiv: gr-qc/9306006].Google Scholar
  35. C. Castro, Chaos, Solitons and Fractals 10, 295 (1999); 11, 1663 (2000); 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, Found. Phys. 30, 1301 (2000).Google Scholar
  36. M. J. Duff, Nucl. Phys. B 335, 610 (1990). M. J. Duff and J. X. Lu, Nucl. Phys. B 347, 394 (1990).Google Scholar
  37. C. Castro and M. Pavšič, Phys. Lett. B 539, 133 (2002) [arXiv:hep-th/0110079].Google Scholar
  38. M. Pavšič, “Clifford space as a generalization of spacetime: prospects for unification in physics,” arXiv:hep-th/0411053.Google Scholar
  39. Castro C. (2005). Found. Phys. 35: 971MATHCrossRefGoogle Scholar
  40. Luciani J.F (1978). Nucl. Phys. B 135: 111CrossRefADSGoogle Scholar
  41. Witten E. (1981). Nucl. Phys. B 186: 412CrossRefADSGoogle Scholar
  42. Kalb M., Ramond P. (1974). Phys. Rev. D 9: 2273CrossRefADSGoogle Scholar
  43. Castro C. (2004). Mod. Phys. Lett. A 19: 19MATHCrossRefADSGoogle Scholar
  44. Aurilia A., Takahashi Y. (1981). Progr. Theor. Phys. 66: 693CrossRefADSGoogle Scholar
  45. P. A. M. Dirac, Proc. R. Soc.(London) A 268, 57 (1962).Google Scholar
  46. A. O. Barut and M. Pavšič, Mod. Phys. Lett. A 7, 1381 (1992).Google Scholar
  47. A. O. Barut and M. Pavšič, Phys. Lett. B 306, 49 (1993); Phys. Lett. B 331, 45 (1994).Google Scholar
  48. G. Savidy “Non-Abelian tensor gauge fields: enhanced symmetries,” arXiv:hep-th/0604118.Google Scholar
  49. Castro C. (1998). Int. J. Mod. Phys. A 13: 1263MATHCrossRefADSGoogle Scholar
  50. C. Castro, Int. J. Mod. Phys. A 21, 2149 (2006). D. V. Alekseevsky, V. Cortes, C. Devchand, and A. Van Proeyen, Commun. Math. Phys. 253, 385 (2004) [arXiv:hep-th/0311107].Google Scholar
  51. C. Castro, J. Math. Phys. 47, 112301 (2006); Z. Kuznetsova and F. Toppan, arXiv:hep-th/0610122.Google Scholar
  52. C. Castro, Ann. Phys. 321, 813 (2006). C. Castro, Found. Phys. 34, 1091 (2004). M. Land, Found. Phys. 35, 1245 (2005) [arXiv:hep-th/0603169].Google Scholar
  53. Castro C. (2006). J. Phys. A: Math. Gen. 39: 14205MATHCrossRefADSGoogle Scholar
  54. Penrose R. (1999). Chaos, Solitons Fractals 10: 581MATHCrossRefGoogle Scholar
  55. F. Smith, Intern. J. Theor. Phys. 24, 155 (1985); 25, 355 (1985); G. Trayling and W. E. Baylis, Int. J. Mod. Phys. A 16, Suppl. 1C (2001) 900; J. Phys. A: Math. Gen. 34, 3309 (2001); G. Roepstorff, “A class of anomaly-free gauge theories,” arXiv:hep-th/0005079; “Towards a unified theory of gauge and Yukawa interactions,” arXiv:hep-ph/0006065; “Extra dimensions: will their spinors play a role in the standard model?,” arXiv:hep-th/0310092; F. D. Smith, “From sets to quarks: deriving the standard model plus gravitation from simple operations on finite sets,” arXiv:hep-ph/9708379. J. S. R. Chisholm and R. S. Farwell, J. Phys. A: Math. Gen. 20, 6561 (1987); 33, 2805 (1999); 22, 1059 (1989); J. S. R. Chisholm, J. Phys. A: Math. Gen. 35, 7359 (2002); Nuov. Cim. A 82, 145 (1984); 185; 210; “Properties of Clifford algebras for fundamental particles”, in Clifford (Geometric) Algebras, W. Baylis, ed. (Birkhauser, 1996), Chapter 27, pp. 365–388. J. P. Crawford, J. Math. Phys. 35, 2701 (1994); in Clifford (Geometric) Algebras, W. Baylis, ed. (Birkhauser, 1996), Chapters 21–26, pp. 297–364; Class. Quant. Grav. 20, 2945 (2003); A. Garrett Lisi, arXiv:gr-qc/0511120.Google Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Jožef Stefan InstituteLjubljanaSlovenia

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