Skip to main content
SpringerLink
Log in

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “Kähler-Dirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons.

The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in K ähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent.

A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cartan, É.: Sur certaines expressions différentieles et le problème de Pfaff. Ann. Éc. Norm. 16, 239–332 (1899)

    MathSciNet  Google Scholar 

  2. Kähler, E.: Einführung in die Theorie der Systeme von Differentialgleichungen. Chelsea, New York (1949)

    Google Scholar 

  3. Kähler, E.: Innerer und äusserer Differentialkalkül. Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math. Phys. Tech. 4, 1–32 (1960)

    Google Scholar 

  4. Kähler, E.: Die Dirac-Gleichung. Abh. Dtsch. Akad. Wiss. Berlin, Kl. Math. Phys. Tech. 1, 1–38 (1961)

    Google Scholar 

  5. Kähler, E.: Der innere Differentialkalkül. Rend. Mat. 21, 425–523 (1962)

    Google Scholar 

  6. Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)

    Google Scholar 

  7. Graff, W.: Differential forms as spinors. Ann. Inst. H. Poincaré 19, 85–109 (1978)

    Google Scholar 

  8. Vargas, J.G., Torr, D.G.: Clifford-valued clifforms: a geometric language for Dirac equations. In: Ablamowicz, R., Fauser, B. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, vol. 1, Ixtapa, June–July 1999. Progress in Physics, vol. 18, pp. 135–154. Birkhäuser, Boston (2000)

    Google Scholar 

  9. Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics. North Holland, Amsterdam (1982)

    MATH  Google Scholar 

  10. Cartan, É.: Leçons sur les Invariants Intégraux. Hermann, Paris (1922)

    MATH  Google Scholar 

  11. Flanders, H.: Differential Forms with Applications to the Physical Sciences. Dover, New York (1989)

    MATH  Google Scholar 

  12. Struik, D.J.: Lectures on Classical Differential Geometry. Addison Wesley, Cambridge (1950)

    MATH  Google Scholar 

  13. Cartan, H.: Differential Forms. Hermann, Paris (1970). Reprinted Dover, New York (2006)

    MATH  Google Scholar 

  14. Lichnerowicz, A.: Éléments de Calcul Tensoriel. Gabay, Paris (1987)

    Google Scholar 

  15. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  16. Slebodzinski, W.: Formes Extérieures et Leurs Applications, vol. 2. Panstwowe Wydawnictwo Naukowe, Warsaw (1963)

    MATH  Google Scholar 

  17. Vargas, J.G., Torr, D.G.: Finslerian structures: the Cartan-Clifton method of the moving frame. J.  Math. Phys. 34(10), 4898–4913 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon, London (1966)

    Google Scholar 

  19. Mead, C.A.: Collective Electrodynamics: Quantum Foundations of Electromagnetism. MIT Press, Cambridge (2000)

    Google Scholar 

  20. Thaller, B.: The Dirac Equation. Springer, New York (1992)

    Google Scholar 

  21. Vargas, J.G., Torr, D.G.: Teleparallel Kähler calculus for spacetime. Found. Phys. 28, 931–958 (1998)

    Article  MathSciNet  Google Scholar 

  22. Vargas, J.G., Torr, D.G.: New perspectives on the Kähler calculus and wave functions. Paper presented at the 7th International Conference on Clifford Algebras, Université Paul Sabatier, Toulouse, 19–29 May 2005. To be published in Advances in Applied Clifford Algebras. www.cartan-einstein-unification.com/pdf/I.pdf

  23. Vargas, J.G., Torr, D.G.: The Kähler-Dirac equation with non-scalar-valued input differential form. Paper presented at the 7th International Conference on Clifford Algebras, Université Paul Sabatier, Toulouse, 19–29 May 2005. To be published in Advances in Applied Clifford Algebras. www.cartan-einstein-unification.com/pdf/II.pdf

  24. Vargas, J.G., Torr, D.G.: The emergence of a Kaluza-Klein microgeometry from the invariants of optimally Euclidean Lorentzian spaces. Found. Phys. 27, 533–558 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  25. Cartan, É.: Sur les équations de la gravitation d’Einstein. J. Math. Pures Appl. 1, 141–203 (1922)

    Google Scholar 

  26. Vargas, J.G., Torr, D.G.: A different line of evolution of geometry on manifolds endowed with pseudo-Riemannian metrics of Lorentzian signature. In: Udriste, C., Balan, V. (eds.) Fifth Conference of Balkan Society of Geometers, Mangalia, August–September 2005. Balkan Geometry Society Proceedings, vol. 13, pp. 173–182. Balkan Geometry Press, Bucharest (2006). www.mathem.pub.ro/dept/confer05/M-VAA.PDF

    Google Scholar 

  27. Vargas, J.G., Torr, D.G.: Klein geometries, Lie differentiation and spin. Differ. Geom. Dyn. Syst. 10, 300–311 (2008). www.mathem.pub.ro/dgds/v10/D10-VA.pdf

    Google Scholar 

  28. Landau, L.D., Lifchitz, E.M.: Mécanique Quantique. Éditions Mir, Moscow (1966)

    MATH  Google Scholar 

  29. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)

    MATH  Google Scholar 

  30. Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. Dover, New York (2002)

    Google Scholar 

  31. Cartan, É.: Sur les variétés à connexion affine et la théorie de la relativité généralisée (suite). Ann. Éc. Norm. 41, 1–25 (1924)

    ADS  MathSciNet  Google Scholar 

  32. Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley, New York (1967)

    Google Scholar 

  33. Schmeikal, B.: Transposition in Clifford algebra: SU(3) from reorientation invariance. In: Ablamowicz, R. (ed.) Conference Proceedings on Clifford Algebras and Their Applications in Mathematical Physics, Cookeville, 2002. Birkhäuser, Boston (2003)

    Google Scholar 

  34. Iwanenko, D., Landau, L.D.: Zur Theorie des magnetischen Elektrons. Z. Phys. 48, 340–348 (1928)

    Article  ADS  Google Scholar 

  35. Donaldson, S.K., Kronheimer, B.P.: The Geometry of Four-Manifolds. Clarendon, Oxford (1990)

    MATH  Google Scholar 

  36. Vargas, J.G., Torr, D.G.: The idiosyncrasies of anticipation in demiurgic physical unification with teleparallelism. Int. J. Comp. Anticip. Syst. 19, 210–225 (2006). www.cartan-einstein-unification.com/pdf/VI.pdf

    Google Scholar 

  37. Cartan, É.: Sur les variétés à connexion affine et la théorie de la relativité généralisée. Ann. Éc. Norm. 40, 325–412 (1923)

    MathSciNet  Google Scholar 

  38. Cartan, É.: La théorie des groupes et les recherches récentes de géometrie differentielle. L’Enseign. Math. 24, 1–18 (1925)

    MathSciNet  Google Scholar 

  39. Cartan, É.: La théorie des groupes et la géometrie. L’Enseign. Math. 26, 200–225 (1927)

    Google Scholar 

  40. Cartan, É.: Le rôle de la théorie des groupes de Lie dans l’évolution de la géometrie moderne. In: C.R. Congrès International, Oslo, pp. 92–103 (1936)

  41. Sharpe, R.W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Springer, New York (1996)

    Google Scholar 

  42. Cartan, É.: La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisées. Exposés de Géométrie. Hermann, Paris (1935)

    Google Scholar 

  43. Vargas, J.G., Torr, D.G.: Of Finsler fiber bundles and the evolution of the calculus. In: Udriste, C., Balan, V. (eds.) Fifth Conference of Balkan Society of Geometers, Mangalia, August–September, 2005. Balkan Geometry Society Proceedings, vol. 13, pp. 183–191. Balkan Geometry Press, Bucharest (2006). www.mathem.pub.ro/dept/confer05/M-VAO.PDF

    Google Scholar 

  44. Einstein, A.: Théorie unitaire du champ physique. Ann. Inst. H. Poincaré 1, 1–24 (1930)

    MathSciNet  Google Scholar 

  45. Debever, R. (ed.): Élie Cartan—Albert Einstein: Letters on Absolute Parallelism. Princeton University Press, Princeton (1979)

    Google Scholar 

  46. Vargas, J.G., Torr, D.G.: The Cartan-Einstein unification with teleparallelism and the discrepant measurements of Einstein’s gravitational constant. Found. Phys. 29, 145–200 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. PST Associates, 48 Hamptonwood Way, Columbia, SC, 29209, USA

    Jose G. Vargas

Authors
  1. Jose G. Vargas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jose G. Vargas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vargas, J.G. The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus. Found Phys 38, 610–647 (2008). https://doi.org/10.1007/s10701-008-9223-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-008-9223-3

Keywords

Search

Navigation