Foundations of Physics

, Volume 23, Issue 9, pp 1203–1237 | Cite as

Clifford algebras and Hestenes spinors

  • Pertti Lounesto
Part I. Invited Papers Dedicated To David Hestenes


This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl 1,3 ≃ M2(H). Hestenes invented first in 1966 hisideal spinors\(\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )\) and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl 1,3 + ≃ M2(C).

This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC ⊗ Cl 1,3 andnot inC ⊗ M4(R)=M4(C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.

Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψγ0ψ=0, ψγ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.

Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμiγμ.


Complex Number Projection Operator Dirac Equation Physical Interpretation Column Spinor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R. Ablamowicz, P. Lounesto, and J. Maks, “Conference Report, Second Workshop on Clifford Algebras and Their Applications in Mathematical Physics” (Université des Sciences et Techniques du Languedoc, Montpellier, France, 1989),Found. Phys. 21, 735–748 (1991).Google Scholar
  2. S. L. Altmann,Rotations, Quaternions, and Double Groups (Clarendon Press, Oxford, 1986).Google Scholar
  3. E. Artin,Geometric Algebra (Interscience, New York, 1957, 1988).Google Scholar
  4. M. F. Atiyah, R. Bott, and A. Shapiro, “Clifford modules,”Topology 3, Suppl. 1, 3–38 (1964). Reprinted in R. Bott,Lectures on K(X) (Benjamin, New York, 1969), pp. 143–178. Reprinted in Michael Atiyah,Collected Works, Vol. 2 (Clarendon Press, Oxford, 1988), pp. 301–336.Google Scholar
  5. I. M. Benn and R. W. Tucker,An Introduction to Spinors and Geometry with Applications in Physics (Adam Hilger, Bristol, 1987).Google Scholar
  6. E. F. Bolinder, “Clifford algebra: what is it?”IEEE Antennas Propag. Soc. Newslett. 29, 18–23 (1987).Google Scholar
  7. N. Bourbaki,Algèbre, Formes sesquilinéaires et formes quadratiques (Hermann, Paris, 1959), Chap. 9.Google Scholar
  8. F. Brackx, R. Delanghe, F. Sommen,Clifford Analysis (Research Notes in Mathematics76) (Pitman Books, London, 1982).Google Scholar
  9. R. Brauer and H. Weyl, “Spinors inn dimensions.”Amer. J. Math. 57, 425–449 (1935). Reprinted inSelecta Hermann Weyl (Birkhäuser, Basel, 1956), pp. 431–454.Google Scholar
  10. P. Budinich and A. Trautman,The Spinorial Chessboard (Springer, Berlin, 1988).Google Scholar
  11. E. Cartan, (exposé, d'après l'article allemand de E. Study), “Nombres complexes,” in J. Molk, ed.:Encyclopédie des sciences mathématiques, Tome I, Vol. 1, Fasc. 4, art. 15 (1908), pp. 329–468. Reprinted in E. Cartan,Œuvres Complètes, Partie II (Gauthier-Villars, Paris, 1953), pp. 107–246.Google Scholar
  12. C. Chevalley,The Algebraic Theory of Spinors (Columbia University Press, New York, 1954).Google Scholar
  13. C. Chevalley,The Construction and Study of Certain Important Algebras (Mathematical Society of Japan, Tokyo, 1955).Google Scholar
  14. J. S. R. Chisholm and A. K. Common, eds.,Proceedings of the NATO and SERC Workshop on “Clifford Algebras and Their Applications in Mathematical Physics”, Canterbury, England, U.K., 1985 (Reidel, Dordrecht, 1986).Google Scholar
  15. W. K. Clifford, “Applications of Grassmann's extensive algebra,”Am. J. Math. 1, 350–358 (1878).Google Scholar
  16. W. K. Clifford, “On the classification of geometric algebras,” in R. Tucker, ed.,Mathematical Papers by William Kingdon Clifford (Macmillan, London, 1982), pp. 397–401. (Reprinted by Chelsea, New York, 1968.) Title of talk announced already inProc. London Math. Soc. 7, 135 (1876).Google Scholar
  17. J. Crawford, “On the algebra of Dirac bispinor densities: Factorization and inversion theorems,”J. Math. Phys. 26, 1439–1441 (1985).Google Scholar
  18. A. Crumeyrolle,Orthogonal and Symplectic Clifford Algebras, Spinor Structures (Kluwer, Dordrecht, 1990).Google Scholar
  19. C. Daviau, “Pourquoi il faut lire Hestenes,”Ann. Fond. Louis de Broglie 16, 391–403 (1991).Google Scholar
  20. R. Deheuvels,Formes quadratiques et groupes classiques (Presses Universitaires de France, Paris, 1981).Google Scholar
  21. R. Delanghe, “On regular-analytic functions with values in a Clifford algebra,”Math. Ann. 185, 91–111 (1970).Google Scholar
  22. R. Delanghe, F. Sommen, and V. Souček,Clifford Algebra and Spinor Valued Functions: A Function Theory for the Dirac Operator (Kluwer, Dordrecht, 1992).Google Scholar
  23. J. Gilbert and M. Murray,Clifford Algebras and Dirac Operators in Harmonic Analysis (Cambridge Studies in Advanced Mathematics26) (Cambridge University Press, Cambridge, 1991).Google Scholar
  24. W. Greub,Multilinear Algebra, 2nd edn. (Springer, Berlin, 1978).Google Scholar
  25. J. D. Hamilton, “The Dirac equation and Hestenes' geometric algebra,”J. Math. Phys. 25, 1823–1832 (1984).Google Scholar
  26. F. R. Harvey,Spinors and Calibrations (Academic Press, San Diego, 1990).Google Scholar
  27. J. Helmstetter,Algèbres de Clifford et algèbres de Weyl, Cahiers Math. 25, Montpellier, 1982.Google Scholar
  28. D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966, 1987, 1992).Google Scholar
  29. D. Hestenes, “Real spinor fields,”J. Math. Phys. 8, 798–808 (1967).Google Scholar
  30. D. Hestenes, “Multivector calculus,”J. Math. Anal. Appl. 24, 313–325 (1968).Google Scholar
  31. D. Hestenes, “Vectors, spinors and complex numbers in classical and quantum physics,”Am. J. Phys. 39, 1013–1028 (1971).Google Scholar
  32. D. Hestenes, “Observables, operators, and complex numbers in the Dirac theory,”J. Math. Phys. 16, 556–572 (1975).Google Scholar
  33. D. Hestenes, “Wherefore a science of teaching?”Phys. Teach. 17, 235–242 (1979).Google Scholar
  34. D. Hestenes, “Space-time structure of weak and electromagnetic interactions,”Found. Phys. 12, 153–168 (1982).Google Scholar
  35. D. Hestenes,New Foundations for Classical Mechanics (Reidel, Dordrecht, 1986, 1987).Google Scholar
  36. D. Hestenes, “The Zitterbewegung interpretation of quantum mechanics,”Found. Phys. 20, 1213–1232 (1990).Google Scholar
  37. D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984, 1987).Google Scholar
  38. B. Jancewicz,Multivectors and Clifford Algebra in Electrodynamics (World Scientific, Singapore, 1988).Google Scholar
  39. M.-A. Knus,Quadratic Forms, Clifford Algebras and Spinors (Universidad Estadual de Campinas, SP, 1988).Google Scholar
  40. T.-Y. Lam,The Algebraic Theory of Quadratic Forms (Benjamin, Reading, Massachusetts, 1973, 1980).Google Scholar
  41. H. B. Lawson and M.-L. Michelsohn,Spin Geometry (Princeton University Press, Princeton, New Jersey, 1989).Google Scholar
  42. R. Lipschitz, “Principles d'un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions,”C.R. Acad. Sci. (Paris) 91, 619–621, 660–664 (1880). Reprinted inBull. Soc. Math. (2) 11, 115–120 (1887).Google Scholar
  43. R. Lipschitz,Untersuchungen über die Summen von Quadraten (Max Cohen & Sohn, Bonn, 1886), pp. 1–147. [The first chapter of pp. 5–57 translated into French by J. Molk: “Recherches sur la transformation, par des substitutions réelles, d'une somme de deux ou troix carrés en elle-même,”J. Math. Pures Appl. (4) 2, 373–439 (1886). French résumé of all the three chapters inBull. Sci. Math. (2) 10, 163–183 (1886).]Google Scholar
  44. R. Lipschitz (signed), “Correspondence,”Ann. Math. 69, 247–251 (1959).Google Scholar
  45. P. Lounesto, “Report on Conference, NATO and SERC Workshop on Clifford Algebras and Their Applications in Mathematical Physics,” University of Kent, Canterbury, England, 1985.Found. Phys. 16, 967–971 (1986).Google Scholar
  46. J. Marsh, Book review: D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus, Am. J. Phys. 53, 510–511 (1985).Google Scholar
  47. A. Micali and Ph. Revoy,Modules quadratiques, Cahiers Math. 10, Montpellier, 1977, reprinted inBull. Soc. Math. Fr. 63, Suppl., 5–144 (1979).Google Scholar
  48. A. Micali, R. Boudet, and J. Helmstetter, eds.,Proceedings of the Second Workshop on “Clifford Algebras and Their Applications in Mathematical Physics,” Université des Sciences et Techniques du Languedoc, Montpellier, France, 1989 (Kluwer, Dordrecht, 1992).Google Scholar
  49. C. Poole, Book review: D. Hestenes,New Foundations for Classical Mechanics, Found. Phys.17, 859–862 (1987).Google Scholar
  50. I. R. Porteous,Topological Geometry (Van Nostrand Reinhold, London, 1969; Cambridge University Press, Cambridge, 1981).Google Scholar
  51. M. Riesz, “Sur certaines notions fondamentales en théorie quantique relativiste,”C. R. 10 e Congrès Math. Scandinaves, Copenhagen, 1946. (Gjellerups, Copenhagen, 1947), pp. 123–148. Reprinted in L. Gårding and L. Hörmander, eds.,Marcel Riesz, Collected Papers (Springer, Berlin, 1988), pp. 545–570.Google Scholar
  52. M. Riesz,Clifford Numbers and Spinors (The Institute for Fluid Dynamics and Applied Mathematics, Lecture Series38) (University of Maryland, University Park, 1958). Reprinted as facsimile by Kluwer, 1993 (E. F. Bolinder and P. Lounesto, eds.).Google Scholar
  53. H. Rothe, “Die Komplexen Zahlensysteme von W. K. Clifford und R. Lipschitz. Die orthogonalen Transformationen vonn Veränderlichen. Die Bewegungen und Umlegungen imn-dimensionalen Euklidischen und nichteuklidischen Raum,”Encykl. Math. Wiss. III AB 11, 1410–1416 (1916).Google Scholar
  54. E. Witt, “Theorie der quadratischen Formen in beliebigen Körpern,”J. Reine Angew. Math. 176, 31–44 (1937).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Pertti Lounesto
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland

Personalised recommendations