Foundations of Physics

, Volume 42, Issue 1, pp 192–208 | Cite as

Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

  • B. J. HileyEmail author
  • R. E. Callaghan


In this paper we show how the dynamics of the Schrödinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, \({\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}\), and \({\mathcal{C}}_{0,1}\). Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time.


Clifford algebras Schrödinger, Pauli and relativistic Dirac-Bohm model Relativistic quantum potential Spin evolution 


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  1. 1.
    Ballentine, L.E.: Quantum Mechanics, p. 129. Prentice Hall, Englewood Cliffs (1990) Google Scholar
  2. 2.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of hidden variables, I. Phys. Rev. 85, 166–179 (1952) CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bohm, D., Hiley, B.J.: An ontological basis for quantum theory: I—non-relativistic particle systems. Phys. Rep. 144, 323–348 (1987) CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bohm, D., Hiley, B.J.: On the relativistic invariance of a quantum theory based on beables. Found. Phys. 21, 243–50 (1991) CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bohm, D., Hiley, B.J.: The Undivided Universe: An Ontological Interpretation of Quantum Theory. Routledge, London (1993) Google Scholar
  6. 6.
    Bohm, D., Schiller, R., Tiomno, J.: A causal interpretation of the Pauli equation (A) and (B). Nuovo Cimento Supp. 1, 48–66, 67–91 (1955) MathSciNetGoogle Scholar
  7. 7.
    Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras: Spinor Structures. Kluwer, Dordrecht (1990) zbMATHGoogle Scholar
  8. 8.
    Dewdney, C., Holland, P.R., Kyprianidis, A.: What happens in a spin measurement? Phys. Lett. A 119, 259–67 (1986) CrossRefADSGoogle Scholar
  9. 9.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) zbMATHGoogle Scholar
  10. 10.
    Eddington, A.S.: Relativity Theory of Protons and Electrons. Cambridge University Press, Cambridge (1936) Google Scholar
  11. 11.
    Emch, G.G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory, p. 32. Wiley-Interscience, New York (1972) zbMATHGoogle Scholar
  12. 12.
    Flanders, H.: Differential Forms with Applications to the Physical Sciences. Academic Press, New York (1963) zbMATHGoogle Scholar
  13. 13.
    Frescura, F.A.M., Hiley, B.J.: The implicate order, algebras, and the spinor. Found. Phys. 10, 7–31 (1980) CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Frescura, F.A.M., Hiley, B.J.: Geometric interpretation of the Pauli spinor. Am. J. Phys. 49, 152–157 (1981) CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Gilbert, J.E., Murray, M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis, p. 92. Cambridge University Press, Cambridge (1991) CrossRefzbMATHGoogle Scholar
  16. 16.
    Gull, S., Lasenby, A., Doran, C.: Electron paths, tunnelling and diffraction in the spacetime algebra. Found. Phys. 23, 1329–56 (1993) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Haag, R.: Local Quantum Physics, p. 109. Springer, Berlin (1992) zbMATHGoogle Scholar
  18. 18.
    Hestenes, D.: Local observables in the Dirac theory. J. Math. Phys. 14, 1–32 (1973) CrossRefGoogle Scholar
  19. 19.
    Hestenes, D.: Observables, operators and complex numbers in the Dirac theory. J. Math. Phys. 16, 556–572 (1975) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Hestenes, D.: Clifford algebra and the interpretation of quantum mechanics. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and their Applications in Mathematical Physics, p. 1. Reidel, Dordrecht (1986) Google Scholar
  21. 21.
    Hestenes, D.: Space-time physics with geometric algebra. Am. J. Phys. 71, 691–714 (2003) CrossRefADSGoogle Scholar
  22. 22.
    Hestenes, D., Gurtler, R.: Local observables in quantum theory. Am. J. Phys. 39, 1028–1038 (1971) CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Hiley, B.J.: Process, distinction, groupoids and clifford algebras: an alternative view of the quantum formalism. In: Coecke, B. (ed.) New Structures for Physics. Lecture Notes in Physics, vol. 813, pp. 705–750. Springer, Berlin (2011) CrossRefGoogle Scholar
  24. 24.
    Hiley, B.J., Callaghan, R.E.: The Clifford algebra approach to quantum mechanics B: the Dirac particle and its relation to the Bohm approach. maths-ph 1011.4033 (2010)
  25. 25.
    Hiley, B.J., Callaghan, R.E.: The Clifford algebra approach to quantum mechanics A: the Schrödinger and Pauli particles. maths-ph 1011.4031 (2010)
  26. 26.
    Horton, G., Dewdney, C.: A non-local, Lorentz-invariant, hidden-variable interpretation of relativistic quantum mechanics. J. Phys. A 34, 9871–9878 (2001) CrossRefzbMATHADSMathSciNetGoogle Scholar
  27. 27.
    Takabayasi, T.: Relativistic hydrodynamics of the Dirac matter. In: Progress of Theoretical Physics, Supplement (4), pp. 2–80 (1957) Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.TPRU, BirkbeckUniversity of LondonLondonUK

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