Foundations of Physics Letters

, Volume 19, Issue 3, pp 225–247 | Cite as

Dirac Equation from the Hamiltonian and the Case With a Gravitational Field

  • Mayeul Arminjon
Original Article


Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation—in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-spacetime Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the static-gravitational case is not equivalent to the standard (Fock-Weyl) gravitational extension of the Dirac equation.

Key words:

Dirac equation classical-quantum correspondence spinor representation 4-vector gravitation curved spacetime 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Mayeul Arminjon
    • 1
    • 2
  1. 1.Dipartimento di FisicaUniversità di Bari Via Amendola 173BariItaly
  2. 2.Laboratoire “Sols, Solides, Structures”Unité Mixte de Recherche of the CNRSGrenoble cedex 9France

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