Foundations of Physics

, Volume 38, Issue 11, pp 1020–1045 | Cite as

Dirac-Type Equations in a Gravitational Field, with Vector Wave Function



An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense.


Quantum mechanics in a gravitational field Classical-quantum correspondence Dirac and Klein-Gordon equations Preferred reference frame Tensor Dirac theory 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Laboratory “Soils, Solids, Structures—Risks”CNRS & University of GrenobleGrenobleFrance

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