Dirac equation from the Hamiltonian and the case with a gravitational field

Details Dirac equation from the Hamiltonian and the case with a gravitational field Dirac equation from the Hamiltonian and the case with a gravitational field

2005-12-07

arxiv.org/abs/gr-qc/0512046v2

Found.Phys.Lett.19:225-247,2006

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

27 pages, standard LaTeX. v2: minor style changes, accepted for publication in Found. Phys. Letters

Scientific paper

10.1007/s10702-006-0514-7

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-space-time 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.

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