2005/12/31 by Mayeul Arminjon
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Causal fermion system #Dirac algebra #Dirac equation #Dirac sea #Dirac spinor #Gamma matrices #Klein–Gordon equation #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Relativistic wave equations #Two-body Dirac equations #gr-qc #quant-ph
paper · pdf · doi:10.1007/s10702-006-0514-7
published as Found.Phys.Lett.19:225-247,2006 · 27 pages, standard LaTeX. v2: minor style changes, accepted for publication in Found. Phys. Letters
arxiv created 2006/01/11 · openalex publication_date 2006/04/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.