On the non-relativistic limit of a spin-12 particle in a classical gravitational field

An external gravitational field modifies the description of a spin-12 particle in various ways. For instance, the inner product of Dirac wave functions, and the equal-time anti-commutation relations and the canonical energy-momentum tensor of the quantized Dirac field are modified. This has the following consequences. The Dirac-Hamiltonian (2.29) of a spin-12 particle in a time-dependent gravitational field is not Hermitian. Furthermore, the Euler-Lagrange equation and the Heisenberg equation for the quantized Dirac field are not consistent. We obviate these deficiencies by the introduction of the η-field as the fundamental variable instead of the Dirac field. At the same time, the non-relativistic limit is most conveniently discussed in the η-description. For this purpose, we introduce a modification of the Foldy-Wouthuysen transformation.