Charged Quantum Particle in Gravitation and Electromagnetic Fields

We describe the interaction of a quantum particle with an electromagnetic field by additional Lagrangian terms in the time dependent phases of the two conjugate wave functions, proportional to the electric charge, with a scalar potential conjugated to time, and a vector potential conjugated to the space coordinates. From the group velocity in the momentum space, we obtain the Lorentz force, as a function of the electric and magnetic fields satisfying the Faraday-Maxwell equation of the electromagnetic induction, and the two Gauss-Maxwell equations for the electric and magnetic fields. When the electromagnetic field is considered as a wave propagating with the maximum relativistic velocity c , the Ampère-Maxwell law and the charge conservation equation are obtained. We consider a gravitational wave for the metric tensor satisfying the D’Alembert equation. For the amplitude tensor of a first-order solution of this equation, we obtain a scalar which describes a rotation of this tensor in a plane perpendicular to the propagation direction of the gravitational wave, with an angular momentum 2, which we call the graviton spin. For the polarization vector we also obtain an invariant describing a matter rotation in this plane, with an integer spin for Bosons, and a half-integer spin for Fermions. The first-order solution describes a particle acceleration in the direction of propagation of the gravitational wave, as the second-order solution describes a harmonic oscillation in this wave. We consider the two propagation wave functions as products of propagation factors, depending only on coordinates and momentum, with time-dependent wave functions satisfying Schrödinger-like time-dependent equations. For a time-dependent wave four-vector, we obtain Dirac-like equations including additional terms explicitly depending on velocity, as it is expected for any relativistic equations. For an electromagnetic decay of a quantum particle, we obtain a redshift depending on the gravitational field. For a free quantum particle with a generalized momentum including the time-dependent component, the time-dependent equations take the form of the quantum field equations with matrix coefficients satisfying the Clifford algebra. We obtain the solutions of these equations for particles and anti-particles, as integrals over the spatial momentum domain, which determines a finite distribution of matter in the coordinate space. Since any matter velocity is equal to the wave velocity, r = ∂/∂p cE , this distribution is invariant.