Unitary Relativistic Quantum Dynamics and Electromagnetic Field

In this chapter, we consider a quantum particle wave function with a bound spectrum of velocity c, and obtain the relativistic momentum based on the group velocity of this wave function. With a space-time isometry condition, the Lorentz transformation and the relativistic dynamics were obtained. Considering a field interacting with a quantum particle as a four-vector conjugated to the space-time vector in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific to the electromagnetic field, while the other equations are general for a field interacting with a charged quantum particle. Considering the time-dependent phase of a quantum particle interacting with an electromagnetic field with a space-time homogeneity condition, we obtain Lorentz transformations for this field. For a quantum particle at a non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor, of a frequency proportional to the rest energy of this particle. From the Schrödinger equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave function into four components, describing a proper rotation of this particle with an angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also calculate electron potential in the magnetic field, and two-electron interaction potential.

Keywords: Action, Ampère-Maxwell equation, Angular momentum, Bohr magneton, Dirac matrices, Electric field, Electric potential, Faradey-Maxwell equation, Four-vector, Gauss equation, Giro-magnetic ratio, Group velocity, Hamiltonian, Kinetic energy, Lagrangian, Liénard-Wiechert potentials, Lorentz transformation, Magnetic field, Magnetic moment, Momentum, Pauli matrices, Potential energy, Rest energy, Rest mass, Spin, Vector potential, Wave function, Wave-packet.