On Lorentz Invariant Complex Scalar Fields

Abstract

We obtain a Lorentz covariant wave equation whose complex wave function transforms under a Lorentz boost according to the following rule, $\Psi \left(x\right)⟶e^{\left(i/\hslash \right)f\left(x\right)}\Psi \left(x\right)$ . We show that the spacetime-dependent phase $f\left(x\right)$ is the most natural relativistic extension of the phase associated with the transformation rule for the nonrelativistic Schrödinger wave function when it is subjected to a Galilean transformation. We then generalize the previous analysis by postulating that $\Psi \left(x\right)$ transforms according to the above rule under proper Lorentz transformations (boosts or spatial rotations). This is the most general transformation rule compatible with a Lorentz invariant physical theory whose observables are bilinear functions of the field $\Psi \left(x\right)$ . We use the previous wave equations to describe several physical systems. In particular, we solve the bound state and scattering problems of two particles which interact both electromagnetically and gravitationally (static electromagnetic and gravitational fields). The former interaction is modeled via the minimal coupling prescription while the latter enters via an external potential. We also formulate logically consistent classical and quantum field theories associated with these Lorentz covariant wave equations. We show that it is possible to make those theories equivalent to the Klein-Gordon theory whenever we have self-interacting terms that do not break their Lorentz invariance or if we introduce electromagnetic interactions via the minimal coupling prescription. For interactions that break Lorentz invariance, we show that the present theories imply that particles and antiparticles behave differently at decaying processes, with the latter being more unstable. This suggests a possible connection between Lorentz invariance-breaking interactions and the matter-antimatter asymmetry problem.