Abstract
We obtain a Lorentz covariant wave equation whose complex wave function transforms under a Lorentz boost according to the following rule, Ψ x ⟶ e i / ℏ f x Ψ x . We show that the spacetime-dependent phase f x 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 Ψ x 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 Ψ x . 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.
Highlights
A complex scalar field ΦðxÞ is usually defined as a function of the spacetime coordinates x = ðct, rÞ such that it remains invariant under a symmetry operation, i.e., ΦðxÞ = Φ′ðx′Þ, where Φ′ðx′Þ is the field after we apply the symmetry operation
Two important issues arise: (1) Is there a fundamental physical process leading to the violation of the Lorentz invariance of the Lorentz covariant Schrödinger theory? What causes it? Can it be traced back to the presence of a background gravitational field? Is this Lorentz invariance-breaking process a feature of the present-day universe or it was relevant in its beginning, being suppressed during its evolution? (2) Can we find a Lorentz invariant theory that gives different masses for particles and antiparticles and, at the same time, leads to an asymmetry in the decay of particles and antiparticles as described above? Can it be done without violating the CPT theorem, or at least without violating its extension, the CPTM theorem as given in this work?
It is clear that whenever bilinear functions of the fields are associated with observable quantities, we only need those bilinears to be invariant under that symmetry operation to obtain invariant physical results
Summary
The Schrödinger field ΨðxÞ illustrates that it is perfectly possible to build a logically consistent field theory assuming a more general transformation law for a complex scalar field under a symmetry operation, where the phase θ depends on the spacetime coordinates The extension of the latter observation to the relativistic domain is the leitmotif of the present work. We observed that particles and antiparticles with the same wave number no longer had the same energies and that we could make them have different momenta as well by properly tuning the free parameters of the field theories here developed Despite all these peculiarities, and as we already noted above, at the level of electromagnetic interactions, the present theories were shown to be equivalent to the Klein-Gordon one. These results, together with the major ones highlighted above, are for ease of access systematically put together below before we start the more technical part of this work
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