Covariant coordinate transformations and scalar-field-matter interactions

We propose a classical Lorentz invariant model of local scalar-field-matter interactions where the zeroth component of the space-time coordinates of a particle is modified by a background Klein-Gordon field. This model is not directly explained in terms of the standard model (SM) because the CPT symmetry is generally broken with a Hermitian Hamiltonian. Our ultimate goal (which is not explored here) is to construct a quantum field theory approach to tackle the matter-antimatter asymmetry problem. To understand the basics of the theory, two examples are discussed only at the quantum mechanical level: (i) a (quantum) harmonic oscillator with the occurrence of negative-energy eigenvalues where the effective energy-masses of the 'particle and antiparticle' are not symmetric with respect to the non-physical zero-energy mode of the oscillator; (ii) a Dirac particle in an electromagnetic (EM) linear potential.