Lorentz-violating scalar Hamiltonian and equivalence principle in a static metric

Abstract

In this paper, we obtain a nonrelativistic Hamiltonian from the Lorentz-violating (LV) scalar Lagrangian in the minimal SME. The Hamiltonian is obtained by two different methods. One is through the usual ansatz $\Phi(t,\vec{r})=e^{-imt}\Psi(t,\vec{r})$ applied to the LV corrected Klein-Gordon equation, and the other is the Foldy-Wouthuysen transformation. The consistency of our results is also partially supported by the comparison with the spin-independent part of the fermion Hamiltonian. In this comparison, we can also establish a relation between the set of scalar LV coefficients with their fermion counterparts. Using a pedagogical definition of the weak equivalence principle (WEP), we further point out that the LV Hamiltonian not only necessarily violates universal free fall, which is clearly demonstrated in the geodesic deviation, but also violates WEP in a semi-classical setting. As a bosonic complement, this method can be straightforwardly applicable to the spin-1 case, which shall be useful in the analysis of atomic tests of WEP, such as the case of the $^{87}\text{Rb}_1$ atom.