Abstract
In this paper, we obtain a nonrelativistic Hamiltonian from the Lorentz-violating (LV) scalar Lagrangian in the minimal standard model extension (SME). The Hamiltonian is obtained by two different methods. One is through the usual ansatz $\mathrm{\ensuremath{\Phi}}(t,\stackrel{\ensuremath{\rightarrow}}{r})={e}^{\ensuremath{-}imt}\mathrm{\ensuremath{\Psi}}(t,\stackrel{\ensuremath{\rightarrow}}{r})$ 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 semiclassical 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 ${^{87}\mathrm{Rb}}_{1}$ atom.
Highlights
Symmetry has been a main theme of physics in the previous century and may continue to be so in the 21st century
This is a little surprising because the results are expected to differ by a pseudounitary transformation; inspecting the CVZ method, we see that it is exactly the pseudounitary transformation, which ensures that the NR Hamiltonian is the same as that obtained with direct Foldy-Wouthuysen transformation (FWT) [38]
Using the test particle assumption, we derive it from two different methods in a static isotropic metric
Summary
Symmetry has been a main theme of physics in the previous century and may continue to be so in the 21st century. If proven to be true, it will definitely be a concrete clue to the physics at Planck scale, an ultrahigh energy scale far beyond any direct experimental access To thoroughly explore this possibility, Kostelecký and collaborators established an effective field theory called Standard Model Extension (SME) [7,8,9], which incorporates SM and GR, with various possible LV operators. Time reversal (CPT) symmetry and has already become a powerful toolbox in both theoretical and phenomenological investigations in this field [10] As another conceptual bridge from special relativity to GR, the equivalence principle (EP), especially the Einstein equivalence principle (EEP), entails a close relationship to Lorentz symmetry and has been broadly tested in various kinds of physical systems [11,12,13,14]. The convention is the same as that in Ref. [8], where diagðημνÞ 1⁄4 ð−1; 1; 1; 1Þ and ε0123 1⁄4 þ1
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