For several years research has been performed on how to map the
field description of the Standard-Model Extension (SME) to a
classical-particle analog. This is carried out to have a base
for incorporating Lorentz violation into gravity. General
Relativity and extensions of it are classical theories.
Therefore, it is reasonable to obtain from the SME how
classical, relativistic pointlike particles are affected by
Lorentz violation. This puts us into a position to couple such a
particle to a curved spacetime and to consider its modified
In the recent article
I obtain the first classical Lagrangians for various sectors of
the nonminimal SME. Recall that the nonminimal SME incorporates
all CPT- and Lorentz-violating contributions based on
higher-dimensional field operators.
In another paper
I collect all classical Lagrangians obtained for the minimal
SME. Each of these Lagrangians is promoted to a
quantum-mechanical Hamilton operator. The latter are shown to
correctly match the low-energy limit of the SME Hamiltonian,
i.e., quantization can be performed consistently. Furthermore,
it is proven that at first order in Lorentz violation and at
second order in the momentum, all minimal Lagrangians are linked
to the Hamilton functions by a simple transformation.