  In this article we investigate whether a theory based on a classical
Lagrangian for the minimal Standard-Model Extension (SME) can be quantized such
that the result is equal to the corresponding low-energy Hamilton operator
obtained from the field-theory description. This analysis is carried out for
the whole collection of minimal Lagrangians found in the literature. The upshot
is that first quantization can be performed consistently. The unexpected
observation is made that at first order in Lorentz violation and at second
order in the velocity the Lagrangians are related to the Hamilton functions by
a simple transformation. Under mild assumptions, it is shown that this holds
universally. This result is used successfully to obtain classical Lagrangians
for two complicated sectors of the minimal SME that have not been considered in
the literature so far. Therefore, it will not be an obstacle anymore to derive
such Lagrangians even for involved sets of coefficients - at least to the level
of approximation stated above.
