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.

35 pages, 1 figure