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Basically, being able to write down a Lagrangian density isn't the same thing as being able to describe the classical theory that is the counterpart of a quantized system. In particular, it seems like this can't possibly work for unstable particles. For example, the Lagrangian density for muon decay has a constant in it, G

_{F}, the Fermi coupling constant. The half-life of the muon goes like [itex]h/G_F^2[/itex]. In the classical limit, the half-life goes to zero, so the classical theory of muons is a theory with no muons in it.