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In relativistic Quantum Field Theory, in contrast, divergences arise. In order to deal with these divergences, one uses renormalization. According to this renormalization, the 'macroscopic' parameters of the lagrangian among which the mass is one, are not the "real" bare quantities. An interacting particle will drag an infinite amount of loop corrections with it in principle, but one has no experimental access to all of these. Instead, all loop corrections above a certain order in perturbation theory cannot be distinguished and are therefore considered as being part of a 'dressed mass'.

But it seems to me the previous implies there is no other fundamental mass constant than the bare mass! The dressed mass depends on the resolution of the measurement, how far you have pushed the renormalization trough. I.e. the dressed mass depends on your knowledge of the system such as for example an entropy does. How can this correspond with the 'electron mass as fundamental constant'?