Generalized Noether Theorems for Field Theories Formulated in Minkowski Spacetime

Abstract

New symmetry theorems are obtained for field theories formulated in Minkowski spacetime, based on the recognition that such theories should be diffeomorphism invariant. These theorems, which are in fact generalized Noether theorems, have several nontrivial ramifications. One immediate consequence is the necessity to re-evaluate some of the default characterizations of the original Noether theorems as general statements - for instance, the statement that every global invariance of the action gives rise to a conserved current. As it turns out, this latter statement remains valid, but apparently only as a consequence of both the generalized first and second Noether theorem (at any rate, it is very far from clear how to avoid such a conclusion). A ramification of a more constructive nature consists in a novel expression for the so-called Belinfante stress-energy tensor, for which it is evident that it generalizes to the correct stress-energy tensor within the context of general relativity. The crucial symmetry property here is diffeomorphism invariance, rather than Poincaré invariance. Using this new form for the stress-energy tensor, an alternative route to general relativity also becomes available, which, in contrast to many standard treatments, is intrinsically based on a variational formulation - i.e. without the need to have prior knowledge of the form of Einstein's equation. The connection of these results to related results previously established in the literature (although in a completely different context) will be pointed out where applicable.