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Lie Derivatives and Symmetry
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[QUOTE="jphsu, post: 6255459, member: 669342"] Yes, there is a link to the external space-time symmetry in inertial frames. However, it is neither simple nor direct. The vanishing Lie derivative of an action is related to the conservation of energy-momentum in the following way: One must consider Lie derivatives in the coordinate expressions (rather than in a general manifold). In this specific case, one can verify and see that the Lie derivatives of general tensors in inertial frames is related to the (infinitesimal) local space-time translation (T_4 group): x'^\mu = x^\mu + \Lambda^\mu(x). The vanishing of the Lie derivative L_{\Lambda} of the action (via Cartan formula) of a gauge symmetry theory is equivalent to the invarince of the theory under the local T_4 translations (with 4 group generators). Hence, it implies the conservation of energy-momentum, which is more direct to see with Fock's approach rather than Noether's approach. Note that if one interprets x'^\mu = x^\mu + \Lambda^\mu(x) as arbitrary coordinate transformations in curved space-time, then there are continuously infinite numbers of generators and, as a result, it is not related to any physical conservation law in the usual sense. (Cf. Noether's theorem II in her famous 1918 paper.) [/QUOTE]
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