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Husserliana97

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From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry groups. A classic, even historical, example (this is the one that Noether had in mind) is the invariance group of Riemannian spacetimes, i.e. the set of spacetime diffeomorphisms of GR. First question: what other groups are possible, apart from general relativity? Does Noether's second theorem also apply to quantum field theory (= the 'gauge groups')? Or is it confined to general relativity?

This theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to these theories satisfy identities, rather than "classical" conservation laws (called "proper conservation laws" by Hilbert), and the quantities thus "improperly" conserved are associated with local symmetries -- and no longer global symmetries, as in Noether's first theorem.

My second question is based on a quotation from an article entitled "On Noether's theorems and gauge theories in Hamiltonian formulation". Here is the extract:

"This brief treatment shows that, when a local group of transformations has a non-trivial global subgroup, it is possible to find some conserved quantities *without requiring Euler- Lagrange equations of motion to be satisfied* and where arbitrary functions are involved. These conservation laws are called improper laws of conservation and we can see the seeds of gauge theories hiding in them."

It's the bit in italics that worries me. Does the second theorem imply the existence, at least as a possibility, of movements that do not respect Lagrange's equations? In other words, violating the principle of stationary action? This is no doubt a naive question (I think I know that you can derive the GR field equation from the action principle), but one that I can't manage to solve on my own.