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In general, we say a transformation is a symmetry of a theory if it leaves the action invariant, i.e. if

$$S \to S' = S,$$

up to, perhaps, a boundary term (b.t.). However, it is known (see e.g. this post) that the equations of motion of a theory are left invariant if a weaker condition is satisfied. Namely if the action is invariant up to a b.t. and a scaling, i.e. if

$$S \to S' = \alpha S + b.t.$$

This generalized notion of symmetry does not fit well in Noether's theorem. In particular, if I understood correctly, there exists a conserved Noether current only when $\alpha=1$ (see this post). My question is: Why do we care? I mean, symmetry is a fundamental, physical notion, while conserved current is only a mathematical artifact that may facilitate computation. So then, why do we stick to Noether symmetries, which have an associated conserved current? Why don't we seem to care for these generalized symmetries, which leave the equations of motion unchanged? (At least at the classical level, since quantization requires the action to be strictly invariant.)

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    $\begingroup$ Your claim is not true. Noether's theorem holds for quasi-symmetries as well. $\endgroup$
    – Gold
    May 13, 2021 at 11:55
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    $\begingroup$ "conserved current is only a mathematical artifact that may facilitate computation". Particle physics would be unthinkable without detailed accounting of conserved quantities, and the Ward identities such conserved charges generate. Indeed, the sublime Lie-algebraic structure of our world would be unthinkable without them. Your particle physics course skipped them? $\endgroup$ May 13, 2021 at 13:43
  • $\begingroup$ @CosmasZachos As you say, particle physics and the "sublime Lie-algebraic structure of our world" is based on Ward identities, which are the quantum analog of conserved currents. My question, if I expressed it correctly, asks why. There are more symmetries (of the e.o.m.) than those associated to conserved currents. $\endgroup$
    – MBolin
    May 13, 2021 at 17:33
  • $\begingroup$ I'm sure the further symmetries have important consequences, and are being investigated as we speak.... (My pet planet is nonlocal Yangian symmetries...) $\endgroup$ May 13, 2021 at 17:36

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