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I am aware of Noether's Theorem which proves that for every differentiable symmetry of a system, there exists a corresponding conserved quantity, e.g. invariance under rotation implies conservation of angular momentum.

I was wondering what conserved quantity is implied by the trivial symmetry (i.e. the symmetry which does nothing).

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    $\begingroup$ Why does this qualify as symmetry? As far as I remember, Noethers theorem does not apply to all symmetries, but only to continuous symmetries. Maybe there are extensions for special cases of discrete symmetries. $\endgroup$
    – Semoi
    Commented Feb 6, 2020 at 22:00
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    $\begingroup$ It is the trivial conserved quantity is $A := 0$. $\endgroup$ Commented Feb 6, 2020 at 22:05
  • $\begingroup$ @Semoi doing nothing is the most continuous thing you can do to something $\endgroup$ Commented Feb 6, 2020 at 22:28
  • $\begingroup$ @Semoi Yes and no. The Noether theorem does not carry over to that case, namely discrete symmetries do not have an associated conserved current. Nevertheless we have the notion of a charge which acts on operators in the same way as the integral of $J^0$ does. In particular it gives the standard Ward identities. $\endgroup$
    – MannyC
    Commented Feb 6, 2020 at 23:04
  • $\begingroup$ @Semoi The trivial symmetry is continuous, in fact differentiable. Unless I am misunderstanding the definition of continuity? $\endgroup$
    – user609020
    Commented Feb 7, 2020 at 2:02

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This is a rather academic discussion, but if you plug the trivial transformation into Noether's theorem, the quantity you get is... zero. Which is indeed conserved.

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