# Conserved quantity corresponding to the identity symmetry

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).

• 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. Feb 6, 2020 at 22:00
• It is the trivial conserved quantity is $A := 0$. Feb 6, 2020 at 22:05
• @Semoi doing nothing is the most continuous thing you can do to something Feb 6, 2020 at 22:28
• @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. Feb 6, 2020 at 23:04
• @Semoi The trivial symmetry is continuous, in fact differentiable. Unless I am misunderstanding the definition of continuity? Feb 7, 2020 at 2:02