Are there conserved quantities in field theory which don't arise from Noether's Theorem?

Question

In some QFT texts one writes down the number operator $N$ for free theories, such that when acting on an $n$-particle state $|n\rangle$ we have

$$N|n\rangle=n|n\rangle$$

In free theories this is a conserved quantity. However I have never seen this quantity derived by using Noether's theorem, i.e. as a consequence of the invariance of the action under some transformation of the fields or coordinates.

Is it possible to derive the number operator via Noether's theorem? If not, is it possible for a theory to have more conserved quantities than just those accessible to Noether's theorem?

Answer 1

There are conserved quantities which don't come from Noether's Theorem. For instance, the topological numbers that characterize the so called topological solutions such as vortices, monopoles, instantons, etc.

In general these topological solutions arise in non-linear, vacuum degenerate and spontaneously broken theories. For gauge theories these topological charges are associated to the topology of the vacuum manifold which can be studied in terms of the gauge group and the spontaneous symmetry breaking pattern.

Answer 2

A simpler example than Diracology's is that any quantity that commutes with the Hamiltonian is conserved. Often, these quantities can be thought of as coming from discrete symmetries, while Noether's theorem is only concerned with continuous symmetries. For example, if the Hamiltonian is parity-invariant (i.e. commutes with the parity operator) than the even- and odd-parity sectors will be conserved.