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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?

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    $\begingroup$ For the generic question of whether every conserved quantity arises from a symmetry, see this question. $\endgroup$
    – ACuriousMind ♦
    Commented Jan 8, 2016 at 21:01
  • $\begingroup$ @AccidentalFourierTransform This isn't true. The Number operator is a conserved quantity in the free real Klein Gordon theory and your transformation is not a symmetry of the corresponding action. Your transformation corresponds to the charge of a complex scalar. $\endgroup$
    – Okazaki
    Commented Jan 8, 2016 at 21:02
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    $\begingroup$ @AccidentalFourierTransform In complex phi four theory, sure. But that's not what I'm asking. Indeed I can get the number operator in the complex case by this transformation, but I'm talking in general. Your transformation doesn't work for real Klein Gordon theory, for example, nor for pure non-interacting Maxwell theory. $\endgroup$
    – Okazaki
    Commented Jan 8, 2016 at 21:14
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    $\begingroup$ @bechira At least in the case of complex Klein Gordon theory the charger operator $Q$ can be derived from Noether's theorem. $Q$ is also equal to $N$ up to an irrelevant constant term. In particular [Q.H]=0 follows from quantizing the classical Noether charge, so it does have a classical counterpart. But Q=N+c from which we have that [N,H]=0. But this seems very odd if we state "something with classical counterpart = something with no classical counterpart + constant". $\endgroup$
    – Okazaki
    Commented Jan 8, 2016 at 21:22

2 Answers 2

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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.

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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.

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