quantum mechanics - Conserved charges and generators

For the Klein Gordon field, the conserved charge for translation in space is given by: $$\vec{P}=\frac{1}{2}\int d^{3}k \, \vec{k}\{a^{\dagger}_{k}a_{k}+a_{k}a^{\dagger}_{k}\}$$

If we were to find the generators for a space translation, we would find that, $$P_{j}=i\partial_{j},$$ where $j=1,2,3$.

If we act both of the above operators on the field $\phi$, the result matches! My question is whether both of these, the generators and the conserved charges of a symmetry are always the same thing? What would be a simple way to see this connection?

Answer

OP is wondering whether the conserved charge associated to a continuous symmetry always generates the symmetry itself. We can say, in full generality, that the answer is

Yes.

Let us see how this works.

Classical mechanics.

We use a notation adapted to classical field theory rather than point-particle mechanics, but the former includes the latter as a special sub-case so we are losing no generality.

Consider a classical system which may or may not include gauge fields and/or Grassmann odd variables. For simplicity, we consider a flat space-time. Assume the system is invariant under the infinitesimal transformation $\phi\to\phi+\delta\phi$. According to Noether's theorem, there is a current $j^\mu$ $$j^\mu\sim \frac{\partial\mathcal L}{\partial\dot\phi_{,\mu}}\delta\phi$$ which is conserved on-shell, $$\partial_\mu j^\mu\overset{\mathrm{OS}}=0$$

This in turns implies that the associated Noether charge $Q$ $$Q\overset{\mathrm{def}}=\int_{\mathbb R^{d-1}} j^{0}\,\mathrm d\boldsymbol x$$ is conserved, $$\dot Q\overset{\mathrm{OS}}=0$$

In Ref.1 it is proved that the charge $Q$ generates the transformation $\delta\phi$,

$$\delta\phi=(Q,\phi)$$

where $(\cdot,\cdot)$ is the DeWitt-Peierls bracket. This is precisely our claim. The reader will find the proof of the theorem in the quoted reference, as well as a nice discussion about the significance of the result.

Furthermore, a similar statement holds when space-time is curved, but this requires the existence of a suitable Killing field (cf. this PSE post).

Moreover, for standard canonical systems, Ref.1 also proves that $(\cdot,\cdot)$ agrees with the Poisson bracket $\{\cdot,\cdot\}$.

Quantum mechanics.

This is in fact a corollary of the previous case. Ref.1 proves that, up to the usual ordering ambiguities inherent to the quantisation procedure, the DeWitt-Peierls bracket of two fundamental fields agrees with the commutator $[\cdot,\cdot]$ of the corresponding operators.

If we assume that the classical conservation law $\partial_\mu j^\mu\equiv 0$ is not violated by the regulator (i.e., if the symmetry is not anomalous), then we automatically obtain the quantum analogue of our previous result, to wit

$$\delta\phi=-i[Q,\phi]$$

as required.

References

1. Bryce DeWitt, The Global Approach to Quantum Field Theory.