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Consider a classical theory on a manifold $M$, together with a smooth action $\theta: G\times M \to M$ of a Lie group $G$. Equivalently, one may regard $\theta: G \to \mathrm{Diff}(M)$. The Lie group action induces a Lie algebra homomorphism $V: \mathfrak g \to \mathfrak X(M)$. Elements of $V(\mathfrak g)$ are called as infinitesimal generators. For each $X\in \mathfrak g$, we have the Noether current $j_X^\mu$ and the Noether charge $Q_X$.

Quantizing the classical theory, we have a Hilbert space $H$. We also expect that the classical $G$-action should be translated to a unitary representation $\hat\theta: \hat G \to U(H)$, and the associated representation $\hat V: \hat{\mathfrak g} \to \mathfrak u(H)$. Here, $\hat G$ and $\hat{\mathfrak g}$ are central extensions of $G$ and $\mathfrak g$, respectively.

Question: What is the most natural choice of $\hat \theta$ and $\hat V$?

I suspect that $\hat V(X)$ should be the quantization $\hat Q_X$ of the classical Noether charge $Q_X$, up to a factor of $\pm i$. Is this correct?

Laplacian
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  • Basically, yes. I gather you have practiced on several simple examples? What detail puzzles you? – Cosmas Zachos Jan 03 '22 at 14:49
  • @CosmasZachos I want a "general" argument that justifies $\hat V(X) \sim \hat Q_X$, not based on simple examples. I also want that the argument is as rigorous as possible. – Laplacian Jan 03 '22 at 15:21
  • Maybe the level of runaway generality practiced in MSE is more appropriate. The standard routine argument is through deformation quantization, of course. I never suggested basing the argument on examples--I merely brought them up to help you concentrate your doubts. – Cosmas Zachos Jan 03 '22 at 16:02
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    Possible duplicate: https://physics.stackexchange.com/q/74780/2451 – Qmechanic Jan 03 '22 at 16:32

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