Hi,

consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to U(\mathcal{H})$ given by acting on the underlying space. From the group representation you get a representation of the Lie algebra of $G$ which maps elements of the algebra generating translations to momentum operators and elements of the algebra generating rotations (use Euler angles) to angular momentum operators. You notice: The elements of the Lie algebra are mapped to the operators which are the quantized conserved quantities corresponding the the element of the Lie algebra (using the Lagrangian of a free particle and Noether’s theorem). Is there any deep mathematical reason why that works? (why you can get the quantized conserved quantity in two ways: 1. use the representation 2. use the free Lagrangian, apply Noether’s theorem, quantize the quantity)

Thanks in advance