Are Noether charges a rep of the generators on the Hilbert space

1. a2009

25
I'm trying to understand the relationship between conserved charges and how operators transform. I know that we can find conserved charges from Noether's theorem. If (for internal symmetries) I call them $$Q^a = \int d^3x \frac{\partial L}{\partial \partial_0 \phi_i} \Delta \phi_i^a$$ then is it always the case that operators transform like

$$\hat O \rightarrow e^{i t_a Q^a} \hat O e^{-i t_a Q^a}$$

i.e. are the conserved charges the rep of the generators on the Hilbert space?

Thanks for any help!

2. samalkhaiat

1,115
Yes, they do generate the correct transformation on the fields AND satisfy the Lie algebra of the symmetry group. More importantly, they ( in the internal case) DON’T need to be CONSERVED to do the job.

sam