I have a question about Noether's theorem in the context of QM, which I'll state in the context of the weak interaction but the basic point could be generalized.

According to Noether's theorem, given an $n$-dimensional Lie group there will be $n$ conserved quantities. $SU(2)$ is 3-dimensional, so that we'd expect 3 such quantities. However, elsewhere the conserved quantities are defined as the 'good quantum numbers', where these are defined as the eigenvalues of the maximal number of commuting generators in the group. In this case, there is just one such generator, and so it seems only one conserved quantity.

Can any one tell me where I'm going wrong?

  • $\begingroup$ Where have you seen that notion of "good quantum numbers?" $\endgroup$ – joshphysics Oct 4 '13 at 18:40

Let $L$ be an $n$-dimensional Lie algebra. The rank $r\leq n$ of $L$ is by definition the dimension of any Cartan subalgebra (CSA) in $L$. (One may show that all CSAs have the same dimension.)

Assume further that the whole Lie algebra $L$ commutes with the Hamiltonian $H$. Then we have $n$ (linearly independent) conserved quantities, but we will only be able to measure a Cartan subalgebra thereof simultaneously, i.e. rank $r$ (linearly independent) conserved quantities.

Nevertheless, we will still say that we have $n$ conserved quantities corresponding to the full Lie algebra $L$.

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