HiLets say I have a Hamiltonian which is invariant in e.g. the

[Niles]

Hi

Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this?


Niles.

 

[Matterwave]

Noether's theorem says that for a group of continuous symmetries in the Lagrangian, there is a related conserved quantity. I'm not sure if there's an analog for this for the Hamiltonian...

 

[nnnm4]

Yes consider an operator that "measures" the spin of the particle. The Sz operator for example. If this operator commutes with the Hamiltonian then it is a conserved quantity. Specifically in terms of the spin indices, for a spin-1/2 particle, then S^2 is conserved as s=1/2 and S_z is conserved as m=1/2 or m = -1/2.

 

[Matterwave]

Whoops...I completely forgot about that lol. Sorry.

 

[dextercioby]

The Hamiltonian of a quantum field is 'ab initio' required to be a Poincare' scalar, in particular a Lorentz scalar. Thus it carries no spinor indices whatsoever.

In general, the spin is not conserved, but the helicity of the massive particle or the polarization of the masseless particle is.