Commutation between Dirac hamiltonian and angular momentum In reading about angular momentum and spin, I came across a derivation showing that the Dirac Hamiltonian does not commute with orbital angular momentum, and hence L is not conserved. What is it about commutation relations which allow us to conclude whether the orbital angular momentum is conserved or not?
 A: A simple answer is to look in the Heisenberg picture with the Heisenberg equation of motion.  In this picture, operators evolve instead of states as in the Schrodinger picture.  Here the evolution of the operator is chosen so that its expectation value has the same time evolution as in the Schrodinger picture.  To do so its time evolution is governed by:
$ \frac{d}{dt}L=\frac{1}{i\hbar}[L,H]$
so the expectation of the time derivitive of L is non 0 if it's commutator with the Hamiltonian is non 0
$ <\frac{d}{dt}L>=\frac{1}{i\hbar}<[L,H]>$
Thus it's not something specific to commutation relations its specific to the commutation relation with the Hamiltonian. For this explanation to be really satisfying you should read about the Heisenberg Picture.
Another way to look at it is that since the two operators don't commute they can't be diagonalized at the same time.  This means an angular momentum eigen state isn't a energy eigen state and therefore you have to decompose the angular momentum eigen state into energy eigen states to get its time evolution. Since this state will be a sum of energy eigen states time will evolve the system away from it and the expectation value of angular momentum will change.
