Dirac equation spin conservation

[Ratnaz]

Hi there, got a question for anyone with some expertise in the Dirac equation.

 

The angular momentum of a free electron is of course conserved, and is made up of the spin and orbital parts. These two are not conserved on their own (they don't commute with the Dirac Hamiltonian)

 

If I work out the time derivative of the spin a.m. operator s (in the Heisenberg picture) I get

 

(d/dt) s_i = epsilon_ijk p_j gamma^0 gamma^k

 

where epsilon if the antisymmetric 3-tensor, gamma^mu is the usual gamma matrix, p is the momentum operator and indices i,j,k = 1, 2 or 3 label spatial coordinates. 0 is the index for the time coordinate. Einstein summation convention applies.

 

This is not identically zero, but if I calculate

 

integral d^3 x psi^dagger (ds_i / dt) psi

 

i.e. the quantum average of ds/dt, where psi is the spinor "wavefunction", I get zero if you expand psi as a general series of momentum eigenstates (I won't do the maths here).

 

The upshot is that the operator ds/dt itself doesn't vanish but any quantum average of it always vanishes. So can't we say spin and orbital a.m. are conserved on their own?