that since β is a constant matrix and m is a constant, the second term in the Hamiltonian drops out of the commutator. Thus, we note that the orbital angular momentum operator does not commute with the Dirac Hamiltonian. Consequently, the total angular momentum which should commute with the Hamiltonian must contain a spin part as well.
To determine the spin angular momentum, we note that (see (2.63))
so that combining this relation with (2.67) we obtain
In other words, the total angular momentum which should commute with the Hamiltonian, if rotations are a symmetry of the system, can be identified with
In this case, therefore, we can identify the spin angular momentum operator with
Note, in particular, that
which has doubly degenerate eigenvalues
2.4Continuity equation
The Dirac equation, written in the Hamiltonian form (see (1.99)), is given by
Taking the Hermitian conjugate of this equation, we obtain
where the gradient is assumed to act on ψ†. Multiplying (2.73) with ψ† on the left and (2.74) with ψ on the right and subtracting the second from the first, we obtain
This is the continuity equation for the probability current density associated with the Dirac equation and we note that we can identify
to write the continuity equation as
This suggests that we can write the current four vector as
so that the continuity equation can be written in the manifestly covariant form
This, in fact, shows that the probability density, ρ, is the time component of Jµ (see (2.78)) and, therefore, must transform like the time coordinate under a Lorentz transformation (as we had alluded to earlier). (We are, of course, yet to show that Jµ transforms like a four vector which we will do in the next chapter.) On the other hand, the total probability
is a constant independent of any particular Lorentz frame. It is worth recalling that we have already used this Lorentz transformation property of ρ in defining the normalization of the wave function.
An alternative and more covariant way of deriving the continuity equation is to start with the covariant Dirac equation
and note that the Hermitian conjugate of ψ satisfies
Multiplying this equation with γ0 on the right and using the fact that
where we have used the property of the gamma matrices that (for µ = 0, 1, 2, 3, see also (2.102) and (2.103) in section 2.6)
Multiplying (2.81) with
This is, in fact, the covariant continuity equation for the Dirac equation and we can identify the conserved current density with
Note from the definition in (2.86) that we can identify
which is what we had derived earlier in (2.78).
Let us conclude this discussion by noting that although the Dirac equation has both positive and negative energy solutions, because it is a first order
