rel="nofollow" href="#ulink_82f1b618-2fb0-5036-8688-52ceaa7d9e22">3.99). Thus, we see that the matrix P projects only on to the space of positive energy solutions and, therefore, we can identify
Similarly, if we define
then, it is straightforward to see that
Namely, the matrix Q projects only on to the space of negative energy solutions with a phase (a negative sign). Hence we can identify
The completeness relation for the solutions of the Dirac equation now follows from the observation that (see (3.105))
In a matrix notation, the completeness relation (3.112) can also be written as
We note here that the relative negative sign between the two terms in (3.112) or in (3.113) can be understood as follows. As we have seen,
These relations are particularly useful in simplifying the evaluations of transition amplitudes and probabilities. For example, let us suppose that we are interested in a transition amplitude which has the form
where M stands for a 4 × 4 matrix (a combination of the 16 Dirac matrices). If the initial and the final states are the same, this may represent the expectation value of a given operator in a given electron state and will have the form (r not summed)
If we are not interested in the expectation value in a particular electron state, but rather wish to obtain an average over the two possible electron states (in experiments we may want to average over the spin polarization states), then we will have
Similarly, if we have a transition from a given electron state to another and if we are interested in a process where we average over the initial electron states and sum over the final electron states (for example, think of an experiment with unpolarized initial electron states where the final spin polarization is not measured), the probability for such a transition will be determined from
The trace is over the 4 × 4 matrix indices and can be easily performed using the properties of the Dirac matrices that we have discussed earlier in section 2.6.
3.5Helicity
As we have seen in section 2.3, the Dirac Hamiltonian
does not commute either with the orbital angular momentum or with spin (rather, it commutes with the total angular momentum). Thus, unlike the case of non-relativistic systems where we specify a given energy state by the projection of spin along the z-axis (namely, by the eigenvalue of Sz), in the relativistic case this is not useful since spin is not a constant of motion. In fact, we have already seen that the spin operator
satisfies the commutation relation (see (2.68))
As a consequence, it can be easily checked that the plane wave solutions which we had derived earlier are not eigenstates of the spin operator. Note, however, that for a particle at rest, spin commutes with the Hamiltonian (since in this frame p = 0) and such solutions can be labelled by the spin projection.
On the other hand, we note that since momentum commutes with the Dirac Hamiltonian, namely,
the operator S · p does also (momentum and spin commute and, therefore, the order of these operators in the product is not relevant). Namely,
Therefore, this operator is a constant of motion. The normalized operator
measures the longitudinal component of the spin of the particle or the projection of the spin along the direction of motion. This is known as the helicity operator and we note that since the Hamiltonian commutes with helicity, the eigenstates of energy can also be labelled by the helicity eigenvalues. Note that
where we have used (this is the generalization of the identity satisfied by the Pauli matrices)
Therefore, the eigenvalues of the helicity operator, for a spin
