Lectures on Quantum Field Theory. Ashok Das

      For completeness we note here that it is easy to check

      for any two spin components of the positive and the negative energy spinors.

      From the form of the equations satisfied by the positive and the negative energy spinors, (3.95) and (3.96), it is clear that we can define projection operators for such solutions as

      These are, of course, 4 × 4 matrices and their effect on the Dirac spinors is quite clear,

      Similar relations also hold for the adjoint spinors and it is clear that Λ₊(p) projects only on to the space of positive energy solutions, while Λ₋(p) projects only on to the space of negative energy ones.

      Let us note that

      where we have used Thus, we see that Λ_±(p) are indeed projection operators and they are orthogonal to each other. Furthermore, let us also note that

      as it should be since all the solutions can be divided into either positive or negative energy ones.

      Let us next consider the outer product of the spinor solutions. Let us define a 4 × 4 matrix P with elements

      This matrix has the property that acting on a positive energy spinor it gives back the same spinor. Namely,

      where we have used (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, and have opposite sign, the latter being negative while the former is positive. Hence, we can think of the space of solutions of the Dirac equation as an indefinite metric space. In such a space, the completeness relation does not involve a sum of terms with positive definite sign, rather it involves a sum with the metric structure of the space built in.

      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 S_z), 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