Lectures on Quantum Field Theory. Ashok Das

(There is no rest frame for a massless particle.) The probability density has to be well defined. Correspondingly, an alternative normalization which works well for both massive and massless particles is given by

      This still behaves like the time component of a four vector (m is a Lorentz scalar). In this case, we will obtain from (2.38) and (2.40) (for the same spin components)

      Correspondingly, in this case, we obtain

      which vanishes for a massless particle. This product continues to be a scalar. Let us note once again that this is a particularly convenient normalization for massless particles.

      Let us note here parenthetically that, while the arbitrariness in the normalization of u(p) may seem strange, it can be understood in light of what we have already pointed out earlier as follows. We can write the solution of the Dirac equation for a general motion (along an arbitrary direction) as

      where a(p) is a coefficient which depends on the normalization of u(p) in such a way that the wave function would lead to a total probability normalized to unity,

      Namely, a particular choice of normalization for the u(p) is compensated for by a specific choice of the coefficient function a(p) so that the total probability integrates to unity. The true normalization is really contained in the total probability.

2.3Spin of the Dirac particle

      As we have argued repeatedly, the structure of the plane wave solutions of the Dirac equation is suggestive of the fact that the particle described by the Dirac equation has spin That this is indeed true can be seen explicitly as follows.

      Let us define a four dimensional generalization of the Pauli matrices as (in this section, we will use the notations of three dimensional Euclidean space since we will be dealing only with three dimensional vectors)

      It can, of course, be checked readily that these matrices are related to the α_i matrices defined in (1.98) and (1.101) through the relation

      where

      We note that so that we can invert the defining relation (2.59) and write

      From the structures of the matrices α_i and we conclude that

      This shows that satisfies the angular momentum algebra (remember ℏ = 1) and this is why we call the matrices, the generalized Pauli matrices. (Note, however, that define a reducible representation of spin generators since these matrices are block diagonal.) Using (2.59) and (2.60), it can also be checked that [α_i, α_j] =

      Let us also note that

      Here we have used the fact that (see (1.101))

      is block diagonal like

      With these relations at our disposal, let us look at the free Dirac Hamiltonian in (1.100) (remember that we are using three dimensional Euclidean notations in this section)

      As we will see in the next chapter, the Dirac equation transforms covariantly under a Lorentz transformation. In other words, Lorentz transformations define a symmetry of the Dirac Hamiltonian and, therefore, rotations which correspond to a subset of the Lorentz transformations must also be a symmetry of the Dirac Hamiltonian. Consequently, the (total) angular momentum operators which generate rotations should commute with the Dirac Hamiltonian. Let us recall that the orbital angular momentum operators are given by (repeated indices are summed)

      Calculating the commutator of this operator with the Dirac Hamiltonian, we obtain

      Here we have used the fact 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