Lectures on Quantum Field Theory. Ashok Das

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 Therefore, we conclude that the particle described by the Dirac equation corresponds to a spin (fermionic) particle.

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 J^µ (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 J^µ 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 we obtain ( so 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 on the left and (2.83) with ψ on the right and subtracting the second from the first, we obtain

      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 equation (particularly in the time derivative), the probability density is independent of time derivative much like in the Schrödinger equation. Consequently, the probability density, as we have seen explicitly in (2.38) and (2.40), can be defined to be positive definite even in the presence of negative energy solutions. This is rather different from the case of the Klein-Gordon equation that we have studied in chapter 1.

2.5Dirac’s hole theory

      We have seen that the Dirac equation leads to both positive and negative energy solutions. In the free particle case, for example, the energy eigenvalues are given by

      Thus, even for this simple case of a free particle the energy spectrum has the form shown in Fig. 2.1. We note from Fig. 2.1 (as well as from the equation above, (2.88)) that the positive and the negative energy solutions are separated by a gap of magnitude 2m (remember that we are using c = 1).

      Even when