Lectures on Quantum Field Theory. Ashok Das

      This is, of course, what we had observed earlier. Namely, the probability current density (see also (2.86)) transforms like a four vector so that the probability density transforms as the time component of a four vector. Finally, we note that in this way, we can determine the transformation properties of the other bilinears under a Lorentz transformation in a straightforward manner.

3.4Projection operators, completeness relation

      Let us note that the positive energy solutions of the Dirac equation satisfy

      where

      while the negative energy solutions satisfy

      with the same value of p⁰ as in (3.92). It is customary to identify (see (2.49), the reason for this will become clear when we discuss the quantization of Dirac field theory later)

      so that the equations satisfied by u(p) and v(p) (positive and negative energy solutions), (3.91) and (3.93), can be written as

      and

      Given these equations, the adjoint equations are easily obtained to be (taking the Hermitian conjugate and multiplying γ⁰ on the right)

      where we have used (γ^µ)^†γ⁰ = γ⁰γ^µ (see (2.84)). As we have seen earlier there are two positive energy solutions and two negative energy solutions of the Dirac equation. Let us denote them by

      where r, as we had seen earlier, can represent the spin projection of the two component spinors (in terms of which the four component solutions were obtained). Let us also note that each of the four solutions really represents a four component spinor. Let us denote the spinor index by α = 1, 2, 3, 4. With these notations, we can write down the Lorentz invariant conditions we had derived earlier from the normalization of a massive Dirac particle as (see (2.50))

      Although we had noted earlier that the last re lation in (3.99) can be checked to be true simply because v(p) = u₋(−p⁰, −p), namely, because the direction of momentum changes for v(p) (see the derivation in (2.52)). This also allows us to write

      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 (

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