Lectures on Quantum Field Theory. Ashok Das

      the solutions take the forms (with p₀ = E_± = ±E = )

      which can be explicitly verified. (The change in the sign in the dependent components in (2.34) compared to (2.30) comes from raising the index of the momentum, namely, p_i = −pⁱ = −(p)_i.)

2.2Normalization of the wave function

      Let us note that if we define

      then, we can write the solutions (2.34) for motion along a general direction as

      Here α and β are normalization constants to be determined. The two component spinors and in (2.21) and (2.27) respectively are normalized as (for the same spin components)

      For different spin components, this product vanishes.

      Given this, we can now calculate

      where we have used the familiar identity satisfied by the Pauli matrices, namely

      Similarly, for the negative energy solutions we have

      It is worth remarking here that although we have seen in (2.37) that, for the same spin components, we have carried along these factors in (2.38) and (2.40) simply because we have not specified their spin components.

      In dealing with the Dirac equation, another wave function (known as the adjoint spinor) plays an important role and is defined to be

      Thus, for example,

      Thus, we see that the difference between the Hermitian conjugate u^† and the adjoint is in the relative sign in the second of the two-component spinors.

      We can also calculate the product

      Similarly, we can show that

      Our naive instinct will be to normalize the wave function, as in the non-relativistic case, by requiring (for the same spin components)

      However, as we will see shortly, this is not a relativistic normalization. In fact, u^†u, as we will see, is related to the probability density which transforms like the time component of a four vector. Thus, a relativistically covariant normalization would be to require (for the same spin components)

      (Remember that this will correspond to the probability density and, therefore, must be positive. By the way, the motivation for such a normalization condition comes from the fact that, in the rest frame of the particle, this will reduce to which corresponds to the non-relativistic normalization (2.45).) The independent wave functions for a free particle, ψ_p(x) = e^−ip·xu(p) with p₀ = ±E, with this normalization condition, would give (for the same spin components)

      With the requirement (2.46), we determine from (2.38) and (2.40) (for the same spin components when (2.37) holds)

      Therefore, with this normalization, we can write the normalized positive and negative energy solutions of the Dirac equation to be

      It is also clear that, with this normalization, we will obtain from (2.43) and (2.44) (for the same spin components)

      This particular product, therefore, appears to be a Lorentz invariant (scalar) and we will see later that this is indeed true.

      Let us also note here that by construction the positive and the negative energy solutions are orthogonal. For example,

      Therefore, the solutions we have constructed correspond to four linearly independent, orthonormal solutions of the Dirac equation. Note, however, that

      While we will be using this particular normalization for massive particles, let us note that it becomes meaningless for massless