Lectures on Quantum Field Theory. Ashok Das

      In this case, the matrix for a finite boost ω can be obtained through exponentiation as

      Furthermore, recalling that

      and, therefore,

      we can determine

      We note here that since

      That is, in this four dimensional space (namely, as 4 × 4 matrices), operators defining boosts are not unitary. This is related to the fact that Lorentz boosts are non-compact transformations and for such transformations, there does not exist any finite dimensional unitary representation. All the unitary representations are necessarily infinite dimensional.

3.3Transformation of bilinears

      In the last section, we have shown how to construct the matrix S(Λ) for finite Lorentz transformations (for both rotations and boosts). Let us note next that, since under a Lorentz transformation

      it follows that

      where we have used the relation (3.58). In other words, we see that the adjoint wave function transforms inversely, under a Lorentz transformation, compared to the wave function ψ(x). This implies that a bilinear product such as would transform under a Lorentz transformation as

      Namely, such a product will not change under a Lorentz transformation – would behave like a scalar – which is what we had discussed earlier in connection with the normalization of the Dirac wavefunction (see (2.50) and (2.55)).

      Similarly, under a Lorentz transformation

      where we have used (3.45). Thus, we see that if we define a current of the form it would transform as a four vector under a proper Lorentz transformation, namely,

      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