Lectures on Quantum Field Theory. Ashok Das

      Let us note here from the form of S(ϵ) that we can identify

      with the generators of infinitesimal Lorentz transformations for the Dirac wave function. (The other factor of is there to avoid double counting.) We will see in the next chapter (when we study the representations of the Lorentz group) that the algebra (2.110) which the generators of the infinitesimal transformations, σ_µν, satisfy can be identified with the Lorentz algebra (which also explains why they are closed under multiplication).

      Thus, at least for infinitesimal Lorentz transformations, we have shown that there exists a S(Λ) which satisfies (3.45) and generates Lorentz transformations and as a result, the Dirac equation is form invariant (covariant) under such a Lorentz transformation. A finite transformation can, of course, be constructed out of a series of infinitesimal transformations and, consequently, the matrix S(Λ) for a finite Lorentz transformation will be the product of a series of such infinitesimal matrices which leads to an exponentiation of the infinitesimal generators with the appropriate parameters of transformation.

      For completeness, let us note that infinitesimal rotations around the 3-axis or in the 1-2 plane would correspond to choosing

      with all other components of ϵ^µν vanishing. In such a case (see also (2.99)),

      A finite rotation by angle θ in the 1-2 plane would, then, be obtained from an infinite sequence of infinitesimal transformations resulting in an exponentiation of the infinitesimal generators as

      Note that since we have S^†(θ) = S⁻¹(θ), namely, rotations define unitary transformations. Furthermore, recalling that

      we have

      and, therefore, we can determine

      This shows that

      That is, the rotation operator, in this case, is double valued and, therefore, corresponds to a spinor representation. This is, of course, consistent with the fact that the Dirac equation describes spin particles.

      Let us next consider an infinitesimal rotation in the 0-1 plane, namely, we are considering an infinitesimal boost along the 1-axis (x-axis). In this case, we can identify

      with all other components of ϵ^µν vanishing, so that we can write (see also (2.99))

      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)

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