Lectures on Quantum Field Theory. Ashok Das

      where S(Λ) is a 4 × 4 matrix, since ψ(x) is a four component spinor. Parenthetically, what this means is that we are finding a representation of the Lorentz transformation on the Hilbert space. In the notation of other symmetries that we know from studies in non-relativistic quantum mechanics, we can define an operator L(Λ) to represent the Lorentz transformation on the coordinate states as (with indices suppressed)

      However, since the Dirac wavefunction is a four component spinor, in addition to the change in the coordinates, the Lorentz transformation can also mix up the spinor components (much like angular momentum/rotation does). Thus, we can define the Lorentz transformation acting on the Dirac Hilbert space (Hilbert space of states describing a Dirac particle) as, (with S(Λ) representing the 4 × 4 matrix which rotates the matrix components of the wave function)

      where the wave function is recognized to be

      so that, from (3.39) we obtain (see (3.37))

      Namely, the effect of the Lorentz transformation, on the wave function, can be represented by a matrix S(Λ) which depends only on the parameter of transformation Λ and not on the space-time coordinates. A more physical way to understand this is to note that the Dirac wave function simply consists of four functions which do not change, but get rotated by the S(Λ) matrix.

      Since the Lorentz transformations are invertible, the matrix S(Λ) must possess an inverse so that from (3.37) we can write

      Let us also note from (3.35) that

      define a set of real quantities. Thus, we can write

      where we have used (3.42).

      Therefore, we see from (3.44) that the Dirac equation will be form invariant (covariant) under a Lorentz transformation provided there exists a matrix S(Λ), generating Lorentz transformations (for the Dirac wavefunction), such that

      Let us note that if we define

      then,

      where we have used the orthogonality of the Lorentz transformations (see (3.16)). Therefore, the matrices γ′^µ satisfy the Clifford algebra and, by Pauli’s fundamental theorem, there must exist a matrix connecting the two representations, γ^µ and γ′^µ. It now follows from (3.46) that the matrix S exists and all we need to show is that it also generates Lorentz transformations in order to prove that the Dirac equation is covariant under a Lorentz transformation.

      Next, let us note that since the parameters of Lorentz transformation are real (namely, (Λ^∗)^µ_ν = Λ^µ_ν)

      Here we have used (3.45) and the relations (γ⁰)^† = γ⁰ = (γ⁰)⁻¹ as well as γ⁰(γ^µ)^†γ⁰ = γ^µ. It is clear from (3.48) that the matrix Sγ⁰S^†γ⁰ commutes with the four Dirac γ^µ matrices and, therefore, with all the 16 basis matrices in the 4 × 4 space given in (2.101) and must be proportional to the identity matrix (this follows simply because each of the sixteen basis matrices in (2.101) consists of products of γ^µ which commute with Sγ⁰S^†γ⁰). As a result, we can denote

      Taking the Hermitian conjugate of (3.49), we obtain

      which, therefore, determines that the parameter b is real, namely,

      We also note that det γ⁰ = 1 and since we are interested in proper Lorentz transformations, det S = 1. Using these in (3.49), we determine

      The real roots of this equation are

      In fact, we can determine the unique value of b in the following way.

      Let us note, using (3.45) and (3.49), that

      which follows since S^†S represents a non-negative matrix. The two solutions of this equation are obvious

      Since we are dealing with proper Lorentz transformations, we are assuming

      which implies (see (3.55)) that b >