Lectures on Quantum Field Theory. Ashok Das

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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 > 0 and, therefore, it follows from (3.53) that

      Thus, we conclude from (3.49) that

      These are some of the properties satisfied by the matrix S which will be useful in showing that it provides a representation for the Lorentz transformations.

      Next, let us consider an infinitesimal Lorentz transformation of the form ( infinitesimal)

      From our earlier discussion in (3.20), we recall that the infinitesimal transformation matrix is anti-symmetric, namely,

      For an infinitesimal transformation, therefore, we can expand the matrix S(Λ) as

      where the matrices M_µν are assumed to be anti-symmetric in the Lorentz indices (for different values of the Lorentz indices, M_µν denote matrices in the Dirac space since S(ϵ) is a matrix in this 4 × 4 space),

      since

      We can also write

      so that

      To the leading order, therefore, S⁻¹(ϵ) indeed represents the inverse of the matrix S(ϵ).

      The defining relation for the matrix S(Λ) in (3.45) now takes the form

      At this point, let us recall the commutation relation (2.107)

      and note from (3.66) that if we identify

      then,

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