Foundations of Quantum Field Theory. Klaus D Rothe

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       Dirac spinors in the Weyl representation

      We now present a derivation of the Dirac equation based on group-theoretical arguments alone. Our fundamental requirement will be that the solution of the “relativistic Schrödinger equation” should belong to a representation of the Lorentz group. In particular consider the irreducible representations (1/2,0) and (0,1/2) in (2.28). Denoting the wave functions in the respective representations by φ(x) and , this means that in analogy to (2.4), under Lorentz transformations,

      The operators acting on these fields are thus given by 2 × 2 matrices. A complete set of such matrices is given by the identity and the three Pauli matrices. As we next show the set of four matrices⁵

      transform as a “4-vector” in the following sense:

      Note that these transformation laws cannot be interpreted as a change in basis. They are easily verified for an infinitesimal Lorentz transformation

      and the corresponding expression for the (1/2,0) and (0,1/2) representations (2.29) and (2.30) with

      We have

      Notice that the rotational part does not care about which of the two representations we are in. Keeping only terms linear in the parameters we have

      (i)for t⁰ = 1,

      (ii)for ti = σi,

      On the other hand we have for an infinitesimal Lorentz transformation ,

      In accordance with our previous parametrization we have

      where we have set t⁰ = 1. This establishes our claim (4.25) for tμ. In a similar way one demonstrates the transformation law (4.26). It now follows from (4.25) and (4.26) that the equations

      transform covariantly under Lorentz transformations.⁶ Indeed, multiplying the first equation from the left with and making the replacement x → x′ = Λx, one has

      or recalling that , we have from (4.25)

      Together with (4.22) this implies

      which proves the covariance of Eq. (4.27). In the same way, one also proves the covariance of the second equation.

      Equations (4.27), (4.28) represent a coupled set of equations, which only decouple in the case of zero-mass fermions. They may be collected into a single equation by defining the 4 × 4 matrices

      where the subscript W stands for “Weyl”.⁷ One explicitly checks that they satisfy the anticommutation relations

      Writing ψ(x) in the form

      the above coupled set of equations takes the form

      Multiplying this equation from the left with the operator (iγμ∂μ + m) and using the anticommutation relations (4.30) we see that ψ is also a solution of the Klein–Gordon equation:

      describing the propagation of a free particle with the correct energy-momentum relation. By further defining the 4 × 4 matrices (in the Weyl-representation)

      

      the transformation laws (4.25), (4.26) can be collected to read

      and

      On this level we now have manifest Lorentz covariance of the Dirac equation (4.8). Note also that the inverse of the matrix is now equivalent to the corresponding “Dirac” adjoint (recall (2.33))

      This will play an important role when we come to define scalar products.

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