Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

Foundations of Quantum Field Theory - Klaus D Rothe

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      We can compactify the notation by introducing the definitions

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      where the subscript D stands for “Dirac representation”. Explicitly we have

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      We may collect these matrices into a 4-tuplet figure. This notation is justified since we shall show later that these matrices “transform” (in a sense to be made precise later) under Lorentz transformations as a “4-vector”. In terms of the matrices (4.6) the Dirac equation takes the compact form2

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      This equation implies that ψ(r, t) is also a solution of the Klein–Gordon equation (3.8). We thus have the following Fourier decomposition into positive and negative energy solutions,

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      where figure, and the sum in σ extends over the two spin orientations in the rest-frame of the particle, as we shall see. The reason for displaying explicitly the factor figure will become clear from the transformation (7.15) and canonical normalization (7.40) in Chapter 7.

      For ψ(x) to be a solution of the Dirac equation (4.8), the (positive and negative energy) Dirac spinors U(p, σ) and V(p, σ) must satisfy the equations

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      Recalling the explicit form (4.7) of the figure-matrices, we obtain for the independent solutions in the Dirac representation,

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      where figure, and N± are normalization constants to be determined below. χ(σ) and χc(σ) denote the spinor and its conjugate in the rest frame of the particle,

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      with c the “charge conjugation” matrix defined by

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      Notice that

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      The matrix c has the fundamental property3

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      The following algebraic relations will turn out to be useful:

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      where

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      with

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      and

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      We further have

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      In the Dirac representation, γ5 is the off-diagonal 4 × 4 matrix

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      In order to fix the normalization constants in (4.11), we need to choose a scalar product. To this end we observe that

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      Hence the Dirac operator iγμ∂μ − m is hermitian with respect to the “Dirac” scalar product

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      Correspondingly we normalize the Dirac spinors by requiring4

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      which finally leads in the Dirac representation to the normalized Dirac spinors

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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 figure, this means that in analogy to (2.4), under Lorentz transformations,

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      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

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

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

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

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      We have

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      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,

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      (ii)for

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