The relation, as it was derived above, holds for Λ close to theidentity. Given the overall covariance
of the tensors of the last line, this relation must now be valid for finite Λ as well. Another way of
writing this relation is by contracting both sides with Λκλ, so that we get
λκλD(Λ)γκD(Λ)−^1 =γλ (22.28)This relation signifies that theγ-matrices areconstantsprovided that we transform both their
vector index and their spinor indices. The Lorentz transformation property of the generatorsSμν
themselves follows from that of theγ-matrices, and we have,
D(Λ)SκλD(Λ)−^1 = ΛμκΛνλSμν (22.29)The structure relations may be recovered from this relation.
22.4 The Dirac equation and its relativistic invariance
We now have all the tools ready to construct relativistic invariant equations for fields which are
spinors of the Lorentz algebra. We shall work in the (reducible) Dirac representation, leaving the
Weyl and Majorana cases as a special case of the Dirac spinors.
The basic Dirac fieldψ(x) transforms as a 4-component representation of the form (^12 ,0)⊕(0,^12 ),
for which we have just constructed the representation matrices. The field transforms as follows,
ψ(x) → ψ′(x′) =D(Λ)ψ(x) (22.30)where the infinitesimal transformations were given in (22.24). We begin by constructing an equation
for afree field, namely a linear equation forψ. Of course, we could write down the free Klein-Gordon
equation forψ,
(∂μ∂μ−m^2 )ψ(x) = 0 (22.31)and this is a perfectly fine free wave equation for a spinor. The correct equation, however, turns
out to be a first order equation, namely thefree Dirac equation,
(γμ∂μ−m)ψ(x) = 0 (22.32)Note that the free Dirac equation implies the free Klein-Gordon equation. This may be seen by
multiplying the Dirac equation to the left by the operator (γν∂ν+m), which gives,
(
γν∂νγμ∂μ−m^2
)
ψ= 0 (22.33)Using the fact that the derivatives are symmetric under interchange ofμandν, and using the
Clifford relation, it is immediate that this equation coincides with the Klein-Gordon equation. The
converse is, however, not true, as a fieldψsatisfying (22.31) may be decomposed into solutions
of (γμ∂μ±m)ψ= 0, with either±sign. In fact, the Dirac equation has only half the number of
solutions of the Klein-Gordon equation (both for 4-component spinors).