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Note that we have omitted the index s, which describes the spin state of the particle because that information is determined by the choice of ξ and η, to which we now turn. It is convenient to choose spin states which are eigenstates of the helicity, the component of angular momentum in the direction of motion of the particle. Therefore we define spinors ξ( p, ±) that satisfy The Dirac field 34 σ · p ξ( p, ±) = ±| p| ξ( p, ±) . 39) Note that σ · p ≡ σ i pi is summed over the spatial components only.

For these reasons the canonical stress tensor needs to be improved, and we now guide the reader through some exercises that accomplish this. 21 It is well known that the Lagrangian density of a field theory can be changed by adding a total divergence ∂μ B μ , since the Euler–Lagrange equations are unaffected. Show that the addition of 12 ∂μ ( ¯ γ μ ) brings the action to the form S =− dD x 1 2 ↔ ¯ γμ∂μ −m ¯ . 61) Note that the antisymmetric derivative is defined as ↔ A ∂ μ B ≡ A(∂μ B) − (∂μ A)B .

9 Prove that U (( )−1 a)U ( ) = U ( )U (a). Verify for operators which are close to the identity that U (a)φ( x + b) = φ( U ( )φ( x + b) = φ( x+ a + b) , x + b). 56) The Lie algebra of the Poincaré group contains the D(D + 1)/2 generators J[μν] , Pμ . The following commutation rules complete the specification of the Lie algebra: [J[μν] , J[ρσ ] ] = ηνρ J[μσ ] − ημρ J[νσ ] − ηνσ J[μρ] + ημσ J[νρ] , [J[ρσ ] , Pμ ] = Pρ ησ μ − Pσ ηρμ , [Pμ , Pν ] = 0. 57). We now discuss the implementation of this Lie algebra on fields.

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