Gravity and Strings

36 Noether’s theorems
which is the Killing equation in Minkowski spacetime. Then, there is invariance under
Poincaré transformations whose generators are ˜δx λ = ˜δ(ν)x λ and ˜δx λ = ˜δ(ρσ )x λ .
On integrating the above variation by parts, using the fact that it vanishes for Poincaré
transformations and using the equation of motion (which implies that ∂µT µ λ = 0), we find
d d
x ∂µ ˜δx λ T µ
ν = 0, (2.62)
and we find automatically the above Noether currents.
This method is clearly inspired by general relativity. We will find more applications for
it soon.
The energy–momentum tensor of a Dirac spinor. The Lagrangian of a massive Dirac spinor
is 8
L = 1
2 (i ¯ψ ∂ψ − i ¯ψ ←
∂ψ) − m ¯ψψ,
¯ψ ←
∂ ≡ ∂µ ¯ψγ µ . (2.63)
It is customary to vary ψ and ¯ψ as if they were independent. This simplifies somewhat
the calculations but we have to bear in mind that they are not independent. The equations
of motion of ψ and ¯ψ are the Dirac conjugates of each other:
(i ∂ − m)ψ = 0,
¯ψ
i ←
∂ + m = 0. (2.64)
Acting with ∂ on the first equation, we find that ψ satisfies the Klein–Gordon equation
∂ 2 + m 2 ψ = 0. (2.65)
The canonical energy–momentum tensor is
Tcan λ µ =− ∂L
∂∂µψ ∂λψ
∂L
− ∂λ ¯ψ
∂∂µ ¯ψ + ηλ µ L
=− i
2 ¯ψγ µ ∂λψ + i
2 ∂λ ¯ψγ µ ψ + ηλ µ
1
2 (i ¯ψ ∂ψ − i ¯ψ ←
∂ ψ) − 2m ¯ψψ , (2.66)
and it is clearly not symmetric. The spin-angular-momentum tensor is
S µ ρσ = 1 ∂L
2
= 1 i
2 2
∂∂µψ Ɣs
¯ψγ µ
1
Mρσ ψ +
2 ¯ψƔ¯s
∂L
Mρσ
∂∂µψ
1
2 γνρ
ψ + 1
2 ¯ψ
− 1
2 γνρ
− i
2 γ µ
ψ
= i
4 ¯ψγ µ νρψ, (2.67)
and it is totally antisymmetric. The spin–energy potential is just
µν ρ =−S µν ρ, (2.68)
8 Our conventions for spinors and gamma matrices are explained in Appendix B.