Gravity and Strings

2.4 The special-relativistic energy–momentum tensor 37
and, after use of the equations of motion, we find the Belinfante tensor
Tλ µ = i
4 ∂ν ¯ψ (γ µ η ν λ + γλη νµ )ψ − i
4 ¯ψ(γ µ η ν λ + γλη νµ )∂νψ
+ ηλ µ 1
2 (i ¯ψ ∂ψ − i ¯ψ ←
∂ ψ) − 2m ¯ψψ . (2.69)
In the case of the vector field, we managed to find the Belinfante tensor by a method
based on the vector transformation law under GCTs. However, it is not clear how to use
this method in the present case. The spinorial character is associated only with Lorentz
transformations and it is not clear what the spinor transformation law should be for other
GCTs. In fact, the only consistent form of dealing with spinors on curved spacetime is to
treat them as scalars under GCTs and to associate the spinorial character with the Lorentz
group that acts on the tangent space at each given point. This is the formalism invented by
Weyl in [954] which we will study later on.
2.4.2 Dilatations
Let is consider now constant rescalings (dilatations) by a factor = eσ :
x ′ µ = x µ ,
ϕ ′ (x ′ ) = ωϕ(x), ⇒
˜δx µ = σ x µ ≡ σ ˜δDx µ ,
˜δϕ = ωσϕ.
The associated conserved current is
(2.70)
jN1 D µ = Tcan µ νx ν + J µ , J µ ≡ ω ∂L
ϕ. (2.71)
∂∂µϕ
If we take the divergence of this current and set it equal to zero, we obtain the identity
Tcan µ µ + ∂µ J µ = 0. (2.72)
It is always possible to find a redefinition of the canonical energy–momentum tensor that
is symmetric, divergenceless, and, furthermore, traceless if there is scale invariance (see
e.g. [204, 247, 491] and [781] and references therein). This redefined energy–momentum
tensor is called the improved energy–momentum tensor and can be constructed systemati-
cally: on rewriting the dilatation current in the form
jN1 D µ
= Tcan µ ν + 2
d − 1 ∂ρ
[µ ρ]
J η ν
x ν − 2
d − 1 ∂ν
[µ ν]
J x , (2.73)
we observe that
T µν = Tcan µν + 2
d − 1 ∂ρ
[µ ρ]ν
J η , (2.74)
is on-shell traceless on account of the identity Eq. (2.72) and also on-shell divergenceless
since the piece that we add to the canonical energy–momentum tensor is of the form
∂ρ [ρµ]ν . Observe that this term can also be obtained directly from the action if we add to
it a total derivative term of the form
S = ω
d − 1
d d x ∂ρ
∂L
∂∂ρϕ ϕ
. (2.75)