Gravity and Strings

130 Action principles for gravity
and, using
we obtain
which add up to
−2∂ν
and so we have (observe the order of indices)
S ρbc ρbc στd = 1
2 στd , (4.91)
2 ∂Lmatter ∂ρbc
∂ωρbc
στd
∂ea στd =−e
µ
ρµ bωρa b ,
∂Lmatter στd
∂στd
ρbc
∂ωρbc ∂∂νea
= ∂ν(e
µ
νµ a),
(4.92)
e∇ν νµ a, (4.93)
Ta µ = Tcan a µ +∇ν νµ a, (4.94)
which is the relation between the Vielbein energy–momentum tensor and the canonical
one. If we substract the antisymmetric part of the Vielbein energy–momentum tensor, we
obtain a symmetric tensor that is conserved when the matter equations of motion hold.
When e a µ = δ a µ, this symmetric tensor becomes the Belinfante tensor, proving the relation
between the Belinfante tensor and the metric (Rosenfeld) energy–momentum tensor that
we mentioned in Section 2.4.1.
Example: a Dirac spinor. Let us now apply this recipe to a Dirac spinor. 10 A Dirac spinor
ψ α has only a spinorial index (which we usually hide). Thus, we are going to assume that it
transforms as a spinor in tangent space and as a scalar under GCTs. Thus, the total covariant
derivative ∇µ coincides with the Lorentz-covariant derivative Dµ acting on it:
∇µψ = Dµψ = ∂µ − 1
4ωµ ab
γab ψ. (4.95)
In the special-relativistic Lagrangian of the Dirac spinor Eq. (2.63) the partial derivative
appears contracted with a constant gamma matrix. Now we have to distinguish between the
derivative index, which is a world-tensor index, and the gamma matrix index, which is a
Lorentz (tangent-space) index and, to contract both indices, we have to use a Vielbein
∇ψ = ea µ γ a ∇µψ. (4.96)
Finally, we also need the covariant derivative on the Dirac conjugate. The Dirac conjugate
¯ψα transforms covariantly (as opposed to the spinor ψ α , which transforms contravariantly).
Then, applying the definitions in Section 1.4,
¯ψ ←
∇µ ≡∇µ ¯ψ = ∂µ ¯ψ + 1
4ωµ ab ¯ψγab.
With all these elements we can immediately write the action
(4.97)
Smatter = d d
1
xe 2 (i ¯ψ ∇ψ − i ¯ψ ←
∇ ψ) − m ¯ψψ . (4.98)
10 The special-relativistic Dirac spinor was studied in Section 2.4.1.