Gravity and Strings

3.1 Scalar SRFTs of gravity 49
via the Noether theorem for global transformations: using the infinitesimal form of the
Poincaré transformations,
we obtain the conservation law
δx µ = σ µ νx ν + σ µ , σ µν =−σ νµ , (3.12)
dJ(σ )
dξ = 0, J(σ ) = PµδX µ . (3.13)
The conserved quantity associated with translations is the linear momentum J(σ µ ) ∼ P µ
and the conserved quantity associated with Lorentz transformations is the angular momentum
J(σ µν ) ∼ M µν .
Observe that the invariance of the action is due to the fact that it depends only on the
derivatives of the coordinates. In particular, the Minkowski metric does not depend on
the coordinates. A better way to express this fact is to say that the Minkowski metric has
d(d + 1)/2 independent isometries that generate the d-dimensional Poincaré group. This
association between spacetime isometries and conserved quantities will still hold in more
complicated spacetimes.
This action is also invariant under non-singular reparametrizations of the worldline ξ ′ (ξ).
These are local (gauge) transformations that infinitesimally can be written δξ = ɛ(ξ).Taking
into account that the X µ s are scalars with respect to these transformations, we find
δξ = ɛ(ξ), δdξ =˙ɛdξ, ˜δX µ = 0, δ ˙X µ =−˙ɛ ˙X µ , (3.14)
and it is a simple exercise to check that ˜δS = 0 identically. If we now consider the variation
of the action under just
δX µ =−ɛ ˙X µ , (3.15)
we find that it is invariant only up to a total derivative
δS = dξ d
Mcɛ
dξ
ηµν
˙X µ ˙X ν
. (3.16)
On varying the action with respect to general variations of the coordinates first and integrating
by parts, we obtain
δS = dξ ɛ δS
δX ν ˙X ν + d
Mcɛ ηµν ˙X µ ˙X ν . (3.17)
dξ
By equating the two results and taking into account that the equation is valid for arbitrary
functions ɛ(ξ),weobtain the gauge identity
δS
δX ν ˙X ν = 0, ⇒ ˙Pν ˙X ν = 0, (3.18)
which is satisfied off-shell (trivially on-shell). Since ˙X ν is proportional to the momentum,
this identity is proportional to
d(P µ Pµ)
= 0. (3.19)
dξ