Gravity and Strings

60 A perturbative introduction to general relativity
3.2.1 Gauge invariance, gauge identities, and charge conservation
in the SRFT of a spin-1 particle
A massive or massless spin-1 particle is described by a vector field A µ . The simplest relativistic
wave equation we could imagine for it would be
∂ 2 + m 2 A µ = 0. (3.62)
However, the energy density of this theory is not positive-definite unless one imposes the
Lorentz or transversality condition
∂µ A µ = 0. (3.63)
Furthermore, just as in the spin-2-particle case, the vector A µ describes spin-1 helicity
states but also spin-0 helicity states. A d-dimensional vector field has d independent
components, but a massive spin-1 particle in d dimensions has d − 1 states (three in
d = 4: sz =−1, 0, 1) and a massless spin-1 particle has d − 2 helicity states (two in d = 4:
sz =−1, +1). It is precisely the unwanted spin-0 helicity states that contribute negatively
to the energy and the Lorentz condition projects them out.
If we couple the massless theory to charged matter, by Lorentz covariance, this has to be
described by a vector current j µ ,sowehave
∂ 2 A µ = j µ
(3.64)
and, by consistency with the Lorentz condition, the vector current has to be conserved,
∂µ j µ = 0, which is, again, a physically meaningful condition that coincides with our experience
with electric charges and currents.
We would like to construct a theory in which the Lorentz condition arises as a consequence
of the equation of motion in the massive case and in which ∂µ A µ is completely
arbitrary in the massless case. These conditions guarantee the removal of the unwanted
helicities. We expect the equation of motion to be of the form
D µ (A) + m 2 A µ = j µ , (3.65)
where, now, by consistency, the massless wave operator D µ (A) has to satisfy off-shell the
identity
∂µD µ (A) = 0, (3.66)
which should arise as the gauge identity associated with some gauge symmetry.
We could proceed as in [739], translating these conditions into a gauge identity for a
general Lagrangian and then trying to find, with as much generality as possible, a gauge
symmetry (forming a group) leading to that gauge identity. As is well known, the result is
the Proca Lagrangian and equation of motion,
S[A] = d d
x − 1
4 F 2 + m2
2 A2
,
(3.67)
where
D µ
(m) (A) = Dµ (A) + m 2 A µ = 0,
D µ (A) ≡ ∂µF µν , Fµν = 2∂[µ Aν], (3.68)