Gravity and Strings

2.5 The Noether method 41
We may expect that this is completely general. We need to know only how the fields
transform under general coordinate transformations, 13 which determines completely the
coupling to gravity in general relativity.
Just as it is possible to give a prescription for how to find the energy–momentum tensor
on the basis of its coupling to gravity through the metric in general relativity, it is possible to
give a definition of the spin–energy potential µν ρ based on its coupling to gravity (maybe
we should say geometry instead of gravity) through the torsion tensor in the framework of
the Cartan–Sciama–Kibble (CSK) theory:
µν ρ =− 2
√ |g|
δS
δTµν ρ
γ =T =0
. (2.88)
The equivalence of this definition and the definition we gave in terms of the spin-angularmomentum
tensor S µ ρσ can also be proven in the CSK theory. In fact, the above definition is
the main characteristic of that theory in which intrinsic (i.e. not orbital) angular momentum
is the source of another field that has a geometrical interpretation (torsion).
2.5 The Noether method
There is a useful recipe for how to find the Noether current associated with global symmetry
transformations of the fields δϕ:ifthe action is invariant under transformations with
constant parameters, then, if we use local parameters, upon use of the equations of motion,
the variation of the action would be proportional to the derivative of the parameters:
δS =− d d x∂µσ I j µ
I , (2.89)
because, by hypothesis, it has to vanish for constant σ I .Uptoatotal derivative, this is
δS = d d xσ I ∂µ j µ
I , (2.90)
that vanishes for constant σ I only if ∂µ j µ
µ
I = 0. Thus the currents j I are the Noether currents
associated with the global symmetry.
The observation that the variation of the action must be of the above form is the basis
of the so-called Noether method which is used to couple fields in a symmetric way. The
simplest example of how this method works is the coupling of a complex scalar field
to the electromagnetic field Aµ. The Lagrangian of the electromagnetic field Eq. (2.56) is
invariant under the transformations with local parameter ,
while the Lagrangian for the complex scalar,
δ Aµ = ∂µ, (2.91)
L = 1
2 ∂µ∂ µ ¯, (2.92)
13 A conformal scalar of weight ω is nothing but a scalar density of weight ω/d.