Gravity and Strings

42 Noether’s theorems
is invariant under phase transformations with a constant parameter σ and a constant g that
infinitesimally look like this:
δ = igσ. (2.93)
These transformations constitute a U(1) symmetry group. g labels the representation of
U(1) corresponding to .Ifσ takes values in the interval [0, 2π], then g can be any integer.
If the conserved current of the scalar Lagrangian is seen as an electric current, it is natural
to couple it to the electromagnetic vector field to obtain the Maxwell equation with sources:
∂ν F νµ = gj µ
N . (2.94)
From a Lagrangian point of view, this equation can be obtained by adding to the free
Lagrangians of the vector and scalar a coupling of the form gAµ j µ
N .However, this term
modifies the equation of motion of the scalar, so the electric current j µ
N is not conserved
on-shell. This renders the above equation inconsistent since the l.h.s. is automatically divergenceless.
Clearly, the addition of a new term to the Lagrangian modifies the Noether
current. The modified Noether current should be conserved on-shell upon use of the modified
equations of motion. It is easy to see that the vector field contributes to it. This is the
Noether current that we should use in the Lagrangian now, and this induces new modifications.
This may go on indefinitely until the new correction does not contribute to the new
Noether current. Observe that the modified Noether current is found using a local phase
transformation according to the above general observation. It should also be stressed that
the physical reason why there was inconsistency is that we did not take into account the
contribution of the vector field to the electric current. Only the total electric current should
be consistently conserved.
The Noether method is essentially a systematic way of performing these iterations emphasizing
the role of symmetry. In the case at hand, the basic idea is that one has to identify
σ with and one has to make the whole system invariant under transformations of the
same form with local. We start by calling L0 the Lagrangian which is the sum of the free
electromagnetic and scalar Lagrangians and using the above general observation: under a
local transformation (σ = ), and up to total derivatives,
δL0 = g∂µ j µ
µ
N , j N =−i
µ
∂ ¯ − ¯∂
2
µ . (2.95)
j µ is the on-shell conserved current associated with the global invariance of the Lagrangian.
The Noether method consists in the addition to L0 of terms that will be of higher order
in the constant g to compensate for the above non-vanishing variation. Typically the first
correction will be of the form
L1 = L0 + gAµ j µ
N . (2.96)
The additional term cancels out the variation of L0 but generates, due to the variation of the
Noether current itself, another term of order O(g2 ).Uptototal derivatives
This variation can be exactly canceled out by
δL1 =−g 2 || 2 Aµ∂ µ . (2.97)
L2 = L1 + 1
2 g2 || 2 A 2 , (2.98)