String Theory Demystified

CHAPTER 3 Symmetries and Worldsheet Currents 55
quantities are and relate them to symmetries by looking at the action S, or more
particularly the lagrangian. This is where Noether’s theorem comes into play—
symmetries in the lagrangian lead to conserved quantities.
You can understand Noether’s theorem with a simple one-dimensional example.
Consider a particle whose motion is described by a lagrangian L( q, q
) where
q
dq
=
dt
The momentum of the particle is given by
L
p =
q
∂
∂
The Euler-Lagrange equations are the equations of motion for this system:
d
dt
∂L
∂L
−
∂q ∂ q
= 0
Now suppose that the lagrangian is invariant under a symmetry. That is, the form
of the lagrangian does not change under a one parameter coordinate transformation
t → s( t):
qt () → qs ()
Saying that the lagrangian is invariant under this symmetry means that
d
Lqs [ ( ), qs ( )] = 0
ds
The symmetry of the lagrangian can be written out explicitly using the chain rule as
d
L dq L dq
Lqs [ ( ), qs (
)] =
ds q ds q
ds
∂
+
∂
∂
∂
= 0
Now let’s get to the central idea. Noether’s theorem tells us that
Q p dq
=
ds
is a conserved quantity, that is,
dQ
dt
= 0