String Theory Demystified

54 String Theory Demystifi ed
of this fact is called Noether’s theorem. Let’s quickly review the most famous
example of a conserved quantity, the conservation of electric charge.
The electromagnetic fi eld tensor F µν is defi ned in terms of the 4-vector
potential via
Fµν =∂µ Aν−∂ ν Aµ
The Maxwell equations with source terms are written as
∂ F = J
µ
µν ν
ν
Now it follows from the defi nition of F µν that ∂ J = 0 because
µν µ ν ν µ
∂ µ F =∂µ ∂ A −∂µ ∂ A ⇒∂ν ∂µ
F
µ ν
νν µ
=∂∂∂A −∂∂∂A
ν µ
ν µ
ν
µν
µ ν
ν µ
=∂ν ∂µ ∂ A −∂µ ∂ν∂ A (partial derivatives commute)
µ ν
µ ν
=∂∂∂ ν µ A −∂∂∂ ν µ A (relabel dummy
indices)
= 0
Hence J µ is a conserved quantity. This is a fact expressed in the famous continuity
equation which tells us that
∂ρ
+∇⋅ J = 0
∂ t
where ρ is the charge density and J is the current density. It follows that charge is
conserved. The charge Q is of course defi ned using
Q= ∫ d x
3 ρ
Using the continuity equation and taking the surface of integration S to be at infi nity
we get
dQ
dt
=
3 ∂ρ
3
dx =− dx∇⋅ J=− J⋅ dA=
0
∂ t
∫ ∫ ∫
Hence, charge is conserved.
We demonstrated that charge is conserved starting with the equation of motion
for the electromagnetic fi eld. More formally, we can determine what the conserved
S