Note that whileDAμis a differential operator, [DAμ,DAν] and thus the curvature
is just a multiplication operator.
The electromagnetic field strengths break up into those components with a
time index and those without:
Definition(Electric and magnetic fields).The electric and magnetic fields are
two functions fromR^4 toR^3 , with components given by
Ej=Fj 0 =−
∂Aj
∂t
+
∂A 0
∂xj
Bj=
1
2
jklFkl=jkl
∂Al
∂xk
or, in vector notation
E=−
∂A
∂t
+∇A 0 , B=∇×A
Eis called the electric field,Bthe magnetic field.
These are invariant under gauge transformations since
E→E−
∂
∂t
∇φ+∇
∂φ
∂t
=E
B→B+∇×∇φ=B
Here we use the fact that
∇×∇f= 0 (45.2)
for any functionf.
45.3 Field equations with background electro-
magnetic fields
The minimal coupling method described above can be used to write down field
equations for our free particle theories, now coupled to electromagnetic fields.
They are:
- The Schr ̈odinger equation for a non-relativistic particle coupled to a back-
ground electromagnetic field is
i
(
∂
∂t
−ieA 0
)
ψ=−
1
2 m
∑^3
j=1
(
∂
∂xj
−ieAj
) 2
ψ
A special case of this is the Coulomb potential problem discussed in chap-
ter 21, which corresponds to the choice of background field
A 0 =
1
r