Quantum Mechanics for Mathematicians

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

, A= 0