Quantum Mechanics for Mathematicians

In this chapter we will see how to make theAμfields dynamical variables,
although restricting to the special case of free electromagnetic fields, without
interaction with matter fields. The equations of motion will be:

Definition(Maxwell’s equations in vacuo).The Maxwell equations for electro-
magnetic fields in the vacuum are

∇·B= 0 (46.1)

∇×E=−

∂B

∂t

(46.2)

∇×B=

∂E

∂t

(46.3)

and Gauss’s law:
∇·E= 0 (46.4)

Digression.In terms of differential forms these equations can be written very
simply as
dF= 0, d∗F= 0

where the first equation is equivalent to 46.1 and 46.2, the second (which uses
the Hodge star operator for the Minkowski metric) is equivalent to 46.3 and
46.4. Note that the first equation is automatically satisfied, since by definition
F=dA, and thedoperator satisfiesd^2 = 0.

Writing out equation 46.1 in terms of the vector potential gives

∇·∇×A= 0

which is automatically satisfied for any vector fieldA. Similarly, in terms of the
vector potential, equation 46.2 is

∇×

(

−

∂A

∂t

+∇A 0

)

=−

∂

∂t

(∇×A)

which is automatically satisfied since

∇×∇f= 0

for any functionf.
Note that since Maxwell’s equations only depend onAμthrough the gauge
invariant fieldsBandE, ifAμ= (A 0 ,A) is a solution, so is the gauge transform

Aφμ=

(

A 0 +

∂φ

∂t

,A+∇φ

)