Quantum Mechanics for Mathematicians

46.2 The Hamiltonian formalism for electromag-

netic fields

In order to quantize the electromagnetic field, we first need to express Maxwell’s
equations in Hamiltonian form. These equations are second-order differential
equations int, so we expect to parametrize solutions in terms of initial data

(A 0 (0,x),A(0,x)) and

(

∂A 0

∂t

(0,x),

∂A

∂t

(0,x)

)

(46.5)

The problem with this is that gauge invariance implies that ifAμis a solution
with this initial data, so is its gauge-transformAφμ(see equation 45.1) for any
functionφ(t,x) such that

φ(0,x) = 0,

∂φ

∂t

(t,x) = 0

This implies that solutions are not uniquely determined by the initial data of
the vector potential and its time derivative att= 0, and thus this initial data
will not provide coordinates on the space of solutions.
One way to deal with this problem is to try and find conditions on the vector
potential which will remove this freedom to perform such gauge transformations,
then take as phase space the subspace of initial data satisfying the conditions.
This is called making a “choice of gauge”. We will begin with:

Definition(Temporal gauge).A vector potentialAμis said to be in temporal
gauge ifA 0 = 0.

Note that given any vector potentialAμ, we can find a gauge transformationφ
such that the gauge transformed vector potential will haveA 0 = 0 by solving
the equation
∂φ
∂t

(t,x) =A 0 (t,x)

which has solution

φ(t,x) =

∫t

0

A 0 (τ,x)dτ+φ 0 (x) (46.6)

whereφ 0 (x) =φ(0,x) is any function of the spatial variablesx.
In temporal gauge, initial data for a solution to Maxwell’s equations is given
by a pair of functions (

A(x),

∂A

∂t

(x)

)

and we can take these as our coordinates on the phase space of solutions. The
electric field is now

E=−

∂A

∂t
so we can also write our coordinates on phase space as

(A(x),−E(x))