If we integrate the last term by parts, (and the fields fall to zero at infinity), then that term contains a∇ ·~ E~
which is zero with no sources in the region. We can therefore drop it and are left with
H=
1
2
(E^2 +B^2 ).This is the result we expected, the energy density and an EM field. (Remember the fields have been decreased
by a factor of
√
4 πcompared to CGS units.)We will study the interaction between electrons and the electromagnetic field with the Dirac equation.
Until then, the Hamiltonian used for non-relativistic quantum mechanics will be sufficient. We have
derived the Lorentz force law from that Hamiltonian.
H=1
2 m(
~p+e
cA~
) 2
+eA 032.4 Gauge Invariance can Simplify Equations
We have already studied many aspects of gauge invariance (See Section 20.3). in electromagnetism
and the corresponding invariance under a phase transformation inQuantum Mechanics. One point
to note is that, with our choice to “treat each component ofAμas an independent field”, we are
making atheory for the vector fieldAμwith a gauge symmetry, not really a theory for the
fieldFμν.
Recall that the gauge symmetry of Electricity and Magnetism and the phase symmetry of electron
wavefunctions are really one and the same. Neither the phase of the wavefunction nor the vector
potential are directly observable, but the symmetry is.
We will not go over the consequences of gauge invariance again here, but, we do want to use gauge
invariance to simplify our equations.
Maxwell’s equation is
∂Fμν
∂xν=
jμ
c
∂
∂xν(
∂Aν
∂xμ−
∂Aμ
∂xν)
=
jμ
c
∂
∂xν∂Aν
∂xμ−
∂^2 Aμ
∂x^2 ν=
jμ
c
∂^2 Aμ
∂x^2 ν−
∂
∂xμ∂Aν
∂xν=−
jμ
cWe can simplify this basic equation by setting the gauge according to theLorentz condition.
∂Aν
∂xν= 0
The gauge transformation needed is
Aμ→Aμ+∂χ
∂xμ