130_notes.dvi

Note that, since we have four independent components ofAμas independent fields, we have four
equations; or one 4-vector equation. TheEuler-Lagrange equation gets us back Maxwell’s
equationwith this choice of the Lagrangian. This clearly justifies the choice ofL.

It is important to emphasize that we have a Lagrangian based, formal classical field theory for
electricity and magnetism which hasthe four components of the 4-vector potential as the
independent fields. We could not treat each component ofFμν as independent since they are
clearly correlated. We could have tried using the six independent components of the antisymmetric
tensor but it would not have given the right answer. Using the 4-vector potentials as the fields does
give the right answer.Electricity and Magnetism is a theory of a 4-vector fieldAμ.

We can also calculate thefree field Hamiltonian density, that is, the Hamiltonian density in
regions with no source term. We use the standard definition of the Hamiltonian in terms of the
Lagrangian.

H=

(

∂L

∂(∂Aμ/∂dt)

)

∂Aμ

∂dt

−L=

(

∂L

∂(∂Aμ/∂x 4 )

)

∂Aμ

∂x 4

−L

We just calculated above that
∂L
∂(∂Aμ/∂xν)

=Fμν

which we can use to get

∂L

∂(∂Aμ/∂x 4 )

= Fμ 4

H = (Fμ 4 )

∂Aμ

∂x 4

−L

= Fμ 4

∂Aμ

∂x 4

+

1

4

FμνFμν

H=Fμ 4

∂Aμ

∂x 4

+

1

4

FμνFμν

We will use this once we have written the radiation field in a convenient form. In the meantime, we
can check what this gives us in general in a region with no sources.

H = Fμ 4

(

F 4 μ+

∂A 4

∂xμ

)

+

1

4

FμνFμν

= −F 4 μ

(

F 4 μ+

∂A 4

∂xμ

)

+

1

4

FμνFμν

= −F 4 μF 4 μ−F 4 μ

∂A 4

∂xμ

+

1

4

FμνFμν

= E^2 −F 4 i

∂A 4

∂xi

+

1

2

(B^2 −E^2 )

=

1

2

(E^2 +B^2 )−iEi

∂(iφ)

∂xi

=

1

2

(E^2 +B^2 ) +Ei

∂φ

∂xi