130_notes.dvi

Note that this is automatically antisymmetric under the interchangeof the indices. As before, the
first two (sourceless) Maxwell equations are automaticallysatisfiedfor fields derived from
a vector potential. We may write theother two Maxwell equationsin terms of the 4-vector
jμ= (~j,icρ).

∂Fμν

∂xν

=

jμ

c

Which is why the T-shirt given to every MIT freshman when they takeElectricity and Magnetism
should say

“... and God said∂x∂ν

(

∂Aν

∂xμ−

∂Aμ

∂xν

)

=jcμ and there was light.”

Of course he or she hadn’t yet quantized the theory in that statement.

For some peace of mind, letsverify a few terms in the equations. Clearly all the diagonal terms
in the field tensor are zero by antisymmetry. Lets take some example off-diagonal terms in the field
tensor, checking the (old) definition of the fields in terms of the potential.

B~ = ∇×~ A~

E~ = −∇~φ−^1

c

∂A~

∂t

F 12 =

∂A 2

∂x 1

−

∂A 1

∂x 2

= (∇×~ A~)z=Bz

F 13 =

∂A 3

∂x 1

−

∂A 1

∂x 3

=−(∇×~ A~)y=−By

F 4 i =

∂Ai

∂x 4

−

∂A 4

∂xi

=

1

ic

∂Ai

∂t

−

∂(iφ)

∂xi

=−i

(

1

c

∂Ai

∂t

+

∂φ

∂xi

)

=−i

(

∂φ

∂xi

+

1

c

∂Ai

∂t

)

=iEi

Lets alsocheck what the Maxwell equation saysfor the last row in the tensor.

∂F 4 ν

∂xν

=

j 4

c

∂F 4 i

∂xi

=

icρ

c

∂(iEi)

∂xi

= iρ

∂Ei

∂xi

= ρ

∇·~ E~ = ρ

We will not bother to check the Lorentz transformation of the fields here. Its right.