Homework 8
1
11/1/2005
Homework VIII
Problems:
VIII.1
Show that the Lagrangian
L = –
¼
F
μ
ν
F
μ
ν
leads to the correct equations of motion for the free
photon field
A
μ
.
Solution:
The EulerLagrange equation from this Lagrangian is:
2
0
AA
µµν
ν
∂
−∂ ∂
=
. With the Lorentz
gauge choice
0
A
µ
∂=
this becomes
2
0
A
∂
=
. This is the wave equation for a free photon with
solution
ipx
A
e
µµ
ε
−
=
. The Lorentz condition limits the
ε
μ
to 3 independent components, and the
remaining gauge freedom
2
, with
0
→+
∂
Λ
∂
Λ
=
to conserve the Lorentz condition,
brings it further down to 2 independent components for
ε
μ
, i.e. two independent polarization di
rections.
VIII.2
Discuss why Dirac's idea for saving his equation with its
E
<0 solutions by assuming these
bothersome states fully filled (the “Dirac sea”), cannot be used to save the Klein Gordon equa
tion.
Solution:
The Dirac idea was to interpret the negativeenergy solutions by means of energy states that are
symmetrically placed around
E
=0. In order to prevent positiveenergy electrons making transi
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 Fall '05
 MichaelRijssenbeek
 Physics, Work, Photon, Quantum Field Theory, Fundamental physics concepts, Dirac sea, klein gordon equation, positiveenergy electrons

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