Single-Particle Electrodynamics - Assassination Science

In terms of ˜J(x), the complete field Lagrangian density (5.47) then takes its
simplest form:
L = 1 4 F αβ F αβ + J α A α − 1 2 ˜J αβ F αβ . (5.49)
5.4.3 The generalised Maxwell equation
We now compute the Euler–Lagrange equations for the electromagnetic potentials
using the Lagrangian density (5.49). The canonical field Π µν (x)
conjugate to A α is given by
Π µν ≡ ∂ ∂ µ A νL = F µν − ˜J µν . (5.50)
(Explicitly computed, only the combination ˜J [µν] contributes to Π µν ; it is
basically for this reason that we were free to arbitrarily define ˜J to be antisymmetric
in (5.48).) Furthermore, we have
∂ A νL = J ν .
The field Euler–Lagrange equations for the fields A µ (x) thus yield
∂ µ ∂ ∂ µ A νL − ∂ A νL = ∂µ( F µν − ˜J
)
µν − Jν = 0,
or, upon reärranging, and employing component-free notation,
∂ ·F = J + ∂ · ˜J. (5.51)
Equation (5.51) is the generalised inhomogeneous Maxwell equation for the
case of both monopolar (i.e., electric charge) and dipolar sources, for applications
in which it is convenient to separate the former from the latter.
(Fundamentally, of course, the dipole moments are implicitly included in the
source term J α (x) of the regular inhomogeneous Maxwell equation (5.7); the
source vector J α appearing in (5.51) should properly be referred to as that
due to free charge only.)
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