Single-Particle Electrodynamics - Assassination Science

the overall Lorentz-gauge four-potential A µ LG(x), now given by
A µ LG(x) = A µ charge (x) + ∂ νÃνµ (x),
where A µ charge (x) is given by equation (5.13) of Section 5.3.3.
5.4.5 The fields for a point particle
We can now determine the retarded fields that are generated by a classical
point particle carrying electric and magnetic dipole moments. As noted
earlier, we are, for practical simplicity, parametrising the dipole moment
four-vectors d α and µ α in terms of the unit four-spin vector Σ α :
d α = d Σ α ,
µ α = µ Σ α ,
where the rest-frame dipole moment magnitudes d and µ are fixed.
The
analogue of (5.4) can then be written down by using equations (5.45), (5.46)
and (5.48):
˜J αβ (x) =
(d g αµ g βν − 1 )∫
2 µ εαβµν dτ δ (4) [ x − z(τ) ] U [µ (τ)Σ ν] (τ). (5.56)
Substituting (5.12) and (5.56) into (5.55), and integrating over d 4 x ′ , we find
à αβ (x) =
( d
2π gαµ g βν − µ
4π εαβµν ) ∫
dτ U [µ Σ ν] ϑ(x 0 − z 0 ) δ [ (x − z) 2] , (5.57)
which is the analogue of (5.15).
We now discard the tensor potential à µν (x)—and, indeed, the Lorentzgauge
four-potential A µ LG(x) itself—in favour of the field strength tensor
F αβ (x). From (5.2) and (5.54) we have
F αβ ≡ ∂ α ∂ µ A µ β − ∂ β ∂ µ A µ α. (5.58)
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