Single-Particle Electrodynamics - Assassination Science

moments of a particle. There are a number of ways of doing this; here, we
follow the same method as was used in the Section 5.3.1: we start with the
interaction Lagrangian for a point particle carrying such moments. From
equation (E.7) of Section E.4, we have
L int = qU µ A µ + µ α ˜F αβ U β + d α F αβ U β ,
We now need to generalise the latter two terms to the case of arbitrary dipole
densities, analogous to the generalisation, in Section 5.3.1, of the first term
from charged current to charged current density. To this end, we define two
antisymmetric density tensors ˜d αβ (x) and ˜µ αβ (x) for the point particle, thus:
∫
˜d(x) ≡ dτ U(τ)∧d(τ) δ (4) [ x − z(τ) ] , (5.45)
˜µ(x) ≡
∫
dτ U(τ)∧µ(τ) δ (4) [ x − z(τ) ] . (5.46)
The combined field–particle Lagrangian density may then be written
L = 1 4 F αβ F αβ + J α A α − 1 2 ˜µαβ ˜F αβ − 1 2 ˜d αβ F αβ . (5.47)
5.4.2 The dipole current density tensor
It is possible to simplify the last two terms of (5.47) a little further. Expanding
out the dual field strength tensor in terms of its definition (B.3),
˜F αβ ≡ − 1 2 ε αβµνF µν , we have
− 1 2 ˜µαβ ˜F αβ − 1 2 ˜d αβ F αβ = 1 4 ˜µαβ ε αβµν F µν − 1 2 ˜d αβ F αβ .
Relabelling indices, this last expression becomes
1
4 ˜µ µνε µναβ F αβ − 1 2 ˜d αβ F αβ .
We now define the dipole current density tensor, ˜J αβ (x), as
˜J αβ (x) ≡ ˜d αβ (x) − 1 2 εαβµν ˜µ µν (x). (5.48)
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