Single-Particle Electrodynamics - Assassination Science

efore neutral particles were discovered, and its use for such particles is messy
and contrived; thus, we shall use the results of (D.15), (D.16) and (D.19),
namely,
M µν ≡ 2mµ
q
˜Σ µν ≡ 2mµ
q
ε µναβ U α Σ β ,
(D.20)
to replace Kolsrud and Leer’s M αβ wherever it appears in favour of µ, U α
and Σ α .
D.6 Verification of the retarded fields
We can now complete the translation of the Cohn–Wiebe field expressions,
(D.7), into the conventions and notation of this thesis.
Concentrating first on the fields generated by the electric charge, we have,
from (D.7) and the relations of Section D.4,
F ′ = qϕ 2 (1 − ˙χ)ζ ∧U + qϕζ ∧ ˙U.
This agrees with the result (5.28) of Chapter 5, and justifies the identification
(D.13).
Turning now to the dipole fields, we need to convert the magnetic dipole
expressions of Cohn and Wiebe into the equivalent expressions for an electric
dipole, since we primarily focussed on the electric case in Chapter 5. To do so,
we need only note that the electric field per electric dipole unit E/d generated
by an electric dipole d α is equal to the magnetic field per magnetic dipole
unit B/µ generated by a magnetic dipole µ α (excluding the extra Maxwell
term for the magnetic dipole—but Cohn and Wiebe do not consider the
worldline fields). Since a duality transformation F αβ → ˜F αβ induces the
transformations E → B, B → −E, we need to take the negative of the dual
of the magnetic dipole part of the Cohn–Wiebe fields (D.7) in order to get
the equivalent electric dipole fields. In other words, we need to compute
×
Fn d ≡ − 1 F (−n) ,
2
(D.21)
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