Single-Particle Electrodynamics - Assassination Science

If we now differentiate (4.63) with respect to v, to obtain the corresponding
contribution to the canonical momentum b, we find
b int = −d ×B.
In other words (adding this result to that of an electric charge), we find
b = p + qA − d ×B. (4.64)
This indicates that perhaps the principle of “minimal coupling”, reviewed in
Chapter 2 for an electric charge, might be extended when an electric dipole
moment is present, to the expression (4.64).
If one computes the Euler–Lagrange equations of motion for the interaction
Lagrangian (4.63), one simply finds the same results as found in Section
4.2.1 via Newtonian mechanics, except that, yet again, the nonrelativistic
analysis does not yield the redshift force.
If we now compute the Hamiltonian for the electric dipole, from the
Lagrangian (4.64), one finds
H =
(b − qA + d ×B)2
2m
− d ·E. (4.65)
Recalling that H is just the zero-component b 0 of the canonical momentum
four-vector b α , we thus obtain the manifestly covariant expression of extended
minimal coupling:
b = p + qA + d ·F,
or, alternatively,
p 2 = (b − qA − d ·F ) 2 = m 2 ,
of which (4.65) is the nonrelativistic limit.
4.3.2 The magnetic-charge dipole
Since one cannot have both a well-defined four-potential A(x) and magnetic
monopoles in the same Universe, without performing serious plastic surgery
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