Gravity and Strings

8.7 Electric-magnetic duality 249
dynamical electric (only) charges. It is necessary to introduce magnetic sources that can
be rotated into the electric ones in order to maintain duality invariance of the Maxwell
equations. We have already seen in Eq. (8.133) that electric–magnetic duality needs the
introduction of magnetic charges into which electric charges can transform. By definition,
then, the magnetic charge is given by25 :
p ≡˜q = d 2 S · ˜
E =
S 2 ∞
S 2 ∞
d 2
S · B =−
S2 F. (8.144)
∞
The simplest electric-charge distribution is a point-like electric charge and its dual is a
magnetic point-like charge, which should be given by a magnetic field obeying
∇ · B = p δ (3) (x3), (8.145)
which is the Dirac monopole equation for the vector potential.
Introducing magnetic sources to preserve electric–magnetic duality is, however, a very
dangerous move: the Bianchi identity is not satisfied at the locations of the magnetic sources
and there the vector potential, the true dynamical field, cannot be defined or, more precisely,
it cannot be defined everywhere: it will have singularities. This may not be as bad as it looks
at first sight, because, after all, the electrostatic potential is not defined at the location of
an electric point-like charge, either. It depends on how bad the singularities of the vector
field are. In the electric case, it is quite benign, since the singularity affects only the particle
that gives rise to the field. Let us see what happens with the vector potential of a point-like
magnetic monopole. First, we have to find it.
Knowing that
2 1
∇
|x3| =−4πδ(3) (x3), (8.146)
we find that the magnetic field is given by
B =− p
∇
1
, (8.147)
4π |x3|
which implies, due to B = ∇ × A, for the Dirac monopole equation
∇ × A =− p
∇
1
, (8.148)
4π |x3|
or, defining, to simplify matters f =−(4π/p) A, the following, standard form:
1
∂m fn − ∂n fm = ɛmnp∂p . (8.149)
|x3|
25 We work again in the standard units of the beginning of Section 8.2.1 and in flat spacetime. At the end of
this section we will say which changes have to be made when using our normalization Eq. (8.58).