–1–
MAGNETIC MONOPOLES
Written in September 2009 by D. Milstead (Stockholm Univ.)
and E.J. Weinberg (Columbia Univ.).
The symmetry between electric and magnetic felds in the
sourceFree Maxwell’s equations naturally suggests that electric
charges might have magnetic counterparts, known as magnetic
monopoles. Although the greatest interest has been in the
supermassive monopoles that are a frm prediction oF all grand
unifed theories, one cannot exclude the possibility oF lighter
monopoles, even though there is at present no strong theoretical
motivation For these.
In either case, the magnetic charge is constrained by a
quantization condition frst Found by Dirac [1].
Consider a
monopole with magnetic charge
Q
M
and a Coulomb magnetic
feld
B
=
Q
M
4
π
ˆ
r
r
2
.
(1)
Any vector potential
A
whose curl is equal to
B
must be singular
along some line running From the origin to spatial infnity. This
Dirac string singularity could potentially be detected through
the extra phase that the waveFunction oF a particle with electric
charge
Q
E
would acquire iF it moved along a loop encircling
the string. ±or the string to be unobservable, this phase must
be a multiple oF 2
π
. Requiring that this be the case For any
pair oF electric and magnetic charges gives the condition that
all charges be integer multiples oF minimum charges
Q
min
E
and
Q
min
M
obeying
Q
min
E
Q
min
M
=2
π.
(2)
(±or monopoles which also carry an electric charge, called
dyons, the quantization conditions on their electric charges can
be modifed. However, the constraints on magnetic charges, as
well as those on all purely electric particles, will be unchanged.)
Another way to understand this result is to note that the
conserved orbital angular momentum oF a point electric charge
moving in the feld oF a magnetic monopole has an additional
component, with
L
=
m
r
×
v
−
4
πQ
E
Q
M
ˆ
r
(3)
CITATION: K. Nakamura
et al.
(Particle Data Group), JPG
37
, 075021 (2010) (URL: http://pdg.lbl.gov)
July 30, 2010
14:34
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Requiring the radial component of
L
to be quantized in half
integer units yields Eq. (2).
If there are unbroken gauge symmetries in addition to
the U(1) of electromagnetism, the above analysis must be
modiFed [2,3].
±or example, a monopole could have both a
U(1) magnetic charge and a color magnetic charge. The latter
could combine with the color charge of a quark to give an
additional contribution to the phase factor associated with a
loop around the Dirac string, so that the U(1) charge could
be the Dirac charge
Q
D
M
≡
2
π/e
, the result that would be
obtained by substituting the electron charge into Eq. (2). On
the other hand, for monopoles without colormagnetic charge,
one would simply insert the quark electric charges into Eq. (2)
and conclude that
Q
M
must be a multiple of 6
.
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 Spring '11
 Kutter
 Charge, Magnetic Field, Magnetic monopole, Magnetic Monopoles, monopoles

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