80
Topological defects
3.7 Magnetic monopoles
It is also possible for point topological defects, magnetic monopoles [14 J, to form
at phase transitions in the early universe. The simplest model exhibiting this is
an SO(3) gauge field theory with SO(3) spontaneously broken to U(l) by the
expectation value of a scalar field • in the three-dimensional representation of
SO(3). The Lagrangian density for this model is
I
"A
2 2
C, = DI-'.· DI-'. - 4~"F: - 8(.·. -" )
(3.69)
where the gauge field strength is
F:" = al-'A~ - a"A~ - gE"bcAtA~.
(3.70)
The covariant derivative of the scalar field is
DI-'fi>Q = al-'fi>" - 8EabcAtfi>c
(3.71)
and a, h, c take the values 1,2,3. Minimization of the effective potential
A
V = _( •.• _ ,,2)2
(3.72)
8
fixes
(3.73)
,., =".
However, because of the SO(3) symmetry, the direction of. is not fixed.
The magnetic monopole solution [16] is a spheric ally symmetric solution for
• of the form
• = ,,/(r);
(3.74)
which is a mapping from ordinary space to the SO(3) space, with the asymptotic
behaviour
...... ,,;
(3.75)
asr-+oo
The spatial variation of. will be determined by the covariant derivative DI-'f
and so g" must enter the r-dependence. On dimensional grounds, we can write,
without loss of generality,
... = H(~)
."
A
(3.76)
-", ~
where
~ == KT/r.
(3.77)
The required behaviour as r -+ 00 is obtained if
H(~) -+ I
as ~ -+ 00.
(3.78)
~