Magnetic mono poles
81
In the absence of gauge fields, the contribution of the scalar kinetic term to the
energy is given by
E =! f a;#/t· a;#/td 3 x
(3.79)
where i = I, 2, 3 is a (summed) spatial index. In spherical polar coordinates
(r, 8, tP),
v _ atPa r ~ atPa j _I _ atPa ~
(3.80)
tPa - ar + r ae + r sin e atP tit·
Since r is a function of e and tP, but not of r, the large r behaviour of the integral
(3.79) for the energy of the monopole is controlled by
f [( ar)2 I (ar)2]
(3.81 )
E ....
dr
ae + sin2 e atP .
Thus, in the absence of a gauge field contribution to E, the energy of the monopole
solution would be infinite.
To find a finite-energy solution, we need the gauge field contribution to the
covariant derivative to produce a cancellation to 'improve' the behaviour of VtPa
for large r. This possibility may be studied by making the following ansatz for
the gauge field expectation value:
A':I = E a il 2
r l (K(~) - I).
(3.82)
I
gr
Then (exercise 7)
K(~)H(~) 2
,
rar;
D;tPa = r4 (r &a; - rar;) + (~H (~) - H(~»4.
(3.83)
g
gr
A dangerous term of the type H(~)r2t5a;/gr4 has cancelled between the a;tPa and
-gEabcA~tPc contributions to D;tPa. For ~H(~) -+ 1 as ~ -+ 00, this term would
have had the unwelcome asymptotic behaviour r- 1 as r -+ 00. The surviving
terms are of order r- 2 as r -+ 00, provided K (~)H (n and ~ H' (~) - H (~) are
finite for ~ -+ 00, and a divergent contribution to the energy of the monopole is
avoided.
With the ansatz (3.82) for the expectation value of the gauge field, the field
strength is given by
K2 - 1
(K'
K2 - 1 )
gFjj = --2-~iaj + "2 - - - 4 - (E;aprprj - Ejaprpr;).
(3.84)
r
r
r
The energy of the magnetic monopole solution may now be written in terms of H
and K as
47f' 710
00
I
g 0
2
E = -
d~ ~-2 [ -(~H' - H)2 + H2K2 + (~K')2
(3.85)
+ ~(K2 - 1)2 + ~(H2 - rh2].
(3.86)
2
Sg