Monopole topological quantum number
83
The surviving U (1) gauge group after spontaneous symmetry breaking by the
scalar field expectation value (3.74) is the group of rotations about the direction
i. The magnetic field should. therefore. be identified with the component B of
Ba in this direction:
B=~
(3.98)
gr2
Thus, the magnetic field is the field of a magnetic monopole with magnetic charge
47r/g.
3.8 Monopole topological quantum number
The asymptotic form t/J = "i of the magnetic monopole cannot be deformed
continuously to the trivial configuration t/J = "i and, for this reason, the magnetic
monopole is topologically conserved once formed. This is reflected in the
existence of a topological quantum number defined as follows. The sphere M
of all solutions for the expectation value of t/J which minimizes the tree-level
effective potential is given by
It/JI = '1.
(3.99)
This defines the surface of a sphere and t/J(r) can be thought of as a mapping from
the surface E of a sphere in coordinate space to the surface M of the sphere in
the space of solutions. If we parametrize the surface E by parameters u and v,
then the element of surface on the space M is
dS =dSn = ( - ot/J x - ot/J) dudv.
(3.100)
ou OV
The unit normal n to the sphere is i and so
( ot/J ot/J)
t/J. - x -
du dv = "dS.
(3.101)
ou ov
As the parameters u and v are varied to allow i to sweep out the surface E of a
sphere in coordinate space, t/J sweeps out the surface M of a sphere in the space
of solutions. For the single monopole solution that we are discussing. as the first
sphere is swept out once so is the second sphere. More generally, for a multimonopole solution, it is possible for the second sphere to be swept out N times,
where N is an integer, as the first sphere is swept out once. Thus,
~ t/J. ( - ot/J x - ot/J) du dv = 47r,,3 N.
(3.102)
1:
OU
OV
If we continuously deform t/J, N cannot change because it is an integer, one in the
case of the single-monopole solution. It is a topological quantum number, which
reflects the fact that a magnetic monopole configuration once formed cannot be
83
The surviving U (1) gauge group after spontaneous symmetry breaking by the
scalar field expectation value (3.74) is the group of rotations about the direction
i. The magnetic field should. therefore. be identified with the component B of
Ba in this direction:
B=~
(3.98)
gr2
Thus, the magnetic field is the field of a magnetic monopole with magnetic charge
47r/g.
3.8 Monopole topological quantum number
The asymptotic form t/J = "i of the magnetic monopole cannot be deformed
continuously to the trivial configuration t/J = "i and, for this reason, the magnetic
monopole is topologically conserved once formed. This is reflected in the
existence of a topological quantum number defined as follows. The sphere M
of all solutions for the expectation value of t/J which minimizes the tree-level
effective potential is given by
It/JI = '1.
(3.99)
This defines the surface of a sphere and t/J(r) can be thought of as a mapping from
the surface E of a sphere in coordinate space to the surface M of the sphere in
the space of solutions. If we parametrize the surface E by parameters u and v,
then the element of surface on the space M is
dS =dSn = ( - ot/J x - ot/J) dudv.
(3.100)
ou OV
The unit normal n to the sphere is i and so
( ot/J ot/J)
t/J. - x -
du dv = "dS.
(3.101)
ou ov
As the parameters u and v are varied to allow i to sweep out the surface E of a
sphere in coordinate space, t/J sweeps out the surface M of a sphere in the space
of solutions. For the single monopole solution that we are discussing. as the first
sphere is swept out once so is the second sphere. More generally, for a multimonopole solution, it is possible for the second sphere to be swept out N times,
where N is an integer, as the first sphere is swept out once. Thus,
~ t/J. ( - ot/J x - ot/J) du dv = 47r,,3 N.
(3.102)
1:
OU
OV
If we continuously deform t/J, N cannot change because it is an integer, one in the
case of the single-monopole solution. It is a topological quantum number, which
reflects the fact that a magnetic monopole configuration once formed cannot be