Exercises
89
monopole is approached, the SU(5) grand unified symmetry is essentially
unbroken in the core of the monopole. Consequently, when a nucleon encounters
a magnetic monopole, decay of the nucleon can be induced by the baryon-number
non-conserving interactions of the lepto-quark fields of the SU(5) grand unified
theory. In this way, the magnetic monopoles collected by stars in the course of
time will cause emission of radiation from neutron stars. This puts severe limits
on the flux of magnetic monopoles. More details of this and other astrophysical
effects may be found elsewhere [19].
3.11 Exercises
I. Calculate the energy of a kink or anti kink soliton solution.
2. Compare the total energy associated with a plane domain wall with 1/ =
lOO Ge V and area Ho2 with the known mass of the universe within a Hubble
volume Ho3.
3. Derive equation (3.16) for the dependence of a global cosmic string on the
cylindrical polar coordinate p.
4. Check that the Lagrangian (3.23) possesses the local UO) gauge symmetry
(3.26) and (3.27).
5. Show that the magnetic flux carried by a local cosmic string is given by
(3.30).
6. Derive the string equation of motion (3.61) and the constraints (3.62) and
(3.63) by varying the action (3.55) with respect to X"" and ha~.
7. Derive the covariant derivative (3.83) for a monopole solution.
8. Derive equations (3.87) and (3.88) for the form of a monopole solution.
9. Derive equation (3.96) for the magnetic field due to a magnetic monopole.
10. Estimate the monopole contribution to Oh 2 in non-supersymmetric and
supersymmetric grand unified theories.
3.12 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Hindmarsh M Band Kibble T W B 1995 Rep. Prog. Phys. 58477
• Vilenkin A 1985 Phys. Rep. 121263
• Kolb E Wand Turner M S 1990 The Early Universe (Reading, MA:
Addison-Wesley)
Bibliography
[1] Kibble T W B 1976 J. Phys. A; Math. Gen. 9 1387
[2] Zeldovich Ya B, Kobzarev I B and Okun L B 1975 SOy. Phys.-JETP 40 1
89
monopole is approached, the SU(5) grand unified symmetry is essentially
unbroken in the core of the monopole. Consequently, when a nucleon encounters
a magnetic monopole, decay of the nucleon can be induced by the baryon-number
non-conserving interactions of the lepto-quark fields of the SU(5) grand unified
theory. In this way, the magnetic monopoles collected by stars in the course of
time will cause emission of radiation from neutron stars. This puts severe limits
on the flux of magnetic monopoles. More details of this and other astrophysical
effects may be found elsewhere [19].
3.11 Exercises
I. Calculate the energy of a kink or anti kink soliton solution.
2. Compare the total energy associated with a plane domain wall with 1/ =
lOO Ge V and area Ho2 with the known mass of the universe within a Hubble
volume Ho3.
3. Derive equation (3.16) for the dependence of a global cosmic string on the
cylindrical polar coordinate p.
4. Check that the Lagrangian (3.23) possesses the local UO) gauge symmetry
(3.26) and (3.27).
5. Show that the magnetic flux carried by a local cosmic string is given by
(3.30).
6. Derive the string equation of motion (3.61) and the constraints (3.62) and
(3.63) by varying the action (3.55) with respect to X"" and ha~.
7. Derive the covariant derivative (3.83) for a monopole solution.
8. Derive equations (3.87) and (3.88) for the form of a monopole solution.
9. Derive equation (3.96) for the magnetic field due to a magnetic monopole.
10. Estimate the monopole contribution to Oh 2 in non-supersymmetric and
supersymmetric grand unified theories.
3.12 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Hindmarsh M Band Kibble T W B 1995 Rep. Prog. Phys. 58477
• Vilenkin A 1985 Phys. Rep. 121263
• Kolb E Wand Turner M S 1990 The Early Universe (Reading, MA:
Addison-Wesley)
Bibliography
[1] Kibble T W B 1976 J. Phys. A; Math. Gen. 9 1387
[2] Zeldovich Ya B, Kobzarev I B and Okun L B 1975 SOy. Phys.-JETP 40 1