Problem 10.2: A magnetic monopole. (30 points) Some useful formulas in spherical coordinates: V = A 1 d rain # Op ' V xX = r sin d 6 (sin 8X) - axe + + sind Op - or ( rx ) 8 + = = (X. ) - ax, Consider the vector potential A g(1 -cos B) (8) r sin 8 where g is a constant. a) First, show that the associated magnetic field B is the field of a magnetic monopole, that is, a Coulomb magnetic field. What quantity plays the role of "magnetic charge"? b) If there is a magnetic Coulomb field, then the divergence of the magnetic field is not zero at the origin. However, we expect that any divergence of a curl of something well-defined is always zero. We should therefore ask what went wrong. What went wrong is that the vector potential is not well-defined everywhere. Where be- sides the origin is the vector potential A singular? Be sure to treat apparently indeterminate quantities carefully. The locus of singularities is called the Dirac string. c) If we have a magnetic monopole we can never get rid of the Dirac string, but gauge transformations can move it around. Define the new vector potential (9) 7 sin & Find a gauge transformation relating ' to A. Where is the Dirac string for A"? Other gauge transformations can move the string to other locations. Since the location of the Dirac string is arbitrary and it only shows up in A, never in B, it is not a physical object. d) Imagine a particle with ordinary electric charge of magnitude q, moving around in the presence of the magnetic monopole; this electrically charged particle has wavefunction v. 2 When using the two different vector potentials ' and A we must use different wavefunctions, w' = he for a function A. What is h in this case? Assuming the wavefunction of is well-defined, the gauge transform ' will only be well defined if h(v = 0) = h(y = 2x), since 1 = 0,2x represent the same point. What constraint must we place on q and g to ensure ' is well-defined? This constraint is called the Dirac quantization condition. If a magnetic monopole exists (none has ever been observed, but a wide class of so-called grand unified theories predicts they might), what must we conclude about every electric charge in the universe