From: merritt@macro.bu.edu (Sean Merritt)
Newsgroups: sci.physics
Subject: Re: Magnetic monopoles?
Date: 27 Jan 93 16:13:03 GMT
	<MERRITT.93Jan25150507@macro.bu.edu>
	<1993Jan27.055742.24399@galois.mit.edu>
Organization: Boston University Physics Department
In-reply-to: jbaez@riesz.mit.edu's message of 27 Jan 93 05:57:42 GMT


#In article <1993Jan27.055742.24399@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:


   In article <MERRITT.93Jan25150507@macro.bu.edu> merritt@macro.bu.edu (Sean Merritt) writes:
   >The precipitating factor for the new searches was more due to the fact
   >that theorist found that monopoles "appeared" in GUT's. I think it
   >was in 1974, (for example see G. t'Hooft, Nucl. Phys. B, 79, 276 (1974)) 
   >that the subject heated up again.

#   You're right.  

   >I still think that the fact that they would add symmetry to Maxwell's
   >equations is a valid reason for the searches.

#   Hmm, I don't.  First, while naive monopoles change dF = 0, *d*F = J to the
#   nicer-looking dF = K, *d*F = J (where F is the field strength, J the
#   electric current, and K is the magnetic current), this destroys the 
#   gauge symmetry of the equations!  Why trade a hefty infinite-dimensional
#   symmetry group for a measly U(1)?  (Of course, one could argue that the
#   infinite-dimensional group, being "gauge" symmetries, is nonphysical and
#   worth less than the puniest group of "physical" symmetries.  But gauge
#   theories have considerable charms.)  Second, and more importantly, in
#   the context of GUTs one is not actually toying with Maxwell's equations
#   by putting in magnetic currents as above: one is going to a wholly more
#   complicated theory in which the Higgs field determines which gauge field
#   counts as the "electromagnetic" one by means of spontaneous symmetry
#   breaking, and monopoles are due to regions of space that can't make up
#   their minds (so to speak).  


Yes, what you say about the Higgs is correct.

if we have a Gauge Group G such that:
  _
G _) SU(3) x SU(2) X U(1),

The fact that G is simple both implies that there is only a single
gauge coupling constant and the quantization of weak hypercharge
(and hence electric charge).

now a Higgs is responsible for the symmeytry breaking, the gauge group
G is broken to a subgroup H by some Higgs field phi. In any finite
energy the asymptotic value of phi in any direction must minimize the 
Higgs potential. There will be topologically non-trivial choices
of phi (phi = infinity,theta,phi) and thus monopole solutions
if the homotopy group is non-trivial pi2(G/H)(has more than one 
element).

Theorem: If G is compact semi-simple, and simple connected:

pi2(G/H) = pi1(H)

pi1(H) is the group whose elements are homotopy classes
of functions from the circle S(superscript1).

This is the theorem that implies if there is a Higgs
then there is a monopole.

pi2(G/H) = pi1 (U(1)e-m x SU(3)color) = Z.

All GUTs have monopoles.

Now you want to validate the GUT (and the Standard Model),

1) You build a machine that can reach the energies of the
   Higgs sector.

2) You look for monopoles which would be relics of the symmetry
   breaking.

3) You do both.

For the details see the above reference by t'Hooft and

H. Georgi and S.L. Glashow , Phys Rev. Lett. 32, 438 (1974)
J. Kim Phys Rev. D23 , 2706 (1981).



-sjm


--
Sean J. Merritt		         |"Road-kill has it's seasons just like     
Dept of Physics Boston University|anything, there's possums in the autumn  
merritt@macro.bu.edu             |and farm cats in the spring." T. Waits