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- Path: sparky!uunet!olivea!mintaka.lcs.mit.edu!bloom-picayune.mit.edu!snorkelwacker.mit.edu!galois!riesz!jbaez
- From: jbaez@riesz.mit.edu (John C. Baez)
- Newsgroups: sci.physics
- Subject: Re: Did electric/magnetic symmetry "break"?
- Message-ID: <1992Sep7.214235.24769@galois.mit.edu>
- Date: 7 Sep 92 21:42:35 GMT
- References: <1992Sep6.185028.16384@galois.mit.edu> <1992Sep7.052822.26368@asl.dl.nec.com>
- Sender: news@galois.mit.edu
- Organization: MIT Department of Mathematics, Cambridge, MA
- Lines: 52
- Nntp-Posting-Host: riesz
-
- To repeat myself, I don't think there is any asymmetry between electric and
- magnetic charges of the form "magnetic charges (if they exist) must be
- 68.5 times as big as electric charges." This seeming asymmetry is
- just an artifact of a particular system of units.
-
- Recall Dirac's argument. If you move an electric charge counterclockwise
- around a loop through a magnetic field its wavefunction is multiplied by
- a phase
-
- exp(ieF/hbar)
-
- where e is its charge and F is the magnetic flux through the surface
- spanned by the loop.
-
- Now suppose our electric charge moves around on the surface of a
- sphere of unit radius enclosing a magnetic monopole of magnetic charge m.
- Then the magnetic field is normal to the surface of the sphere and of
- strength m. If we move our charge around a loop enclosing an area A,
- the flux enclosed is mA so the phase shift is
-
- exp(iemA/hbar)
-
- But, as I explained more carefully in my posts on braids (hint,
- hint! - they're free!), we can equally well think of this loop as going
- clockwise around an area 4pi - A. Here 4pi is the area of the sphere, and
- the point is that if you take a bike tour counterclockwise
- around the border of Russia you can equally well say you're going
- clockwise on a tour around the border of "all of the world
- that's NOT Russia". So we can alternatively compute the phase shift
- to be
-
- exp(iem(A - 4pi)/hbar)
-
- These two have to agree so we must have
-
- exp(4 pi iem/hbar) = 1
-
- or in other words, 2em/hbar must be a multiple of an integer. Note that
- this condition is PERFECTLY SYMMETRICAL BETWEEN MAGNETIC AND ELECTRIC
- CHARGES. Indeed, we could equally have ran the argument the other way
- and thought about moving a magnetic monopole around a electric charge.
-
- Okay, so I got that weird factor of 2 that's floating around. Where's
- the fine structure constant coming from? That must be an artifact of
- your units! The argument does NOT say that magnetic charges are
- bigger than electric ones! It says that the bigger the fundamental
- unit of electric charge is, the smaller the fundamental unit of
- magnetic charge is.
-
- In case anyone feels like flaming, yes, I know that nobody has seen a
- magnetic monopole, and no, I don't believe in them. This is a summary
- of some speculations by Dirac on WHAT IF there were magnetic monopoles.
-