Sunday, October 21, 2007

What is a magnetic monopole?

All the known magnetic fields in nature are produced by moving or fluctuating electric charges. When a voltage is applied to the ends of a wire, the free electrons flowing through the conductor produce a magnetic field around the wire. Similarly, individual atoms and molecules possess magnetic fields due to the quantum fluctuations of bound electrons. These magnetic fields can be aligned to produce an aggregate magnetic field in something like a bar magnet.

Whilst all known magnetic fields are produced by electric charges, the Maxwell equations for an electromagnetic field can be easily, and naturally modified to incorporate magnetic charges and magnetic currents. Moreover, even if one restricts attention to the standard Maxwell equations, there are solutions which describe a purely magnetic field. These solutions seem to require that space, and space-time, possess a non-trivial topology. Starting with Minkowski space-time, R4, it is necessary to remove a timelike curve to obtain a manifold with topology S2 × R2. The curve removed is considered to be the worldline which would be occupied by the magnetic charge.

In the gauge field formulation of electromagnetism, an electromagnetic field is specified by an object called a 'connection one-form' ω upon a U(1)-principal fibre bundle over space-time M. A magnetic monopole field is specified by a connection one-form ω upon a U(1)-principal fibre bundle over S2 × R2. To this connection there corresponds a 'curvature two-form', which induces an electromagnetic field strength tensor Fμν upon space-time M. The following integral

Sr 1/2π Fμν

over Sr, a sphere of radius r around the monopole, specifies the charge of the monopole.

Selecting different U(1)-principal fibre bundles over S2 × R2 enables one to define connections corresponding to different magnetic charges. If P is U(1)-principal fibre bundle corresponding to a magnetic charge of 1, then for any positive integer n, the cyclic group Zn enables one to construct another U(1)-principal fibre bundle Pn, over which P provides an n-fold cover via the n-fold covering map U(1) → U(1)/Zn. The bundle Pn can then be equipped with connections corresponding to magnetic charges of n.

In each case, 1/2π Fμν identifies the '1st Chern class' of the principal fibre bundle. This is an equivalence class of closed 2-forms on the space-time manifold. The Chern class selected is independent of the choice of connection; it is a topological invariant of the principal fibre bundle. The set of equivalence classes of closed 2-forms has the structure of a group, and is called the 2nd cohomology group of the space-time manifold M. Magnetic monopoles of different charges thereby correspond to different elements of the 2nd cohomology group, by virtue of the fact that they correspond to inequivalent U(1)-principal fibre bundles over space-time M.

Magnetic monopoles thereby require a space-time topology with non-trivial cohomology, and this rules out Minkowski space-time, R4. Note carefully, however, that whilst S2 × R2 has a non-trivial cohomology, it remains simply connected. The existence of magnetic monopoles does not, therefore, require space-time to have a non-simply connected topology.

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