Types of quantum field thories
The electromagnetic field is is a gauge field which unifies the electric field and the magnetic field. A configuration of the electromagnetic field on a space in the absence of magnetic charge is modeled by a cocycle in degree 2 ordinary differential cohomology.
This may be realized in particular equivalently by
In the presence of magnetic charge the electromagnetic field is modeled by a cocycle in differential twisted cohomology, where the twist is given by the differential 3-cocycle that models the magnetic current.
…historical section eventually goes here..
…electricity and magnetism were discovered independently, Maxwell's equations in classical vector analysis which allows the formulation as a tensor as below, and “magnetism is a consequence of electrostatics and covariance, hence the composite noun electromagnetism”
We describe how this identification arises from experimental input.
The input is two-fold
The Dirac charge quantization argument shows that in order for the electromagnetic field to serve as the background gauge field to which a charged quantum mechanical particle couples (for instance an electron), it must be true that this 2-form is the curvature 2-form of a circle bundle with connection.
We say this now in more detail.
In modern language, the insight of (Maxwell, 1865) is that locally, when physical spacetime is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional Minkowski space , the electric field and magnetic field combine into a differential 2-form
in and the electric charge density and current density combine to a differential 3-form
in such that the following two equations of differential forms are satisfied
where is the de Rham differential operator and the Hodge star operator. If we decompose into its components as before as
then in terms of these components the field equations – called Maxwell’s equations – read as follows.
magnetic Gauss law:
The first equation with the Poincare lemma implies that one may find
a collection of differential 1-forms, such that
and a collection of real valued functions on double overlaps such that
The forms are called a vector potential or the electromagnetic potential for the electromagnetic field.
Notice that it follows that on triple overlaps we have
which means that on that overlap the function
is constant. If one requires these constants all to be inside a discrete subgroup , then the data defines a degree 2-cocycle in Cech-Deligne cohomology on with coefficients in . Below we see that experiment demands that such a subgroup exists and is given by the additive group of integers.
Therefore the above data is subject to the additional constraint that it induces well-defined -valued holonomy – this is Dirac’s quantization condition, a necessary requirement for the existence of quantum mechanical particles on that are charged under the background electromagnetic field.
Concretely: for any smooth curve and any cover of refining the pullback of the cover to , and for every triangulation of subordinate to , i.e. such that there is an index map such that and
(where if is the final vertex of and otherwise)
has to be a well defined element in (independent of all the choices made).
This implies in particular that cancelling from the triangulation an edge of vanishing length must have no effect on the formula, which in turn means that for all we have
In short: the holonomy of the constant path on a point must be , but if that path sits in a triple intersection then the holonomy is equivalently given as the exponentiated sum of the three transition functions. This forces the sum to land in .
In total this says precisely that the data
Dirac originally presented the following reasoning, which captures the main point of the above considerations.
He considered to be without the origin,
He imagined a situation with a magnetic charge supported on the point located at the origin and removed that point in order to keep the field strength to be a closed 2-form on all of .
(Indeed, if one does not remove the support of magnetic charge, the argument becomes much more sophisticated and involves higher differential cocycles given by bundle gerbes. This was not understood before Dan Freed’s Dirac charge quantization and generalized differential cohomology.)
Then he considered a single coordinate patch
given by minus the right half of the first coordinate axis.
Traditionally physicist try to give that half-line a physical interpretation by imagining that it is the body of an idealized infinitely-thin and to one side infinitely-long solenoid. Indeed, such a solenoid would have a magnetic monopole charge on each of its ends, so if the one end is imagined to have disappeared to infinity, then the other one is the magnetic charge that Dirac imagines to sit at the origin of our setup.
In this context the half-line is called a Dirac string. While there is the possibility to sensibly discuss the idea that this Dirac string actually models a physical entity like an idealized solenoid, its main purpose historically is to confuse physics students and keep them from understanding the theory of fiber bundles. Therefore here we shall refrain from talking about Dirac strings and consider as exactly what it is, by itself: an open subset that is part of a cover of . Unfortunately, of course, Dirac didn’t mention the other open subsets in that cover (at least one more is needed for a decent discussion), so that the Dirac string keeps haunting physicists.
…running out of time…just quickly now
…Dirac effectively considered the overlap cocycle condition , found that by the requirement that has well defined holonomy it follows that there must be a function with values in such that , then did away with the -patch (considering a kind of limit as we encircle the half axis) and concluded that must be the log-differential of a -valued function, whose winding number around the half-axis he identified with the magnetic charge, which in terms of bundles one identifies with the Chern-class of the bundle in question
…have to run
that is supported in some compact spatial region with boundary sphere .
For a closed but contractible trajectory of an electrically charged particle, the action functional is
by the Stokes theorem, for some 2-disk cobounding the circle. If now approaches a constant path and the 2-disk is taken to wrap the 2-cycle , then this becomes
Which implies that with the magnetic charge being quantized, also the electric charge is.
Maxwell's equations originate in
Dirac’s quantization argument appeared in
Discussions of the basic geometry behind Maxwell equations can be found in
For undergraduate lectures including experimental material see