The

Dirac Magnetic Monopole is an example in

physics where it is crucial to understand the

topological structure of the

space (

manifold) on which you're working.

To give a brief description of what a magnetic monopole is (or would be), it helps to know a little bit about electric and magnetic fields. Very roughly, electric fields arise from the presence of point charges (like electrons or protons), and exert forces on other point charges. Magnetic fields arise from the *motion* of these same charges, and similarly only affect moving charges. For this reason, it is possible to think of magnetism as a purely relativistic effect; it is ultimately caused by the motion of the reference frame of the particle on which it's exerted. Thus, there are electric and magnetic fields, but only electric charges. There are no magnetic charges, as far as we know. However, there is nothing in the universe which bars the possibility of the existence of such a charge, and as we shall see, it leads to some nice symmetries, and finally to some incredible conclusions.

The reader is expected to be familiar with electromagnetism, and a very small amount of quantum mechanics. In addition, it will be helpful to have read the manifold node to get a true feeling for the power that topology can have in physics.

The essential physics comes from Maxwell's Equations for electromagnetism (I have suppressed factors of c, ε_{0}, and μ_{0}):

∇ • **E** = ρ

∇ • **B** = 0

∇ x **E** + d**B**/dt = 0

∇ x **B** - d**E**/dt = **J**

Where ρ and **J** are electric charge and current distributions, respectively. When Paul Dirac looked at these equations, he noticed an incredible amount of potential symmetry in this theory, particularly if we were to set ρ = **J** = 0. Then a transformation of (**E**,**B**) → (-**B**,**E**), for example, is a symmetry of the theory. The electric and magnetic fields suddenly seem very interchangeable. Unfortunately, ρ and **J** are not always zero. However, this potential symmetry was enough to convince Dirac to explore the possibilities.

Motivated by this observation, Dirac studied Maxwell's Equations augmented to include a magnetic charge and current distribution:

∇ • **E** = ρ_{E}

∇ • **B** = ρ_{M}

∇ x **E** + d**B**/dt = **J**_{M}

∇ x **B** - d**E**/dt = **J**_{E}

Now the symmetry is present in our theory. Does it describe the real world?

Let's follow Dirac's lead and study the new equation,

∇ • **B** = ρ_{M},

in the simplest case; a single magnetic monopole sitting at the origin (a direct analog of a single electric charge, like an electron).

ρ_{M} = gδ^{(3)}(r), where g is the monopole's magnetic charge, and δ is the Dirac delta function.

Using the spherical symmetry of the problem, it is easy to show that the magnetic field everywhere is the same as an electric field would be for the analogous electric charge:

**B** = g/r^{2} (in the radial direction)

In ordinary electromagnetism, we would introduce an auxiliary variable, the magnetic vector potential, **A**, with the relationship:

**B** = ∇ x **A**

Why do we normally do this? Because in the original Maxwell Equations, **B** had zero divergence (∇ • **B** = 0), which means it could be written as a curl, at least locally. Physically, **A** is not just useful for calculations, but also plays a direct role in the quantum mechanical description of charged particles.

Notice, though, that in our case, ∇ • **B** *does* equal zero on R^{3} - {0}, all of space minus the origin (and we don't really worry about the magnetic field at the origin anyway), meaning we can still write **B** = ∇ x **A**!

So, let's find an appropriate **A** so that ∇ x **A** = **B** = g/r^{2} (in the radial direction).

Sparing you the details of solving this equation, we end up with the following (in spherical coordinates):

A_{r} = 0, A_{θ} = 0, A_{φ} = g(a - cosθ)/(r sinθ), where a is some integration constant.

This equation for **A** blows up at θ = 0 and θ = π. Thus, for general values of the integration constant a, our vector potential is singular along the z-axis. For special values of a, we can do somewhat better. If we choose a = 1, then **A** is well-behaved at θ = 0, since the numerator vanishes faster than the denominator. This will give us a singular vector potential on just the negative z-axis. If we choose a = -1, we find that **A** is singular on the positive z-axis.

The bottom line is, the magnetic vector potential must blow up somewhere on R^{3} - {0}.

This is no good. One might argue "Yeah, it sucks that it blows up, but its curl is well-behaved, and that's all that matters, right?" Not really. In quantum mechanics, Schrodinger's Equation for a charged particle in a magnetic field looks like:

((1/2m)(∇ - e**A**/c)^{2}) ψ = ih dψ/dt

Meaning **A** appears directly in this equation. If **A** blows up, this equation makes no sense. And we want this equation to make sense, believe me.

At this point, Dirac tried to play a complicated mathematical game of introducing "Dirac String Singularities" along which the wave function vanished, making the equation somewhat sensible again. This doesn't get to the heart of the issue. Allow me to get to the heart of the issue.

We're working on R^{3} - {0}. To help our understanding of the space, we consider any fixed r, and notice that this looks like we're working on the sphere, S^{2}. We've been trying to cover the entire space with a function, and find that we can't make it look sensible everywhere.

I claim that our problems are rooted in the fact that you can't cover S^{2} with just one coordinate patch. The offending singularity will, in fact, go away if we appropriately cover S^{2} with two patches, instead of incorrectly trying to use just one patch. Thus, our "singularity" in **A** is really what is known as a coordinate singularity, which results from trying to use an inappropriate set of coordinates.

We choose two patches:

U_{1} = {0 <= θ < π - &epsilon},

U_{2} = {ε < θ <= π},

for some small ε. Note that these two patches cover all of R^{3} - {0}. We now look at our solution:

A_{φ} = g(a - cosθ)/(r sinθ).

In U_{1}, choose A_{φ}^{(1)} = g(1 - cosθ)/(r sinθ), which is nonsingular in all of U_{1}.

In U_{2}, choose A_{φ}^{(2)} = g(-1 - cosθ)/(r sinθ), which is nonsingular in all of U_{2}.

In the overlap region between the two patches, the two solutions must be physically identical, and hence agree. However,

**A**^{(1)} - **A**^{(2)} = 2g/(r sinθ) (in the φ-direction).

These two vector potentials differ from each other by what is known as a gauge transformation. On the surface, there is nothing too physically significant to this transformation, but let's look at it more carefully. A gauge transformation of the vector potential can always be written **A'** = **A** + ∇Λ, where Λ is an arbitrary scalar function. In our case,

Λ = 2gφ.

Thus, the difference in our vector potentials is definitely a gauge transformation. However, now we have a Λ which is not single-valued! As φ goes from zero to 2π, Λ increases by 2g. The physical significance of this shows up in quantum mechanics. In order to make Schrodinger's Equation consistent, ψ must also undergo a phase transformation:

ψ → ψ' = e^{ieΛ/hc}ψ = e^{2iegφ/hc}ψ

Thus, the phase of ψ is not single-valued as φ goes from zero to 2π. This will not do. Although the actual phase of the wavefunction does not have direct physical significance, if we want to calculate the interference between two waveforms (which is indeed physically significant!), the answer will be ambiguous. Thus, we want this to be single-valued. It will be single-valued if the phase returns to 2π at the same time as φ. In other words, we resolve this problem by requiring:

2eg(2π)/hc = 2πn, or

g = (hc/2e)n, where n is an integer.

This can be thought of as an equation for quantizing g, but a far more interesting way to think of it is as an equation for quantizing e:

e = (hc/2g)n.

Do you realize what we've just done? We started by noticing an interesting symmetry of Maxwell's theory, assuming we introduce the concept of a magnetic monopole. We carried through the calculations, and after resolving some topological issues we found that the physics only makes sense if the electric charge is quantized in units proportional to 1/g, where g is the magnetic charge! All that is required is that there exists a single magnetic monopole somewhere in the universe, and electric charge everywhere becomes quantized. This incredible fact comes directly from the topology of R^{3} - {0}.