In the absence of charges Maxwell's equations of electromagnetics are skew symmetric under interchange of the Electric and Magnetic field.

div B = 0
div E = 0
curl B =  dE/dt
curl E = - dB/dt

Introducing a charge changes this as there are no observed sources for the Magnetic field. (hence div B = 0 ). We have sources for the electric field (electrons and other charged particles) so with sources:

div B = 0
div E = e
curl B = j + dE/dt
curl E = - dB/dt

If a source for the magnetic field were to exist then it would be a magnetic monopole and
the symmetry would be restored to the equations. Lots of people have thought that this would be a good idea. Paul Adrian Maurice Dirac proved that if even one magnetic monopole existed then electric charge has to be quantized! (That's an awesome thing to prove). To date no one has found a magnetic monopole. They are expected in some GUT's and their absence was explained by Alan Guth using inflation.

The magnetic monopole problem is that it really offends our artistic sensibilities that they don't seem to exist. Classically, Maxwell's Equations are completely symetric for electric fields and magnetic fields, except that the source term for the magnetic field is always zero.

In special relativity, we find that magnetic fields are just Lorenz Transformations of electric fields, and thus there is no compelling need for monopoles. If the speed of light were infinite, there wouldn't even be any magnetic fields.

String Theory and other grand unified theories again have a beautiful mathematical symmetry showing the possibility of existence of magnetic monopoles. Also, theories of dark matter and other cosmological theories involve magnetic monopoles.

In short, magnetic monpoles are beautiful and elegant, but don't seem to exist. Mother nature is a bitch.

Well I dont know the exact argument used by Dirac, but by assuming quantization of angular momentum and the existence of a magnetic monopole it is possible to show that electric charge must be quantized

The essential argument is as follows. Consider a magnetic monopole and an electric charge. We know that the resultant crossed electric and magnetic fields have a linear momentum
p =epsilon0 * (E x B) and
and an angular momentum L = r x p

It turns out that the total angular momentum is proportional to the product of q(the magnitude of the charge) and g(the magnitude of the magnetic monopole). As we know that angular momentum comes in bundles of h/(2*pi) we can conclude that q must be quantized(If the product of q and g is quantized then q must be quantized).

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, ε0, and μ0):

∇ • E = ρ
∇ • B = 0
∇ x E + dB/dt = 0
∇ x B - dE/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 = ρE
∇ • B = ρM
∇ x E + dB/dt = JM
∇ x B - dE/dt = JE

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 = ρM,

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

ρM = 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/r2 (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 R3 - {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/r2 (in the radial direction).

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

Ar = 0, Aθ = 0, Aφ = 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 R3 - {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)(∇ - eA/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 R3 - {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, S2. 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 S2 with just one coordinate patch. The offending singularity will, in fact, go away if we appropriately cover S2 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:
U1 = {0 <= θ < π - &epsilon},
U2 = {ε < θ <= π},

for some small ε. Note that these two patches cover all of R3 - {0}. We now look at our solution:

Aφ = g(a - cosθ)/(r sinθ).

In U1, choose Aφ(1) = g(1 - cosθ)/(r sinθ), which is nonsingular in all of U1.

In U2, choose Aφ(2) = g(-1 - cosθ)/(r sinθ), which is nonsingular in all of U2.

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' = eieΛ/hcψ = e2iegφ/hcψ

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 R3 - {0}.

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