Nobody has observed any fundamental magnetic monopoles. Nevertheless, theorists are hard at work to propose testable laws that describe the nature of a magnetic monopole.

## One Magnetic Monopole

$$ \nabla\cdot\mathbf{B} = q_b\delta(\mathbf{r}) $$

Dirac said we only need one magnetic monopole to describe charge quantization. So the simplest model is that there is one, at the center of the universe.

This is falsifiable: all you need is to go to the center of the universe and measure the magnetic field there.

## Vector Scalar

$$ \nabla\cdot\mathbf{B} = \frac{4\pi}{c}\mathbf{J} $$

Since currents create magnetic fields, its logical to believe that they should be the charge density of \(\mathbf{B}\). Some also call this field the “1.5 rank” tensor.

## Vector Scalar, Covariant Formulation

$$ \partial_\zeta \left(\frac{1}{2}\epsilon^{\zeta\xi\upsilon\nu}F_{\upsilon\nu}\right) = O^{\xi\mu} $$ Here we take the convention that \(\epsilon_0 = 4\pi = 1\) .

## Vector Scalar In Free Space

$$ \nabla_\mu\cdot^\mu\mathbf{B} = \mathbf{0} $$

There are no monopoles in free space. In addition, some people believe that the above formulations are “abuses of notation” due to confusion as to what is a vector, scalar, and tensor. Here the summation between the \(\nabla\) and the \(\cdot\) is explicit.

## Commutation

$$ \nabla\cdot\mathbf{B} = \mathbf{B}\cdot\nabla $$

In quantizing the magnetic field, many assume \([\nabla,\mathbf{B}] = 0\). In addition, some people also assume \([\nabla,\cdot] = [\cdot,\mathbf{B}] = 0\):

$$ \nabla\cdot\mathbf{B} = \nabla\mathbf{B}\cdot = \cdot\nabla\mathbf{B} $$

## Unit Scaling

$$ \nabla\cdot\mathbf{B} = \nabla\mathbf{B} $$

Here we use the unit convetion where \(\cdot = 1\). It’s really tiny when you write it so its contribution is negligible but non-zero.

## Quanitzed Monopoles

$$ \nabla\cdot\mathbf{B} = \frac{1}{\sqrt{2}}\left(|+\rangle \pm |-\rangle \right) $$

Magnetic monopoles must obey quantum mechanics. This is an example for one spin \(1/2\) particle.

## Background Monopoles

$$ \nabla\cdot\mathbf{B} = 4 $$

If there are magnetic monopoles at every location in space, then clearly we can’t measure them because all of our measurements are biased. Some people believe that there are 7 monopoles per cubic foot, and not 4 (using the unit convention \(4 = 7\)).