1.1 Electric and magnetic charges

Consider a $U\left(1\right)$ gauge field, described by the gauge potential $A=A_{\mu }d{x}^{\mu }$ and the field strength $F=\frac{1}{2}{F}_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$, where $F_{\mu \nu }={\partial }_{[\mu }{A}_{\nu }$. This is invariant under the gauge transformation

where $g$ is a map from the spacetime to complex numbers with absolute value one, $\left|g\right|=1$. We can write $g=e^{i\chi }$ with a real function $\chi$, and we then have a more familiar

but it will be important for us that $\chi$ can be multi-valued, so that we identify

Consider a field $\varphi$, with the gauge transformation given by

We require here that $g$ specifies the transformations of all fields in the system uniquely. Then $n$ needs to be an integer; fractional powers are not uniquely defined.

The covariant derivative given by

and the kinetic term $|D_{\mu }\varphi {|}^2$ is gauge-invariant. We write the action of the gauge field as

The coefficient $2$ in the denominator is slightly unconventional, but this choice removes various annoying factors later. Then the force between two particles obtained by quantizing the field $\varphi$ is proportional to $e^2{n}^2$. In phenomenological literature the combination $en$ is often called the electric charge, but in this lecture note we call the integer $n$ the electric charge. It might also be tempting to rescale $F$ to eliminate the factor of $e^2$ from the denominator above. But we stick to the convention that the periodicity of $\chi$ is $2\pi$, see (1.1.3).

An electric particle with charge $n$ in the first quantized setup, Wick-rotated to the Euclidean signature, couples to the gauge field via

where $L$ is the worldline. The integrality of $n$ in this approach can be seen as follows. Due to the periodicity of $\chi$ (1.1.3), the line integral $\int \; <!--nolimits-->d{x}^{\mu }{A}_{\mu }$ is determined only up to an addition of an integral multiple of $2\pi$. Inside the path integral, $e^{iS}$ needs to be well defined. Then $n$ needs to be an integer.

Adding (1.1.6) and (1.1.7) and writing down the equation of motion for $A_{\mu }$, we see that

where $E_i=F_{0i}$ are the electric field components, $S^2$ is the sphere at infinity,

where

is the dual field strength. We also use the notation $\tilde{F}=\star F$ interchangeably.

Next, consider a space with the origin removed. Surround the origin by a sphere. The gauge fields $A_{N,S}$ on the northern and the southern hemispheres are related by gauge transformation:

on the equator. Then we have

where $m$ is an integer. We call $m$ the magnetic charge of the configuration. The energy contained in the Coulombic magnetic field diverges at the origin; but you should not worry too much about it, as the quantized electric particle also has a Coulombic electric field whose energy diverges. They are both rendered finite by renormalization. When $m$ is nonzero, the configuration is called a magnetic monopole. Usually we simply call it a monopole.

Put a particle with electric charge $n$, and another particle with magnetic charge $m$ on two separate points. The combined electric and magnetic field generate an angular momentum around the axis connecting two points via their Poynting vector, see Fig. 1.1. A careful computation shows that the total angular momentum contained in the electromagnetic field is $\hslash nm∕2$, which is consistent with the quantum-mechanical quantization of the angular momentum.

More generally, we can consider dyons, which are particles with both electric and magnetic charges. If we have a particle with electric charge $n$ and magnetic charge $m$, and another particle with electric charge $n^{\prime }$ and magnetic charge $m^{\prime }$, the total angular momentum is $\hslash ∕2$ times

We call this combination the Dirac pairing of two sets of charges $\left(n,m\right)$ and $\left(n^{\prime },m^{\prime }\right)$.