# Where did the idea that there *might* be magnetic monopoles come from? [duplicate]

I know that we declare that there exist no magnetic monopoles. I see it all the time in my E&M class. $$\vec{\nabla}\cdot\vec{B}=0$$ So there exists no point for $$\vec{B}$$ to originate from and no point for it to terminate at. We see that the monopole term in the multipole expansion: $$\vec{A}(\vec{r})=\frac{\mu_oI}{4\pi}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\oint(\vec{r}')^nP_n(cos(\theta'))d\vec{l}'$$ Is zero.

But I always hear my professors laughingly say unless you ask a string theorist? What makes some theorists predict the existance of magnetic monopoles? Feel free not do dumb it down if its an advanced topic (im sure it is). It can be fun to lean into ones own ignorance.

• In addition to Dirac's charge-quantization argument, there is also the simpler argument that Maxwell's Equations are more symmetric (and thus more beautiful) with magnetic charge. – G. Smith Dec 3 '18 at 4:08
• @G. Smith If there was a magnetic monopole we would have to append maxwells equations right? Similar to how $\vec{\nabla} \times \vec{E} \neq \vec{0}$ but is actually $\vec{\nabla} \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}$ its just that we usually talk about situations where $\frac{\partial\vec{B}}{\partial t}=\vec{0}$. Right? – Alex Sampson Dec 3 '18 at 4:15
• A new answer by DanielSank shows you what the equations would look like. I think they look much nicer. – G. Smith Dec 3 '18 at 4:19
• Nonabelian gauge theories with spontaneous symmetry breaking can have monopoles. See en.wikipedia.org/wiki/%27t_Hooft–Polyakov_monopole. I'm not sure whether this is the same as what Dan Yand was talking about. – G. Smith Dec 3 '18 at 4:22
• Possible duplicates: physics.stackexchange.com/q/4784/2451 and links therein. – Qmechanic Mar 9 '19 at 10:51

I can think of three reasons:

1. Maxwell's equations (in dimensions where the usual $$\epsilon_0$$ and $$\mu_0$$ constants don't show up) \begin{align} \nabla \cdot E &= \rho \\ \nabla \cdot B &= 0\\ \nabla \times E &= - \frac{\partial B}{\partial t} \\ \nabla \times B &= J + \frac{\partial E}{\partial t} \end{align} are tantalizingly asymmetric. It looks like they really ought to be \begin{align} \nabla \cdot E &= \rho_\text{electric} \\ \nabla \cdot B &= \color{red}{\rho_\text{magnetic}} \\ \nabla \times E &= \color{red}{J_\text{magnetic}} - \frac{\partial B}{\partial t} \\ \nabla \times B &= J_\text{electric} + \frac{\partial E}{\partial t} \, . \end{align}

2. Furthermore (as noted in another answer) it can be argued that the existence of magnetic monopole would "explain" the quantization of charge.

3. Finally, note that in fact our known theory of electromagnetism does work if we add in magnetic monopole terms (i.e. magnetic charge and current) as long as we also put in the requirement that the ratio of electric and magnetic charge is constant. That point is argued in the famous electrodynamics book by J. D. Jackson.

In a paper published in 1931 Paul Dirac showed that if magnetic monopolies exist then electric charge must be quantized.

• So magnetic monopoles must exist then? since charge is quantized (right?) also what paper was that? Was it "Quantised Singularities in the Elctromagnetic Field"? – Alex Sampson Dec 3 '18 at 3:02
• @AlexSampson Yes that is the paper – Lewis Miller Dec 3 '18 at 3:08
• Isn't that also his paper where he predicted the existence of the positron? – Alex Sampson Dec 3 '18 at 3:11
• @AlexSampson "If magnetic monopole exists, then electric charge is quantized." However, if electric charge is quantized, it does not necessarily imply magnetic monopole exists (at least logically). – K_inverse Dec 3 '18 at 3:18
• Dirac's equation was introduced in 1928 and that led to the prediction of the positron. – Lewis Miller Dec 3 '18 at 3:20

The other answers have given good accounts of the initial reasons why we first thought monopoles might exist. However, modern physicists who believe monopoles exist tend to use a different argument.

In 1981 't Hooft and Polakov showed that certain classes of gauge field theory have vacuum solutions that carry a magnetic charge. In plain language, this means that there are solutions to the equations that describe the theory that behave like magnetic monopoles.

The Standard Model, which is the gauge theory that fits best to experimental data so far, is not one of these theories. However, it can be shown that if the Standard Model is part of any higher gauge theory (a Grand Unified Theory), then this theory must admit monopole solutions. Furthermore, we expect that if this is true, the conditions shortly after the big bang would have produced monopoles in copious amounts.

This raises the question as to where all these monopoles are. Some theorists think that this is evidence that no Grand Unified Theory exists. However, it is also widely believed that cosmic inflation may have diluted the abundance of monopoles so there is practically zero chance of observing them in the present day.

There's a good review for non-experts here: https://physicstoday.scitation.org/doi/10.1063/PT.3.3328

And a more technical one here: http://puhep1.princeton.edu/~kirkmcd/examples/EP/goddard_rpp_41_1357_78.pdf