Physicists have been hunting for one particle longer than perhaps any other.
It’s not the tachyon or some supersymmetric particle.
It’s the magnetic monopole - and of all the fantastical beasts of particle physics,
this is perhaps the most likely to actually exist.
So where are they all?
Let’s try an experiment.
Take a metal bar and force all the electrons to one end.
The electric field of the bar now looks like this - that’s a dipole field.
Now cut the bar in half and you get a pair of electric charges - one negative and one
positive, both of which have electric fields that radiate straight out.
Now take a metal bar and intsead magnetize it.
You get a dipole magnetic field that’s very similar to the dipole electric field.
So if we cut this bar in half surely we get a pair of magnetic charges similar to our
electric charges, right?
Wrong.
The ends of the split magnet still have north and south poles, and still generate a dipole
field.
And according to classical electromagnetism, it doesn’t matter how many times you slice
it - you’ll never get isolated magnetic charges - what we call magnetic monopoles.
This magnet-slicing experiment was first performed by French scholar and Beauxbatons Academy
Professor Petrus Peregrinus de Marincourt way back in 1269.
That was before we knew what caused the magnetism in magnets.
These days we know where magnetism comes from we’re not so surprised that a halved magnet
just makes two smaller magnets.
In a ferromagnet, the field is the sum of the countless tiny aligned dipole fields of
electrons in the magnet’s atoms.
The other popular way to make a dipole magnetic field is the electromagnet - where were push
electrons around in a circle
In both cases - electron spin or or a circular electric current there’s a sense of electric
charge in motion.
And according to classical electrodynamics, moving electric charge is the source of the
magnetic field.
If that’s true then, why should we even expect there to be isolated magnetic charges - magnetic
monopoles?
Well, according the classical theory we shouldn’t.
The non-existence of magnetic monopoles is codified in the mathematics of electrodynamics.
In particular, Gauss’s law for magnetism, one of the four Maxwell’s equations.
It states that the divergence of a magnetic field is zero.
The divergence is just this mathy term for the amount that a field points inward toward
a sink or outward toward a source.
Zero divergence means no source and no sink.
Magnetic field lines can form loops or head out toward infinity, but they never end.
According to this law there are no magnetic monopoles.
On the other hand, Gauss’ law for electric fields tells us that the divergence of the
electric field is not zero - it’s equal to the electric charge density.
That charge density is where the electric field lines can end - it forms their source or their sink.
So there are such things as isolated electric charges.
Let’s take a quick gander at Maxwell’s equations.
This is them without any charges - electric or magnetic.
E is the electric field and B is the magnetic field.
There’s a near perfect symmetry between electricity and magnetism which only gets
screwed up when you put in the electric charge - here in the form of charge density and current
density.
You could also have symmetry between these equations if there was such a thing as magnetic
charge.
If you add magnetic charges to these equations then you get a magnetic force that looks exactly
like the electrostatic force.
The physicist Murray Gell-Mann said that "Everything not forbidden is compulsory."
meaning that if the math of our physical theory allows it, then it exists in nature.
There’s nothing in Maxwell’s equations that really says magnetic monopoles can’t
exist except for the fact that James Clark Maxwell set the magnetic charge to zero because
he didn’t believe it existed.
But in principle it could exist, and so could magnetic monopoles.
At least according to classical theory.
But what about quantum mechanics?
When quantum theory first appeared it quickly revolutionized our understanding of electromagnetism
by explaining it in terms of quantum fields rather than charges and forces.
We talked previously how electromagnetism arose automatically from requiring that the
equations of quantum mechanics had a particular symmetry - the measurements they predict are
unaltered by changes in one simple property - the phase of the wavefunction.
Electromagnetism pops into the equations as soon as we require this - but in that version
of electromagnetism, the electric and magnetic fields are VERY different from each other,
and not at all interchangeable as they are in Maxwell’s equations.
In particular, the magnetic field emerging from the quantum theory must have zero divergence
- its field lines can never end - so it can’t have its own charge, unlike the electric field.
So perhaps here we have our reason for the apparent non-existence of magnetic monopoles.
Quantum mechanics, as the saying goes, forbids it.
Well, not so fast.
Don’t underestimate the power of the obsessed physicist.
The great Paul Dirac had a habit of discovering particles just by staring at the math.
In 1928 he predicted the existence of antimatter this way, as we’ve discussed in a previous
episode.
But then in 1931, just before his antimatter thing was verified, Dirac made another prediction
- of the existence of magnetic monopoles.
His argument goes something like this.
If you start with a dipole magnetic field, you can approximate a monopole by moving the
ends far enough apart and somehow vanishing the connecting field lines.
And there is a way to do that.
If you build a solenoid - just a coil carrying an electric current - you get a dipole field
whose connecting field lines are constrained within the coil.
So make the width of the coil much smaller than the length, and it looks like two isolated
magnetic charges.
This construction is called the “Dirac string”, and Dirac’s argument is that if the string
part of the Dirac string is fundamentally undetectable, then magnetic monopoles can
exist.
The second part of the argument is under what conditions that string is undetectable.
So magnetic fields affect charged particles.
In quantum mechanics, this works by shifting the phase of the particle’s wavefunction.
Imagine a charged particle - say an electron - passing by a Dirac string.
To plot that trajectory you add up all possible paths of the electron, including paths to
the left and to the right at the string.
The presence of the string, with its magnetic fields, should introduce different phase shifts
depending on which side of the string the electron passes - and that would actually
have a noticeable effect on the path of the electron.
In other words, the string would be detectable.
But there’s one scenario where the string can never be detected.
The amount of the phase shift is proportional to the electric charge.
For the right value of that charge, the phase shift induced between the different sides
of the string is exactly one wave cycle - which means no observable difference.
So for the Dirac string to be undetectable then electric charge can only exist in integer
multiples of that basic charge
This is a very loose form of the argument - and you can get to it in different ways.
But the upshot is that the string connecting monopoles is fundamentally unobservable, and
Dirac argued that this makes it a mathematical figment, kind of like virtual particles.
Reality should only be assigned to the monopoles themselves.
On the one hand this was taken as a prediction of the quantization of electric charge - electric
charge has to be discrete if there’s even a single magnetic monopole in the entire universe.
And of course we know that electric charge really is quantized - it can only be integer
multiples of the charge of the electron.
Or maybe of quarks - a third the electron charge.
But instead of taking this as a prediction of charge quantization, you can also flip
it: magnetic monopoles are possible if electric charge is quantized.
Charge turns out to be quantized, so quantum mechanics doesn’t actually forbid monopoles.
Let’s fast forward 40 more years.
In the early 70s physicists had managed to explain the weak nuclear force and unify it
with electromagnetism.
We talked about that before - about how the breaking of the symmetry of the Higgs field
separated the weak and electromagnetic forces.
With that squared away, physicists were working to bring the strong nuclear force into the
fold with the so-called Grand Unified Theories.
These involved slightly more complicated symmetry breakings.
Well it turns out that magnetic monopoles are inevitable in all “GUTs”.
Let me try to give you a sense of why - and we have to talk about the Higgs field to do
this.
We’ve done a couple of Higgs episodes before if you want to get all the nitty gritty.
But in electroweak theory, the Higgs field is a scalar field - it takes on a numerical value
everywhere in the universe, but with no direction - it’s not a vector.
In fact it really takes on two complex values everywhere, and the interplay of those two
“degrees of freedom” gives the Higgs its power.
In the simplest grand unified theory, the Higgs field has three degrees of freedom instead
of two.
That means the field can sort of act like a vector, even though it really isn’t one.
It can have a little internal arrow that points in a particular direction - not pointing in
physical space, but in the space of those 3 degrees of freedom.
Now the laws of physics shouldn’t care about the relative internal values of the Higgs
field - what matters is the absolute length of that internal vector - not the direction
it’s pointing.
There should be no noticeable effect even if the direction of the Higgs field changes
smoothly across space.
Except in one very special case.
If the direction of the Higgs field varies smoothly from one point to the next, it can
still have these sorts of knots - places where the field arrows all point away from that
point - in what was called a hedgehog configuration.
These are topological discontinuities - points that can’t be removed by a smooth defomation
of space.
And it turns out these knots in the Higgs field in GUT theories behave as massive particles
with magnetic charge - magnetic monopoles.
The hedgehog solution was figured out in 1974, simultaneously by Gerard t’Hooft and Alexander
Polyakov.
It turns out that GUT theories generically predict these magnetic monopoles, and that
they should be A) very massive, and B) should form spontaneously in extremely high-energy
environments like in the very early universe.
The fact that magnetic monopoles should exist in these theories was both exciting and problematic.
GUTs predict that monopoles should be produced in enormous numbers in the very early universe
- as abundantly as protons and electrons.
So where are they all?
They should also be very massive - quadrillions of times the mass of the proton - and so should
have quickly recollapsed the universe.
This conflict with reality might have ruled out both monopoles and the grand unified theories that predict them.
Except that both of these ideas are saved by yet another speculative idea - cosmic inflation.
Many physicists think that a period of prodigious exponential growth kicked off the expansion
of our universe.
This should have happened after the production of magnetic monopoles, and so should have
thrown these things far apart that there may be very few remaining in our entire observable
universe.
That would be a bummer for our hopes of detecting these things.
But that hasn’t stopped physicists from trying.
There are various approaches.
If you have one of these magnetic monopoles in your lab it wouldn’t be too hard to spot
- for example a monopole would excite an electric current if passed through a conducting coil.
Back in 1982, physicist Blas Cabrera Navarro set up a superconducting coil in his Stanford
lab and managed to detect what looked like a monopole with the same charge predicted
by Paul Dirac.
But no one ever saw such a thing ever again.
That includes at the Large Hadron Collider, where a couple of different experiments have
failed to spot monopoles created in the collider.
That's not so surprising given that the LHC reaches energies about 100 billion times
lower than is needed to produce the monopoles predicted by grand unified theories.
People also look for magnetic monopoles coming from space - typically using cosmic ray observatories
- or contributing to the Earth’s magnetic field - and in a number of other ways.
But again, no convincing evidence as of yet.
We have been hunting for magnetic monopoles for longer than just about any particle.
Their discovery would mean cracking open the grand unified theories and revealing mysteries
far beyond.
And so many of us remain obsessed with this elusive beast, and convinced of its inevitability
according to the symmetries of space time.
If you've joined us on Patreon, I can't emphasize enough how helpful your support is in keeping
this show going.
You don't have to do it - you could keep watching for free anyway - but that makes it all the
more awesome of you.
Today I want to give a huge shoutout to our Big Bang supporter Kyle Bulloch.
Kyle, we are taught by Paul Dirac that if there's even a single magnetic monopole in
the entire universe then electric charge must be quantized.
We don't know where that single magnetic monopole would be, but it's rare and special and so
we decided to name it after you.
So now Kyle, the magnetic monopole Kyle - is doubly important - sure, it ensures the quantization
of all charge, but it also serves as a testament to the generosity of Kyle.
Of Kyle the human.
Kyle, you have our thanks and now you also have a monopole.
Last week we talked about how quantum spin leads to the universe as we know it - for
example all the structure of solids, via the Pauli exclusion principle.
4fmagnet points out a potentially misleading point in that episdoe.
We represented very distinctly separated electron energy levels in our explanation of how fermion’s can’t occupy
identical quantum states.
4fmagnet notes that energy levels in atoms can actually hold 2 electrons, not one, because
it’s possible to have two electrons at the same energy but different quantum states by
allowing for opposite spin orientations.
That's a fair call - if I was being more careful I would have said that we were trying to represent
separate quantum states, not energy levels in an atom.
David points out that we don’t need to keep talking about spin as this incomprehensible
quantum property that has no intuitive analogy - reminding us that Hans Ohanian, in his book
“what is spin?”
showed that spin can be described as a circular charge current in the Dirac field.
We actually did mention this result in our episode on what is spin.
And it may be that people should talk about this result more - Ohanian’s calculation is not
generally accepted as “true”, so I’m not sure we can be so confident in it,
although as far as I know no-one has actually disputed the math.
Still, it would be nice to have a physical picture of spin that doesn’t involve me
taking off my belt.
Flosi Lyons asks what happens when the degeneracy pressure - the structural support against
collapse produced by the Pauli exclusion principle - is overcome.
For example, if a collapsing stellar core is massive enough it smashes through the degeneracy
pressure that supports white dwarf stars and instead produces a black hole or neutron star.
So the answer is that the Pauli exclusion principle is never violated.
Fermions never occupy the same quantum states.
When degeneracy pressure is broken it’s because new space in quantum states opens
up.
In the case of the collapsing star, densities and energies become high enough for electrons
to be captured by protons, converting them to neutrons.
So the loss of electrons reduces the degeneracy pressure, allowing gravitational collapse
to continue.
Tom Rostrom asks whether any of this spin statistics stuff explains why USB plugs need
to be turned 540° to return to the correct orientation?
Fantastic question, Tim.
The answer is no.
USB plugs have spin 2/3s - a 360 degree rotation gets you 2/3 of the way around.
They’re neither fermions nor bosons, so way outside the standard model.
Also, remember that you can undo fermion rotations by translation of the ends - for USB cables
its the opposite- twists approach infinity as you move the ends around, especially in
the presence of other USB cables.
Physicists hope to understand these things someday, perhaps with twistor or string theory.
Or maybe you could just use USBC - they’re spin-2 bosons like gravitons, so easily understood with
a basic theory of quantum gravity.