Why Magnetic Monopoles Should Exist (Script) (Patreon)

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 really exist. So where are they?

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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 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, 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.

B= 0

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.

E=0r

That charge density is where electric field lines can end - it forms their source and sink. There ARE such things as isolated electric charges.

Let’s take a quick gander at Maxwell’s equations, shall we? This is them without any charges - electric or magnetic. E is the electric field and B is the magnetic field.

E= 0,

E= -Bt,

B= 0,

B=Et.

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.

E= 0,

E= -Bt,

B= 0,

B=0j + 1c2Et.

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.

E = qr2 F=q(E + v B)

B = gr2 F=g(B - v E)

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. And 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’s 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 the 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. 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.

Phase() q

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

Phase() nq; n = 1,2,...

- or more accurately some basic electric times the basic magnetic charge. This is a 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.

q=ne; e = 1.60 10-19C; n = 1,2,3,...

q=n3e; e = 1.60 10-19C; n = 1,2,3,...

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 the nitty gritty. 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 could rule out both monopoles and grand unified theories that predict them. Except that both 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 them 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 got 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 conduction 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. Which isn’t 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 detections as of yet.

We have been looking for magnetic monopoles for longer than just about any particle. Their discovery could 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.