In the normal language of quantum gravity, the spin connection is an insanely complicated bit of mathematics that is used to connect the spin (of electrons and suchlike) to gravity (described by curvature of spacetime). In the normal run of things, if you can’t talk about the spin connection, then nobody will take you seriously in the quantum gravity world. Just to give you a bit of context, the spin connection is part of the subject of differential geometry. When I was a final-year student in Cambridge, I started going to a course of lectures on differential geometry, but gave up because I considered it to be too hard. And when I was a professor in QMUL, and I tried to talk to another professor in QMUL about quantum gravity (about which they knew a great deal) the conversation ended rather quickly when I couldn’t talk the talk about spin connections.
But now I find myself in an Aladdin’s Cave of treasures, and I swept with the magic Broom and found the spin connection staring me in the face. To be fair, I did already know more or less what the spin connection is, and what it does and how it works, but now I have a complete description, that unifies all the forces, and quantises them all, so I know exactly what the spin connection is, exactly what it does and exactly how it works. So now I can explain it to an 8-year-old. Are you ready for this?
Forget any idea you may have that mathematics is hard. It isn’t. You just have to look at it in the right way. The right way to think about the spin connection is to take a trip to the Lake District (any lake will do, as long as it’s nice and smooth and has lots of flat stones near it). Take a flat stone, spin it, and watch it skip over the surface of the lake. Now explain how this works, to your 8-year-old. That is how the spin connection works.
Did you see the spin flip as the stone bounced? To explain this, I first need to know whether you are right-handed or left-handed – because left-handed stones (or electrons) spin the opposite way to right-handed stones (or electrons). Anyway, let’s suppose that as you follow the path of the stone, from the perspective of the stone itself, it hits the water spinning clockwise. When it bounces, then from the perspective of the stone, it is sent travelling backwards, which means it is spinning anti-clockwise. That is a spin-flip, OK? Now look at the next bounce. Again, it hits the water spinning clockwise. WTF? How the hell did it do that? How did that spin “spontaneously” change from anti-clockwise to clockwise? What the hell is going on?
Those are the questions that physicists have been asking themselves for 100 years, and they still haven’t got any answers. But you know the answers, don’t you? I know the answer, and any other reasonably intelligent 12-year-old can explain it. It is because gravity affects the spin – when it is going up, it is spinning anti-clockwise, and when it is coming down it is spinning clockwise. So if you want to know anything at all about how electron spins work, you have to consider it in the context of gravity. If you don’t you won’t understand how the spins flip, and you certainly won’t understand why they don’t flip between a pair of entangled particles.
OK, so to explain entanglement, I need you to take a pair of 10-year-old twins to the lake, and make sure one of them is left-handed and the other right-handed. Now get them to skip stones together, so that their spins are opposite. Then every time you measure the spin (that is, when the stone interacts with the water), you will always measure the two to be opposite. But if you measure the spin vertically instead of horizontally, for example by putting a sticky tar-covered wall in the path of the stone, there’s a 50/50 chance of it coming up heads or tails. Of course, in this experiment, you can’t get everything exactly opposite, so the two 50/50 probabilities are independent. That’s what is called decoherence in quantum mechanics (I prefer to call it incoherence, but they consider that to be rude). But if you imagine that you can take away all the differences between the twins, all the air resistance, and everything else except gravity, you will find that the tides will determine whether they come out heads or tails, and because the tides don’t vary much on this scale, the two measurements will still be opposite to each other, almost all the time.
That is how entangled electrons work. It is also how entangled photons work. And it completely explains all of the experimental properties of measurements of entangled particles. If you take away all other interactions, so that you maintain coherence between the two particles, and you measure spins in the same direction, they will always be opposite. That is, if you measure both horizontally, or both vertically, they will always be opposite. For the horizontal measurement, you use up-down axis of gravity to determine the difference between clockwise and anti-clockwise spins. For the vertical measurements, you use the left-right axis instead, and you have to use the tides to tell you which way the electrons are falling (sideways). But it’s still gravity, so it’s still in the spin connection.
So, anyway, I still can’t talk the talk about spin connections. But those people who can talk the talk, can’t walk the walk. If they could, they’d have a theory of quantum gravity that works. What you have to judge is, can I walk the walk? Can I explain how gravity works at a quantum level?
I have shown you how an electron looks like a coin (via Lorentz contraction), spins like a coin, and flips like a coin. It is a well known principle that something that looks like a duck, swims like a duck and quacks like a duck, is a duck. This principle is a form of Occam’s Razor, which used to be widely used in physics. If an electron is a spinner, then there is no need to invent spinors to explain how the electron behaves. The spinor is dead, long live the spinner!