courtesy SMBC |

Somehow, the refusal to address the complications of quantum and to just skate on by, has led to all sorts of weird mysticism stuff like quantum healing. Today most non-physicists have misunderstandings of entanglement and many-worlds and why Schrodinger hated cats so much. And very sadly, most physicists have no ability to correct them, as all they can do is draw squiggly tridents and funny S's and say "here is the answer". That's all we're

*taught*! "It's a mystery, no one knows so shut up and calculate!"
The result is that no one really knows anything. Physicists have a blackbox of expectation values and non-physicists have neat anecdotes for cocktail parties.

Maybe a year ago I wrote a post on this blog trying to derive a set of Newton-esque equations that, if enacted, would simulate a game of chess. Since the writing of that post I think I spent about four weeks locked away in my parent's lakehouse, reformulating it all in terms of much deeper principles of geometry and symmetry (it's such a cool idea and I wish I had time to finish it). Anyway, the original post is kind of neat and I'm pretty proud of it. It's the sort of thing I'd like to do more of.

An anonymous reader left a comment linking to a press-release of a theoretical study of lattice-spin on graphene. I looked up the original article, but here is the press release, which, you can see why it was posted.

It was a fascinating article, and I'm very glad to have read it. It isn't so much about chessboard-ness (though it may have implications I haven't considered). What it is about, however, is the nature of quantum mechanical spin.

courtesy Abstruse Goose |

Spin is one of those things that, for whatever reason, is over mystified. If you've ever read a book by Stephen Hawking or any other popular science author, you've heard the description of the book-belt thing.

Spin-1/2 particles require 720 degrees of rotation to return to their original orientation. I tell you this, maybe at a dinner party. You say "Surely, you're joking!" I don't feel like teaching you group theory and irreducible representations over dinner (or a blog post without LaTeX), so I give you the book-belt trick, showing how the same behavior can be exhibited even in everyday objects. Here it is:

You have a book. You close its cover on a belt. You hold on to one end of the belt. You twist the book through 360 degrees. The book is back where it started, but now there's a topological distortion in the belt (i.e. it's twisted). So you twist the book through another 360 degrees, and now the bend is gone. (Or not "gone" per se, but "goneable".)

It has relatively nothing to do with spin. It reveals relatively nothing about spin. It even mis-reveals spin, so that people may think spin is also a topological phenomenon, like the belt loop. All the book-belt thing shows is a normal object needing to rotate through 720 degrees to return to its original state. To Hawking's credit, he explicitly states his example has nothing to do with spin except by analogy, but not all authors are as forthcoming.

The graphene thing shows what spin is much, much more simply.

Here's what spin is: it's a degree of freedom.

Maybe that doesn't help much.

Hold out your arm, and point in some direction. Any direction; go ahead, hold your arm out and point.

When you picked your direction to point in, you had some considerations. You had to pick how far up you pointed; you might have pointed straight level ahead, or straight above, or slightly upwards, or at your neighbor's shoes, or wherever. That's a degree of freedom It is a freedom to pick orientation. In particular, it's a continuous degree of freedom, as you have the whole 180 degrees of up-down range to pick in.

You also had to pick how far to the left/right you pointed; straight right, off to your left, whatever. That's another degree of freedom; another continuous degree of freedom.

You only had two; this is because your arm is fixed length. If you could adjust your arm's length, then that's a third continuous degree of freedom. Since you can't stretch your arm (by much), then determining up/down and left/right also fixes forward/backward; however, forward/backward is still part of your pointing, just like up/down, left/right.

When you picked your direction to point in, you had some considerations. You had to pick how far up you pointed; you might have pointed straight level ahead, or straight above, or slightly upwards, or at your neighbor's shoes, or wherever. That's a degree of freedom It is a freedom to pick orientation. In particular, it's a continuous degree of freedom, as you have the whole 180 degrees of up-down range to pick in.

You also had to pick how far to the left/right you pointed; straight right, off to your left, whatever. That's another degree of freedom; another continuous degree of freedom.

You only had two; this is because your arm is fixed length. If you could adjust your arm's length, then that's a third continuous degree of freedom. Since you can't stretch your arm (by much), then determining up/down and left/right also fixes forward/backward; however, forward/backward is still part of your pointing, just like up/down, left/right.

Those degrees of freedom of where to point your arm are explicitly rotational degrees of freedom, and are directly related to angular momentum. If you move your arm from one place to another -- that is, change the degrees of freedom -- then you are applying angular momentum (technically torque, but bear with me). That's the connection, simply; in terms of algebra, it's deeper than that, but whatever.

If I take all the different ways you can move your arm from pointing in one direction to the next, I might call these a "group"; the group of ways to wave your arm around. Physicists call this group SO3. If I take your arm and my arm, the same group of movements describe how we might move from pointing one way to another. Or anyone's arm. Or even non-arms. Where you point pencils, or rulers, or antennae, are described by the group SO3.

An electron likewise has two degrees of freedom. They are likewise rotational degrees of freedom, meaning that spin corresponds to an angular momentum. The interesting thing about electrons is that their internal degrees of freedom do not rotate the same way as everything else.

These spin degrees of freedom come from wavefunctions. So, you know an electron can behave like a wave or a particle; the wave part of it is described by the wavefunction. Particles with zero spin only have one wavefunction. But electrons have two. At each location and time, an electron can choose which of its two wavefunctions it can follow; one wavefunction is spin up, the other is spin down. Or, because electrons are quantum things, it could be 45% spin up and 55% spin down. Or whatever. It has a degree of freedom as to which wavefunction to choose.

That's what gives an electron spin.

What is amazing is that this spin follows the same kind of transformation laws as vectors (the group of movements for our arms).

I know this sounds like a contradiction of what I said earlier, that electrons don't rotate the same way as everything else. They do and they don't.

The group of rotations we considered with our arms is a particular "representation" of SO3 (the group of rotations in three dimensions), that applies to objects with three components (up/left/forward). But spin is only two components (up/down); it transforms according to a different "representation" of SO3. When I say that it transforms according to SO3, I mean that rotating it (like your arm) causes its components (up/down) to get mixed up; yet because it isn't a true vector, they aren't mixed up the way most everything else is. And that's where it gets its weird properties.

I know this sounds like a contradiction of what I said earlier, that electrons don't rotate the same way as everything else. They do and they don't.

The group of rotations we considered with our arms is a particular "representation" of SO3 (the group of rotations in three dimensions), that applies to objects with three components (up/left/forward). But spin is only two components (up/down); it transforms according to a different "representation" of SO3. When I say that it transforms according to SO3, I mean that rotating it (like your arm) causes its components (up/down) to get mixed up; yet because it isn't a true vector, they aren't mixed up the way most everything else is. And that's where it gets its weird properties.

The example of the graphene spin demonstrates this wonderfully. There, they exhibit a kind of "pseudospin" that comes from a lattice degree of freedom. Graphene is a totally awesome material made of basically a single-layer hexagonal lattice of carbon atoms. It's electrical and heat conduction are phenomenal. It's material strength is beyond anything else in existence. It's great, they should make more of it, go graphene.

People who study graphene break it up in to unit cells, which are repeated throughout. For graphene, the cell is hexagonal. In this unit cell, there are two atoms, both carbon, located at sites A and B. The designation of A and B is arbitrary, but you have to distinguish them. People who study graphene also typically use a tight-binding approximation, assuming that electrons are located "around" one of the carbon atoms, probably with some harmonic potential. So an electron in the unit cell can be at an A carbon, or at a B carbon. Each of those gives it a different wave function.

Hence, breaking the graphene up in to different lattice sites gives the electrons an effective "spin", called "pseudospin", based on the degeneracy of which lattice site the electron occupies.

If the sites are very far away, then this difference between A and B is more pronounced and the spin disappears. But if they are close together, this difference produces pseudospin. The paper linked above basically demonstrates that this pseudospin is an actual angular momentum, just as spin is an actual angular momentum. From this, it suggests perhaps a discretization of space could account for the appearance of electron spin.

Whether discretization of space is or is not where electrons gain spin, electrons certainly do gain spin by having two possibilities for wavefunction at each location. And if you give them two more possibilities of wave functions (site A or site B) they get an additional pseudospin.

Anyway, I wanted to share that.

There are good explanations for quantum mechanical behavior, without recourse to mysticism, nor with throwing up our hands and saying "whatever, just calculate". I think this paper was a great way to illustrate that. For more on non-mystic, non-fuzzy understandings of quantum phenomena that are completely intuitive, free of paradoxes, and full of much more insightful mechanics, please read up on the Bohm-de Broglie interpretation, or Bohmian pilot wave mechanics, or just Bohmian mechanics. I've linked to several physics-undergrad-level papers in this post, or maybe find a more toned-down explanation.

Thanks to the anonymous commenter for sharing the paper, and giving me this chance to rant about quantum spin. Hope this had anything to do with your intentions when you linked me.

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