Dirac’s Belt Trick

Is classical physics boring? In his preface to Volume 1 of The Feynman Lectures on Physics, Richard Feynman worries about students’ enthusiasm:

… They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many would be very discouraged because there were really very few grand, new, modern ideas presented to them. They were made to study inclined planes, electrostatics, and so forth, and after two years it was quite stultifying.

Inclined planes, electrostatics, and so forth might also refer to the theory of the rotations of a rigid body. Mathematically challenging – Euler angles and all – but not very exciting and ‘modern’, compared to the weird behavior of particles in quantum physics. But Dirac’s belt trick shows that (classical) rotations are indeed ‘weird’, and you do not need to invoke quantum objects to feel the weirdness. Paul Dirac was one of the founders of quantum mechanics. Reading his Principles of Quantum Mechanics, it may come as a surprise that a ‘belt trick’ is named after him. But Dirac was originally trained as an engineer, and maybe more hands-on than we think.

There are many versions of this trick, using belts, cups, or plates. In any case, something is rotated and you keep track of how often you rotate it. All rotations by multiples of 360° (2π) are equivalent, aren’t they? The belt trick shows, that they are not. As with some abstract mathematical objects used to describe particles in quantum mechanics, you may need two full rotations to really restore your original state.

One end of the belt remains fixed – e.g. by some weight as I sketch below. The other end is twisted – turned around to its original position once or twice. Roger Penrose shows this trick in his book The Road to Reality (without attributing it to anybody). He uses the free end like a bookmark in a book – so that the book makes it easier to count the rotations. I marked one side of the belt with a letter.

If you rotate the end of the belt by 2π, the twist of the belt cannot be undone – unless you rotate it back. Of course, it cannot be undone by a pure translation. But for a 2π rotation this is possible:

You move the end of belt over the belt, without rotating it. Then the belt has two twists, but in the opposite sense, so the belt can be flattened without having to rotate the end. As an intermediate step I enlarge the twist next to the free end, while tightening the other. The large ‘twist’ now looks like the looping of a roller coaster. The take-off ramp – the free end – resides to the right of the looping. I lift the take-off ramp up, then move it sideways, to the left, then lower it to the same height as before. Now the take-off ramp resides to the left of the looping.

This trick is not just a vague metaphor for ‘quantum behavior’, but classical rotations really distinguish between 2π and 4π. Dirac said about his experiment: [*]

I used it to illustrate a property of rotations, that two rotations of a body about an axis can be continuously deformed, through a set of motions which each end up with the original position, into no motion at all.

Penrose provides the underlying reason [#]: All rotations – all positions of a rigid body after a rotation – can be represented by a peculiar ball with radius π. Each point in the ball represents a rotation: The axis of rotation is parallel to the line connecting the point to the origin. The distance of the point from the origin indicates the angle of rotation. The zero point means zero rotation. Moving from the center to the surface of the ball, the angle increases until it reaches its maximum of π. Moving in the other direction is equivalent with rotating more and more in the other sense, until the angle reaches -π. But π is the same as -π: If you reach the sphere (the boundary of a ball is a sphere) you are suddenly teleported through a wormhole and end up at the antipodal point. Each point on this sphere is connected with its antipodal point which makes this topology different from an ordinary sphere in Euclidean space.

A topology is sort of the minimum structure you need to speak about points being in a neighborhood of other points – it is a set of open sets of things that are not ‘sorted’ or ‘related’ in any other way. A topology does not include a notion of distance: The iconic example is a coffee mug being isomorphic to a torus and anything without a handle or hole being equivalent to a sphere. A torus and a sphere are fundamentally different because any loop you can draw on a sphere can be shrunk continuously until it becomes a point. You can also draw such shrinkable loops on a torus – as long as you do not embrace the torus by this ‘yarn’. But by feeding the yarn through the ‘hole’ in the middle of the torus you create a loop that cannot be shrunk.

Dirac said that rotations of a body about an axis can be continuously deformed into no motion at all. This can be shown by considering loops within traversing the ball that represents the topology of rotations: A rotation by 2π is a loop that starts in the center of the ball – the body is in original position, then you move away from the center – increasing the angle of rotation. On the surface (π) you need to use the wormhole connection and continue the path through the ball on the antipodal point. The loop has to end at the origin – the original position of the body. This point is fixed, and you need to touch the surface of the ball twice – these are constraints like the hole of the torus; the loop can never shrink.

A loop for a 4π rotation is different – this sketch compares the 2π loop (first image to the left), to a 4π loop and its deformation (other three images). Antipodal points are marked with black and white playing pieces.

When the surface is reached (2 – white round head), we wormhole again to the antipodal point (3-  black round head). From there we move / rotate by another (about) π until we hit the surface again at (4 – white square head). We can arrange this part of the path so that the initial – unshrunk – loop passes through the origin, but this point is not fixed in the way the final destination is. From (4) we reach its antipodal point (5 – black square head) and only then we loop back to the fixed origin. The four points on the surface can move, as long as the antipodal relations are maintained. The right part of the loop inside the ball is free to move up to the surface. Then it can jump the left side – through the wormhole – and form one loop without any wormhole connections with the fixed part of the loop. This remaining loop can now be shrunk to a point.


[*] In this discussion https://www.physicsoverflow.org/41945/did-dirac-ever-publish-anything-about-his-string-trick a reader quotes from Riddles of the sphinx, and other mathematical puzzle tales by Martin Gardner who had contacted Dirac. Dirac sent a brief response to Gardner. Other than that, Dirac did not mention the trick in any of his scientific papers it seems.

[#] Here is a comparable explanation of the topology of rotations: https://www.damtp.cam.ac.uk/user/examples/D18S.pdf

3 Comments Add yours

  1. Great content :) Thank you

  2. Peter Mander says:

    Your excellent drawings of the belt with two clockwise twists +1 +1 clearly indicate that the manoeuvre of lifting the end of the belt over the axis simply reverses the direction of the second twist, so +1 -1 = 0. There doesn’t seem to be anything mysterious about this. Or am I missing something?

    1. elkement says:

      Yes exactly, it does reverse the direction of the twist! But I found it interesting that this can be be done without rotating the end but just translating it, and that this is only possible for two full rotations (it also does not work for 3 rotations).

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