Four-dimensional Topology of the Universe

Werner Ahrendt and Tom Becker attribute the puzzling phenomenon that most elementary particles are identical to themselves only after a 720° rotation to a four-dimensional shape of the universe.

The starting point for our considerations is the extremely strange and previously misunderstood property of fermions that they are identical to themselves only after a rotation of 720°. After a rotation of 360°, the wave function of the particle has the opposite polarity - only after a further rotation of 360° is the initial state reached again. This peculiarity occurs not only when the observer is fixed and the particle is rotated around its own axis, but also when the particle is fixed and the observer moves in a circle around the particle. Therefore, this strange phenomenon cannot be due to the fermions, as is usually assumed, but must be related to the topological properties of the universe, which we are also subject to when we rotate around another object as an observer.

Are there mathematical structures whose topology is such that you need to rotate twice to return to the initial state?

Such mathematical structures do exist. A simple version is known as a Möbius strip. A Möbius strip can be made by gluing a strip of paper upside down. If an object moves along the strip, after one complete revolution it is upside down and at the back of the strip; only after another revolution is it back in its original position. Thus, motion on such a structure has the same strange property that we observe with fermions. So we should consider the crazy idea that our universe has a topological structure that resembles a Möbius strip. However, our universe cannot be a two-dimensional surface connected at the ends to form an endless ribbon in three-dimensional space, because to describe the universe we need (at least) four dimensions - three space-like dimensions and one time-like dimension.

Mathematicians also have suitable objects in their drawers for this case. Alexander Unzicker offers a promising candidate in his book Die mathematische Realität – warum Raum und Zeit eine Illusion: The four-dimensional sphere - mathematicians call it the S3 sphere. Unfortunately, our imagination is limited to three dimensions, so we can only understand the properties of a four-dimensional sphere by analogy. Mathematicians define a spherical surface as the set of all points that are at the same distance from the center of the sphere. This definition can be applied to any space, whether it has two, three, four, or more dimensions. For example, a two-dimensional sphere is an ordinary circle whose boundary is a curved line. A three-dimensional sphere is an ordinary sphere whose boundary is a curved surface.

A four-dimensional sphere is therefore bounded by a curved three-dimensional space. If the radius of a sphere is quite large and therefore the curvature is quite small, the surface looks flat locally. A circle can be approximated locally by a tangent, a sphere by a tangential surface and a four-dimensional sphere by a plane Euclidean space. So if our universe really were a four-dimensional sphere, it would look locally approximately like a plane Euclidean space. It would therefore not be at all surprising that in the course of evolutionary history our brain has developed the visual form of a plane three-dimensional space, because this allows us to grasp the everyday world of experience simply and with sufficient accuracy.

What is interesting in our context is the property of a four-dimensional sphere that it takes a rotation of 720° to reach the starting point. To understand this, let's use another analogy: On a two-dimensional circle, you return to the starting point after a full 360° rotation. A three-dimensional sphere can now be thought of as having another circle attached to each point of the circle. On a sphere, you can also return to your starting point after a 360° rotation, although you can use different paths to do so. On a four-dimensional sphere, however, it takes two full 360° rotations to get back to the starting point! This is because a four-dimensional sphere can be thought of as a circle with two more circles attached to each point. To get back to the starting point, you have to go through the two circles like a figure of eight - i.e. after a 360° rotation you are back at the starting point, but "upside down" like a Möbius strip.

But why don't we realize in everyday life that you have to rotate twice around your own axis in the universe to return to your starting position?

In our opinion, the fact that material bodies look the same after a 360° rotation is due to the fact that the physical interactions are mediated by bosons, which are identical to themselves after a 360° rotation. All atoms and molecules that make up material bodies are essentially held together by electromagnetic interactions. For photons as interacting particles of electromagnetism, the world looks identical again after a 360° rotation. Therefore, when we interact with matter, we don't usually realize that it would actually take us two full rotations to get back to the starting point.