Hopf fibration and quantum mechanics
There is a very interesting topic in mathematics called the Hopf fibration. In 1931, Heinz Hopf published his work on the construction he discovered in topology, which received the name “Hopf fibration” in history. The essence of this design was based on the geometric developments of William Kingdon Clifford.
However, it first came to the attention of theoretical physicists only more than forty years later, in the 1970s, due to the direct and immediate mathematical relationships between the Hopf fibration and gauge symmetries in quantum field theory.
This short article discusses some of the main points related to my site, which discusses visualization of the Hopf fibration.
And here videowhich shows the use of the site to visualize the Hopf fibration.
The meaning of the Hopf fibration is as follows. An ordinary circle can be called a one-dimensional sphere in two-dimensional space – that is, on a plane. The two-dimensional sphere must already be located in three-dimensional space, and the three-dimensional sphere must already be located in four-dimensional space. But instead of the usual four-dimensional space, it is convenient to consider a two-dimensional complex space. It is, in fact, also four-dimensional.
The entire three-dimensional sphere can be filled with circles. No two such circles intersect. This partition of a three-dimensional sphere into circles is called a Hopf fibration. These circles themselves are called Hopf circles or Clifford parallels.
To visualize the Hopf fibration each Hopf circle is compared dot but already on two-dimensional sphere called the Riemann sphere. It is also often called the Bloch sphere or the Riemann-Bloch sphere.
The only question is how to make this comparison. You can watch the lecture Alexey Savvateev or at least the first seven minutes of the academician’s lecture Anatoly Fomenko.
But all this is quite confusing. Therefore, we will do things a little differently. To do this, remember that the Bloch sphere is used to represent the state vector of a quantum system and in fact the state vector is in two-dimensional complex space. All this is discussed in sufficient detail on my website. Therefore, we can assume that each point on the Bloch sphere corresponds to one Hopf circle. Note that the name “Bloch sphere” and the name “Riemann sphere” are two names for the same mathematical object. By changing the Bloch vector we can define a new point on the Bloch sphere and, therefore, a new Hopf circle.
We have generally understood the correspondence between Hopf circles and points on the Riemann-Bloch sphere. It remains to understand how to draw circles on the display screen. Usually quaternions are used for this. But their application is quite difficult to understand the essence of the visualization process.
Therefore, let us remember about the global phase in electron spin mechanics. The quantum state will not change if this phase is changed. Let the global phase vary from 0° to 360°. Thus, we obtain a certain circle that we can compare with the Hopf circle. This is all shown in sufficient detail on the website. Here I just want to focus on this.
You should also pay attention to the fact that the Riemann-Bloch sphere is completely abstract mathematical an object. Therefore, just in case, I will say that we should not think that by taking a certain point on this sphere, we can attach a circle directly at this point for visualization. Everything is much more complicated and is also shown on the website.
If we set the position of a point in an arbitrary place on the Riemann-Bloch sphere, then we will not see all the beauty of the Hopf fibration. But if we use quantum rotations around the spatial axis given on the Riemann-Bloch sphere, we will see tori consisting of Hopf circles. If quantum rotation is carried out around the vertical Z axis, then the tori will be symmetrical. If the rotation is carried out around a spatial axis that does not coincide with the Z axis, then the tori will turn out to be asymmetrical. Why this happens is easy to understand by turning to the next page of my other site The Bloch sphere and the Stern-Gerlach experiment.
In fact, the Stern-Gerlach experiment plays a fundamental role in understanding the mathematics of quantum mechanics. Many books on quantum mechanics published in English in recent years begin by describing how this experiment can explain the fundamental mathematics of electron spin.
There is a close connection between the Bloch sphere and the Stern-Gerlach experiment. You can build the following sequence:
Stern-Gerlach experiment ⇒ Riemann-Bloch sphere ⇒ quantum rotations ⇒ Hopf fibration
Short video about the connection between the Stern-Gerlach experiment and the Bloch sphere and quantum rotations.
Without understanding how quantum rotations occur on the Riemann-Bloch sphere, it is very difficult to understand the Hopf fibration – at least with the approach I used to create my visualization program.
