# Hopf fibration, Bloch sphere and quantum rotations - interactive site

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TL;DR Summary
quantum; math; Hopf fibration;
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 in history was called "Hopf fibration".
The essence of this design was based on the geometric designs of William Kingdon Clifford.

However, it first came to the attention of theoretical physicists only more than forty years later,
in the 1970s, because of the direct and immediate mathematical relationships between the Hopf fibration
and symmetries in quantum field theory.

This short article discusses some of the main points related to my site, which discusses
the visualization of the Hopf fibration.

In order to visualize the Hopf fibration, an interactive program was made that allows you to set
the values of various parameters of this fibration. Since it is not possible to place this program
in the text of the article, I made a special website on which not only this program was placed,
but also a lot of other information. Therefore, this article highlights only some points related to the Hopf bundle.
I put all the basic information on the website https://vlad0007.github.io/Hopf-fibration-quantum-rotations/.

And here is a video that shows the use of the site to visualize the Hopf fibration.

The meaning of the Hopf fibration is as follows. The two-dimensional sphere
should be located in three-dimensional space, and the three-dimensional sphere 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 division of a three-dimensional sphere into circles is called a Hopf fibration.
These circles themselves are called Hopf circles or Clifford parallels.

In order to visualize the Hopf fibration, each Hopf circle is mapped to a point on a two-dimensional
sphere called the Riemann sphere. It is also often called the Bloch sphere or the Riemann-Bloch sphere.

Remember that the Bloch sphere is used to represent the state vector of a quantum system and
in fact the state vector is located in a 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 single 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 set a new point
on the Bloch sphere and, therefore, thereby a new Hopf circle.

We have figured out the correspondence of Hopf circles to points on the Riemann-Bloch sphere in general terms.
It remains to figure out how to draw circles on the display screen.
Quaternions are usually used for this purpose. But their application is quite difficult
to understand the essence of the visualization process. Therefore, let us recall the global phase
in the mechanics of electron spin. The quantum state will not change if this phase is changed.
Let the global phase vary from 0° to 360°. Thus, we will get some kind of circle that
we can match to the Hopf circle. All this is shown in sufficient detail on the my website.
Here I just want to focus on this.

It should also be noted that the Riemann-Bloch sphere is a completely abstract mathematical 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 right 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 of 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 the 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 asymmetric. Why this is happening is not difficult to understand
by referring to my site.

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.

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This all looks more mathematical - which is why we moved it to Differential Geometry - than it looks physical. Are there other physical interpretations than that ##SU(2)## and ##U(1)## are involved and therefore the SM?

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