Hopf fibration

The Hopf fibration (after Heinz Hopf ) is a specific figure in the mathematical branch of topology. This is a picture of the 3- sphere, which can be thought of as a three-dimensional space along with a point at infinity, in the 2-sphere, ie a sphere:

Description of the illustration

They are obtained as follows: First, as the unit is embedded in the sphere. By pairs of complex numbers are on their quotient in ready. After forming the pixel from the inverse Stereographic projection with respect to the North Pole to the. To specify the illustration specifically in the formulas, there are various possibilities.

With real numbers

The figure

With

Forms the three - sphere onto the 2-sphere. This limitation is the Hopf map.

With complex numbers

The 3- sphere will as the subset

The two-dimensional complex space considered, the 2-sphere as a Riemann number sphere. Then the Hopf map is by

Given. Summarizing the Riemann sphere as the projective number on line, so you can imaging using homogeneous coordinates as

. Write

With Lie groups

The 3- sphere is diffeomorphic to the Lie group Spin ( 3) as a superposition of the rotation group SO (3 ) acts on the 2-sphere. By this operation, we obtain identifications

Example from quantum physics

As a natural perception of the Hopf fibration to quantum states not relativistic electrons on the unit sphere can be represented.

Here, the state vector: given with. Further, the shape is the unit sphere of the 2-dimensional Hilbert space

From the scalar product of the quantum state

Follows

This corresponds to the 3- sphere.

Two quantum states are equivalent if there exists a complex number or a representative of the unitary group, which meets the requirement. Looking at the entire union of the equivalence class

On the sphere

As the group on the unit sphere surgery. The amounts of fiber are also known. This amount of the fiber is represented as follows

Properties

• The Hopf map is a fiber bundle with fiber ( even a - principal bundle ).
• The two fibers form a Hopf entanglement.
• The Hopf mapping generates the homotopy group.

Generalizations

The above description uses complex numbers can be carried out by analogy with quaternions or Cayley numbers; then we obtain fibrations

Which are also referred to as the Hopf - fibrations.

History

Heinz Hopf was this figure in 1931 in his work over the images of the three-dimensional sphere to the spherical surface and showed that it is not null-homotopic (more precisely, that its Hopf invariant is equal to 1).