Hopf fiber

The Hopf fiber (according to Heinz Hopf ) is a specific mapping in the mathematical sub-area of topology . It is a mapping of the 3-sphere , which can be imagined as the three-dimensional space together with an infinitely distant point, into the 2-sphere, i.e. a spherical surface:

${\displaystyle \eta \colon S^{3}\to S^{2}.}$

Description of the illustration

It is obtained as follows: First, the is embedded in the as a unitary sphere . By pairs of complex numbers on their quotient in mapped. Then the image point is mapped onto the with the inverse stereographic projection with respect to the north pole . There are various options for specifying the figure specifically in formulas. ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle (z_{1},z_{2})\mapsto (z_{1}/z_{2})}$ ${\displaystyle \mathbb {C} \cup \infty =\mathbb {R} ^{2}\cup \infty }$ ${\displaystyle S^{2}}$

With real numbers

The image

${\displaystyle \mathbb {R} ^{4}\to \mathbb {R} ^{3},\quad (x_{1},x_{2},x_{3},x_{4})\mapsto (y_{1},y_{2},y_{3})}$

With

${\displaystyle y_{1}=2(x_{1}x_{3}+x_{2}x_{4})}$

${\displaystyle y_{2}=2(x_{2}x_{3}-x_{1}x_{4})}$

${\displaystyle y_{3}=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}}$

maps the 3-sphere onto the 2-sphere . This limitation is the Hopf map. ${\displaystyle \{x\in \mathbb {R} ^{4}\mid x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\}}$ ${\displaystyle \{y\in \mathbb {R} ^{3}\mid y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\}}$

With complex numbers

The 3-sphere is said to be the subset

${\displaystyle \{(z,w)\in \mathbb {C} ^{2}\mid |z|^{2}+|w|^{2}=1\}}$

of two-dimensional complex space, the 2-sphere as a Riemannian number sphere . Then the Hopf mapping is done

${\displaystyle (z,w)\mapsto {\frac {z}{w}}}$

given. Summarizing the Riemann sphere as a projective line , so can the image using homogeneous coordinates as ${\displaystyle \mathbb {C} P^{1}}$

${\displaystyle (z,w)\mapsto [z:w]}$

write.

With lie groups

The 3-sphere is diffeomorphic to the Lie group Spin (3) , which operates as a superposition of the rotation group SO (3) on the 2-sphere . This operation provides identifications

${\displaystyle S^{2}=SO(3)/SO(2)=Spin(3)/Spin(2)=S^{3}/S^{1}}$.

Example from quantum physics

As a natural view of the Hopf fibers, quantum states of non-relativistic electrons can be represented on the unit sphere .

Here, the state vector: with given. Furthermore, let the shape of the unit sphere of the 2-dimensional Hilbert space be ${\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\end{pmatrix}}}$ ${\displaystyle \psi _{1},\psi _{2}\in \mathbb {C} }$

${\displaystyle S(\mathbb {C} ^{2})=\{\psi \in \mathbb {C} ^{2}:||\psi ||=1\}}$

From the scalar product of the quantum state

${\displaystyle |\psi \rangle ={\begin{pmatrix}\alpha _{1}+i\beta _{1}\\\alpha _{2}+i\beta _{2}\end{pmatrix}}}$

follows

${\displaystyle \langle \psi |\psi \rangle =\alpha _{1}^{2}+\beta _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}=1}$

This corresponds to the 3-sphere.

Two quantum states are equivalent if there is a complex number or a representative of the unitary group that fulfills the requirement. If one considers the entire union of the equivalence class ${\displaystyle \psi _{a},\psi _{b}\in S(\mathbb {C} ^{2})}$ ${\displaystyle \lambda \in U(1)}$ ${\displaystyle \psi _{a}=\lambda \psi _{b}}$

${\displaystyle [\psi ]:=\{\lambda \psi :\lambda \in U(1),|\lambda |=1\}}$

on the sphere

${\displaystyle S(\mathbb {C} ^{2})=\bigcup _{S(\psi \in \mathbb {C} ^{2})}[\psi ]}$

so the group operates on the unity sphere. The amounts of are also called fiber. This amount of fiber is represented as follows ${\displaystyle U(1)}$ ${\displaystyle [\psi ]}$ ${\displaystyle U(1)}$ ${\displaystyle U(1)}$

${\displaystyle S(\mathbb {C} ^{2})/U(1)}$

properties

• The Hopf figure is a fiber bundle with fiber (even a - main fiber bundle ). ${\displaystyle S^{1}}$ ${\displaystyle S^{1}}$
• Every two fibers form a Hopf loop .
• The Hopf mapping creates the homotopy group . ${\displaystyle \pi _{3}(S^{2})\cong \mathbb {Z} }$

Generalizations

The above description using complex numbers can also be carried out analogously with quaternions or Cayley numbers ; fibers are then obtained

${\displaystyle S^{3}\to S^{7}\to S^{4}}$or , ${\displaystyle S^{7}\to S^{15}\to S^{8}}$

which are also referred to as Hopf fibers.

history

Heinz Hopf indicated this figure in 1931 in his work About the images of the three-dimensional sphere on the spherical surface and showed that it is not null homotop (more precisely: that its Hopf invariant is equal to 1).

literature

• Heinz Hopf : About the images of the three-dimensional sphere on the spherical surface. Math. Ann. 104 (1931), 637-665 ( PDF )
• Eberhard Zeidler : Quantum Field theory I - Basics in Mathematics and Physics. Springer Verlag, 2006, ISBN 3-540-34762-3 , pp. 269ff