Hopf fibration
- How does the Hopf fibration relate to the Dirac equation and magnetic monopoles?
K-theory formulation of Bott periodicity (Husemoller. Fibre Bundles. Chapter 11. Bott Periodicity in the Complex Case. 1.2 Theorem.)
1.2 Theorem. Let {$X$} be a compact space. The external cup products {$K(X) \otimes
K(S^2) \rightarrow K(X \times S^2)$} in complex {$K$}-theory and {$KO(X) \otimes KO(S^8)\rightarrow
KO(X \times S^8)$} in real {$K$}-theory are isomorphisms. In addition, {$K(S^2)$} is a free
abelian group on two generators {$1$} and {$\eta$} class of the complex Hopf bundle, and
{$KO(S^8)$} is the free abelian group on two generators {$1$} and {$\eta_8$}, where {$\eta_8$} is the class of the real eight-dimensional Hopf bundle.
{$S_0\hookrightarrow S_{1} \rightarrow S_1$}
{$S_1\hookrightarrow S_{3} \rightarrow S_2$}
{$S_3\hookrightarrow S_7 \rightarrow S_4$}
{$S_7\hookrightarrow S_{15} \rightarrow S_8$}
Simon Davis. Supersymmetry and the Hopf fibration The Serre spectral sequence of the Hopf fibration {$S_{15}\overset{S_7}{\rightarrow} S_8$} is computed. It is used in a study of supersymmetry and actions based on this fibration.
Wu-Yang Dictionary
- Wu-Yang dictionary Tsai Tsun Wu and C.N.Yang translates the concepts of gauge theory and differential geometry.
- Gauge fields are exactly connections on fiber bundles.
- It allows the understanding of monopole quantization in terms of Hopf fibrations.
- Jim Simons (differential geometry) and C.N.Yang (particle theory) in the 1970s
Potential | Connection |
Field tensor (interaction) | Curvature |
Field tensor-potential relation | Structural equation |
Gauge transformation | Change of bundle coordinates |
Gauge group | Structure group |
Original version for electromagnetism |
gauge (or global gauge) | principal coordinate bundle |
gauge type | principal fiber bundle |
gauge potential {$b_\mu^k$} | connection on principal fiber bundle |
gauge transformation brings the electron wave function from one configuration to the other {$S_{ba}\psi_a=\psi_b$} | transition function |
phase factor {$\Phi_{QP}$} | parallel displacement |
field strength {$f_{\mu \nu }^{k}$} | curvature |
source {$J_{\mu }^{k}$} | electromagnetism |
connection on a {$U(1)$} bundle |
isotopic spin gauge field | connection on a {$SU(2)$} bundle |
Dirac's monopole quantization | classification of {$U(1)$} bundle according to first Chern class |
electromagnetism without monopole | connection on a trivial {$U(1)$} bundle |
electromagnetism with monopole | connection on a nontrivial {$U(1)$} bundle |
Ideas
Gauge theory
- Slack is the structure of goodness. Gauge theory models slack. Gauge invariance reflects a redundancy in the description of the system. (Likewise, for general relativity, diffeomorphism invariance reflect a redundancy in the description of the system.) Slack is associated with the gauge symmetry, the gauge boson, for example, the photon.
Perspectives and circles
- perspective - circle, perspective on perspective - sphere, perspective on perspective on perspective - 3-sphere
- a circle models a perspective but also models recurring activity (CT group)
- a linear complex structure looks like disjoint circles. Multiply it by {$t$} and consider powers. When you have a second linear complex structures, pairs of circles get linked.
Self
- In the stereographic projection, the missing point on the sphere is the self, the vantage point upon everything. This defines the self as a point on a sphere and gives meaning to all points as expressing selves.
- A perspective is a circle, thus an excursion from the self and back, to the default point and back, as with the knowledge of anything, heading out from the vantage point upon everything.
读物
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