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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

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  • 读物 书 影片 维基百科

Introduction E9F5FC

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Bott periodicity, Investigating Bott periodicity

How does Hopf fibration model perspective, perspective on perspective, and perspective upon a perspective on perspective?


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
Summary
PotentialConnection
Field tensor (interaction)Curvature
Field tensor-potential relationStructural equation
Gauge transformationChange of bundle coordinates
Gauge groupStructure group
Original version for electromagnetism
gauge (or global gauge)principal coordinate bundle
gauge typeprincipal 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 fieldconnection on a {$SU(2)$} bundle
Dirac's monopole quantizationclassification of {$U(1)$} bundle according to first Chern class
electromagnetism without monopoleconnection on a trivial {$U(1)$} bundle
electromagnetism with monopoleconnection 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.

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This page was last changed on January 31, 2025, at 12:25 PM