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geometric_algebra [2020/06/13 08:44]
pbk [Wikipedia links]
geometric_algebra [2020/06/20 00:46] (current)
pbk [Articles]
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   * [[http://​link.springer.com/​content/​pdf/​10.1007%2Fs00006-016-0700-z.pdf|Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity]] (2016) - //Anthony Lasenby//   * [[http://​link.springer.com/​content/​pdf/​10.1007%2Fs00006-016-0700-z.pdf|Geometric Algebra as a Unifying Language for Physics and Engineering and Its Use in the Study of Gravity]] (2016) - //Anthony Lasenby//
 Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics,​ physics and engineering. In this contribution,​ we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism,​ quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach. Geometric Algebra (GA) is a mathematical language that aids a unified approach and understanding in topics across mathematics,​ physics and engineering. In this contribution,​ we introduce the Spacetime Algebra (STA), and discuss some of its applications in electromagnetism,​ quantum mechanics and acoustic physics. Then we examine a gauge theory approach to gravity that employs GA to provide a coordinate free formulation of General Relativity, and discuss what a suitable Lagrangian for gravity might look like in two dimensions. Finally the extension of the gauge theory approach to include scale invariance is briefly introduced, and attention drawn to the interesting properties with respect to the cosmological constant of the type of Lagrangians which are favoured in this approach.
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 +  * [[https://​arxiv.org/​pdf/​1504.02906.pdf|Skew ray tracing in a step-index optical fiber using Geometric Algebra]] (2015) - //Angeleene S. Ang, Quirino M. Sugon, Daniel J. McNamara//
 +We used Geometric Algebra to compute the paths of skew rays in a cylindrical,​ step-index multimode optical fiber. To do this, we used the vector addition form for the law of propagation,​ the exponential of an imaginary vector form for the law of refraction, and the juxtaposed vector product form for the law of reflection. In particular, the exponential forms of the vector rotations enables us to take advantage of the addition or subtraction of exponential arguments of two rotated vectors in the derivation of the ray tracing invariants in cylindrical and spherical coordinates. We showed that the light rays inside the optical fiber trace a polygonal helical path characterized by three invariants that relate successive reflections inside the fiber: the ray path distance, the difference in axial distances, and the difference in the azimuthal angles. We also rederived the known generalized formula for the numerical aperture for skew rays, which simplifies to the standard form for meridional rays.
  
   * [[https://​arxiv.org/​ftp/​arxiv/​papers/​1502/​1502.02169.pdf|Geometric algebra, qubits, geometric evolution, and all that]] (2015) - //Alexander Soiguine//   * [[https://​arxiv.org/​ftp/​arxiv/​papers/​1502/​1502.02169.pdf|Geometric algebra, qubits, geometric evolution, and all that]] (2015) - //Alexander Soiguine//
geometric_algebra.txt ยท Last modified: 2020/06/20 00:46 by pbk