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What is new with Koide sum rules?

  1. Nov 4, 2018 #201

    ftr

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    Oh Good, I hate loosing unconventional talents.I see that you have been working very hard behind the scenes .Good luck and be strong:biggrin:
     
  2. Nov 6, 2018 #202

    Hans de Vries

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    Science Advisor
    Gold Member

    Thanks, indeed, with many new insights.



    Insights on the spinor level:

    1) How to calculate all three spin-vectors
    How to calculate all three spin-vectors ##s_x,~s_y## and ## s_z## of a spinor and how to do so with a single matrix multiplication. The sum of the three vectors is the total spin ##s##: T
    he precessing spin 1/2 pointer.

    2) A second triplet of spinor rotation generators
    These generators rotate the spinor in its local reference frame instead of in world coordinates. This uncovers the (infinitesimal) rigid-body aspect of field theory with generators that rotate a spinor around its own three principle axis.

    Insights on the fermion field level:

    1) A single fermion field

    The two light-like chiral components ##\xi_L## and ##\xi_R## each get two orthogonal polarization states, with the orientation of the states defined by spinors.
    $$\mbox{Dirac field}~~
    \left(\!

    \begin{array}{c}
    \xi_{_L} \\ \xi_{_R}
    \end{array}
    \!\right)
    ~~~~\Longrightarrow~~~~
    \left(\!\!\!
    \begin{array}{rc}
    \xi_{_{L}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{L}} \\ \pm~~\,\xi_{_{R}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{R}}
    \end{array}
    \!\!\right)
    ~~\mbox{Unified Fermion field}$$

    2) A Standard Model fermion generator.
    All standard model fermions, three generations of leptons and quarks and their anti-particles are the eigen-vectors of a single generator with only the charge and its sign as input. All fermions obtained this way posses all the right electroweak properties corresponding with a ##\sin^2\theta_w## of 0.25

    3) A single electroweak fermion Lagrangian.
    The many different electroweak-fermion pieces of the Lagrangian can be replaced by:
    $$\mathcal{L} ~~=~~ \bar{\psi}\,\check{m}\big(\,\gamma^\mu_{_0}\partial_\mu+\mathbf{U}-\check{m}\,\big)\,\psi,~~~~~~~~
    \mathbf{U} ~=~\tfrac{\,g'}{\,2\,}\gamma^\mu_{_o}\gamma^5_{_o}Z_\mu + \tfrac{g}{2}\gamma^\mu_{_1}A_\mu + \tfrac{g}{2}\gamma^\mu_{_2}W^1_\mu + \tfrac{g}{2}\gamma^\mu_{_3}\gamma^5_{_o}W^2_\mu$$

    4) A single bilinear field matrix
    This matrix contains all bilinear field components as well as all source currents for all electroweak bosons. The matrix is calculated with a single matrix multiplication.

    Insights on the electroweak boson level.

    1) The fundamental representation of the electromagnetic field.
    This representation uses the operator fields acting on the fermion field:
    $$\begin{array}{lrcl}
    \mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A_\mu \\
    \mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\
    \mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu~j_\mu \\
    \end{array}$$
    We now obtain the fundamental covariant description of the electromagnetic field:
    $$/\!\!\! \partial\mathbf{A} = \mathbf{F}~~~~~~ ~~~/\!\!\!\partial\mathbf{F} = \mathbf{J}$$
    In the first step we have applied the conservation law ##\partial_\mu A^\mu\!=\!0## on the diagonal and the second step involves all four of Maxwell's laws, the inhomogeneous ##\partial_\mu F^{\mu\nu}\!=\!j^\nu## as well as the homogeneous ##~\partial_\mu\! *\!\!F^{\mu\nu}\!=\!0##.

    2) A single electroweak boson field
    As given in the Lagrangian above. Note that each electroweak boson has its own set of gamma matrices.
    $$\mathbf{U} ~=~\tfrac{\,g'}{\,2\,}\gamma^\mu_{_o}\gamma^5_{_o}Z_\mu + \tfrac{g}{2}\gamma^\mu_{_1}A_\mu + \tfrac{g}{2}\gamma^\mu_{_2}W^1_\mu + \tfrac{g}{2}\gamma^\mu_{_3}\gamma^5_{_o}W^2_\mu$$


    The documents, mathematica files and the stand alone matlab executable are available here,
    but look at the video for the best introduction.
     
    Last edited: Nov 6, 2018
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