# What is new with Koide sum rules?

1. Nov 4, 2018

### ftr

Oh Good, I hate loosing unconventional talents.I see that you have been working very hard behind the scenes .Good luck and be strong

2. Nov 6, 2018

### Hans de Vries

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