Wednesday, October 31, 2018


A single, very simple, Unified Fermion Lagrangian, produces the many separated, 
particle dependent, pieces of the Electroweak Fermion Lagrangian.

 All the Standard Model fermions, three generations of leptons and quarks, are found to be different excitations of a single unified field, as the eigenvectors of a single generator function with the charge  as only variable. The field's content determines the type of fermion and its characteristics.


Mathematica code for the Unified Field Theory

You can now download the Mathematica code for the Unified Field Theory.  The ZIP file contains the following notebook files:

(1)  FermionField_definitions_V1.00.nb  This file contains the definitions of the Generators used plus a set of useful functions. This file is called from the other two files when these are executed.
(2)  BilinearFields_V1.00.nb  The Bilinear fields are calculated for any Standard Model fermions.
(3)  Bosons_Tests_V1.00.nb  Tests on the Unified E.W. boson fields and their generators.

Tuesday, September 25, 2018

Download the United Fermion Field Explorer software

You can now download the UNIFIED FERMION FIELD software package
(see the instructions at the bottom of the post) 

I'm planning to make some youtube movies with instructions and demos.

-1  The software requires a 4k monitor or 4k TV (at windows 100% or 125% settings)
-2  The software is MATLAB code and compiled to a run-time executable.
-3  Download the zip-file and unpack it in the folder of your choice.
-4  Download and install the MATLAB run time package 9.0.1  (R2016a, 64 bit Windows)
-5  You may need to add the path: SET PATH=%PATH%;C:\Program Files\MATLAB\R2016a\runtime\win64
-6  Included in the zip-file is a bat file which does this for you: Add_Path_to_MATLAB_Runtime.bat


Saturday, September 8, 2018

I'm making earlier work available via this blog.

This includes chapters from my book and a number of papers of interesting papers.

Relativistic Quantum Field Theory

Part I   Relativistic foundations of light and matter Fields

             Chapter 1:    Elementary solutions of the classical wave equation 
             Chapter 2:    Lorentz contraction from the classical wave equation
             Chapter 3:    Time dilation from the classical wave equation

             Chapter 4:    Non-simultaneity from the classical wave equation

Part II   Advanced treatment of the EM field

             Chapter 5:    Relativistic formulation of the electromagnetic field
             Chapter 6:    The Chern-Simons EM spin and axial current density
             Chapter 7:    The EM stress energy tensor and spin tensor  
             Chapter 8:    Advanced EM treatment of Spin 1/2 fermions  

Part III   The relativistic matter wave equations

             Chapter 9:     Relativistic matter waves from Klein Gordon's equation
             Chapter 10:   Operators of the scalar Klein Gordon field
             Chapter 11:   EM Lorentz force derived from Klein Gordon's equation
             Chapter 12:   Klein Gordon transition currents and virtual photons
             Chapter 13:   Propagators of the real Klein Gordon field
             Chapter 14:   Propagators of the complex Klein Gordon field
             Chapter 15:   The self propagator of the Klein Gordon field

             Chapter 16:   The Poincare group and relativistic wave functions
             Chapter 17:   The Dirac Equation 
             Chapter 18:   Transformations of the bilinear fields of the Dirac electron
             Chapter 19:   Gordon decomposition of the vector/axial currents
             Chapter 20:   Operators and Observables of the Dirac field
             Chapter 21:   The EM interactions with the Dirac field
             Chapter 22:   The Hamiltonian and Lagrangian densities

             Chapter 28:   Full Gordon decomposition of all bilinears




Sunday, September 23, 2012

The maximal Gordon decomposition

                       Download the PDF                  Download the PDF from GOOGLE                           

We present a novel and elegant expression for the maximal Gordon composition for an electron field that can vary locally in all of it's 16 field description parameters.

The 16 bilinear Dirac field components

The fully extended Gordon decomposition provides a powerful tool to obtain a thorough understanding of the fundamental behavior of the interacting electron and in general the interacting fermion described by the Dirac equation. I turns out that, despite the long and painstaking calculations, all the results can be arranged in a compact manner which is easy accessible for interpretation. We'll apply the decomposition on all the Dirac bilinear fields (16 components in total)

The bilinear Dirac fields

$\begin{aligned} &&\mbox{Scalar}~~~~ && ~~\bar{\psi}\psi & ~~~~~~~~\mbox{1 component} \\ &&\mbox{Vector}~~~~ && ~~\bar{\psi}\gamma^\mu\psi & ~~~~~~~~\mbox{4 components} \\ &&\mbox{Antisym.Tensor}~~~~&& ~~\bar{\psi}\sigma^{\mu\nu}\psi & ~~~~~~~~\mbox{6 components} \\ &&\mbox{Axial vector}~~~~ && ~~\bar{\psi}\gamma^\mu\gamma^5\psi & ~~~~~~~~\mbox{4 components} \\ &&\mbox{Pseudo scalar}~~~~ && ~i\bar{\psi}\gamma^5\psi & ~~~~~~~~\mbox{1 component} \\ \end{aligned}$ 

These fields are, due to their Lorentz transform, associated with: The invariant mass (scalar), The charge/current density (vector), the spin density (axial vector) and the magnetization/polarization tensor.

The 16 fermion field description parameters

By applying the decomposition we determine how these fields depend on the first order derivatives of, not only, the magnitude and phase of the field but on how they depend on the first order derivatives of a systematically complete set of 16 field description parameters including for instance the local rotation and local boost of the field.The Magnitude, Phase, Balance and Phase skew are the single component field descriptors which transform like Lorentz scalars. The Boost, Rotation, Magnetization and Polarization are all 3-component field descriptors which transform like tensor fields. The generators of the 16 fermion field description parameters and their relations can be compactly defined by.

The field description parameters
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~~~~\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} \vec{\sigma}~~~~~~0 \\ 0~~~~~~\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} i\sigma^o~~~0~~ \\ 0~~~~i\sigma^o  \end{array}\!\!\!\right)$}}~~~~~~~~
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~~~~i\vec{\sigma}  \end{array}\!\right)$}}
\\ \\
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~-\!\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} \vec{\sigma}~~~~~~0 \\ 0~~~-\!\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&\mbox{Phase skew:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} i\sigma^o~~~0~~ \\ 0~-\!i\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~-\!i\vec{\sigma} \end{array}\!\right)$}}

The single component field descriptors are associated with $\sigma^o$ while the 3-component field descriptors are associated with the spatial matrices $\vec{\sigma}$. All generators of the set, acting in spinor space, are systematically scaled with the same factor 1/2 familiar from the boost and rotation generators. This means they are scaled to express their effect in Minkowsky space. The Magnetization and Polarization generators are the same as those from the $\sigma^{\mu\nu}$ tensor. The Phase skew generator occurs in Electroweak interactions where the intermediate vector boson fields act asymmetrically on the left and right chiral components $\psi_L$ and $\psi_R$

The Gordon decomposition method

The expressions for the bilinear fields do not contain any partial differentials. It is the Dirac equation which links the values of these fields to the first order differentials of the field. We write the Dirac equation like this,

\psi_L  ~~=~~\tfrac{\hbar}{mc}(~i \,\sigma^\nu \,\partial_\nu ~)\,\psi_R
\psi_R  ~~=~~\tfrac{\hbar}{mc}(~i \,\tilde{\sigma}^\nu\,\partial_\nu ~)\,\psi_L

to show us how to substitute $\psi_L$ and $\psi_R$ by differentiated terms. We then take the bilinear product terms and substitute both $\psi^*$ and $\psi$ one at a time and average the results, for instance.

\psi_R^*\psi_L \longrightarrow  ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\tilde{\sigma}^\nu \,\partial_\nu ~\psi_L]^*\psi_L ~~+~~
\psi_R^*[~i \,\sigma^\nu\,\partial_\nu ~\psi_R]~  \Big)
\psi_L^*\psi_R \longrightarrow  ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\sigma^\nu \,\partial_\nu ~\psi_R]^*\psi_R ~~+~~
\psi_L^*[~i \,\tilde{\sigma}^\nu\,\partial_\nu ~\psi_L]~  \Big)

Then we consider $\psi_L$ and $\psi_R$ as exponentials $\exp (\hat{G}\cdot \mathcal{G})$ of the whole set of generators $\hat{G}$ defined above. We can now define $\partial_\mu\psi_L$ and $\partial_\mu\psi_R$ as a function of the first order derivatives $\partial_\mu\mathcal{G}$ of the field description parameters $\mathcal{G}$ associated with the set of generators $\hat{G}$. When we apply this on all Dirac bilinears we can reorganize the results to obtain the following result.

The results of the complete Gordon decomposition

&\tfrac{2mc}{\hbar}~\bar{\psi}\psi &=&-
&\bar{\psi}\gamma^\mu\psi~~\Big(~\partial_\mu\,\phi~\,-~\partial_\alpha\, \overset{\hspace{-6pt}\star}{{\cal P}^\alpha_{~\mu}}~\, \Big) && \mbox{Phase}  \\
& &&+~~
&\bar{\psi}\gamma^\mu\gamma^5\psi~~\Big(~\partial_\mu\,{\cal P}~-~\partial_\alpha\,\overset{\hspace{-6pt}\star}{J^{\alpha}_{~\mu}}~~ \Big) && \mbox{Phase skew}
\\ \\
&\tfrac{2mc}{\hbar}~\bar{\psi}\gamma^\mu\psi &=&+
&\bar{\psi} \sigma^{\mu\nu}\psi~~\Big(~\partial_\nu {\cal M}\,+~\partial_\alpha\,J^{\alpha}_{~\nu}~~ \Big) && \mbox{Magnitude} \\
& &&-~~
&\bar{\psi}\psi~~\Big(~\partial^\mu\,\phi~\,-~\partial_\alpha \overset{\hspace{-6pt}\star}{{\cal P}^{\alpha\mu}}~ \Big) && \mbox{Phase} \\
& &&-~~
& i\bar{\psi}\gamma^5\psi~~\Big(~\partial^\mu\,\vartheta~\,+~\partial_\alpha {\cal P}^{\alpha\mu}~ \Big) && \mbox{Balance}  \\
& &&+~~
& \bar{\psi}\overset{\hspace{-9pt}\star}{\sigma^{\mu\nu}}\psi~~\Big(~\partial_\nu\,{\cal P}~-~\partial_\alpha\,\overset{\hspace{-6pt}\star}{J^{\alpha}_{~\nu}}~~ \Big) && \mbox{Phase skew}
\\ \\
&\tfrac{2mc}{\hbar}~\bar{\psi}\sigma^{\mu\nu}\psi &=&-
&\bar{\psi}\gamma^\mu\psi\,\bigcirc\hspace{-10.40pt}\wedge~\Big(~\partial^\nu {\cal M} +~\partial_\alpha\,J^{\alpha\nu}~ \Big) && \mbox{Magnitude} \\
& &&-~~
&\bar{\psi}\gamma^\mu\gamma^5\psi\,\overset{\hspace{+2pt}\star}{\bigcirc\hspace{-10.3pt}\wedge}\, \Big(~\partial^\nu\,\phi~\,-~\partial_\alpha \overset{\hspace{-6pt}\star}{{\cal P}^{\alpha\nu}}~ \Big) && \mbox{Phase}  \\
& &&+~~
&\bar{\psi}\gamma^\mu\gamma^5\psi\,\bigcirc\hspace{-10.30pt}\wedge~ \Big(~\partial^\nu\,\vartheta~\,+~\partial_\alpha {\cal P}^{\alpha\nu}~ \Big) && \mbox{Balance}  \\
& &&+~~
&\bar{\psi}\gamma^\mu\psi\,\overset{\hspace{+2pt}\star}{\bigcirc\hspace{-10.4pt}\wedge}\, \Big(~\partial^\nu\,{\cal P}~-~\partial_\alpha\, \overset{\hspace{-6pt}\star}{J^{\alpha\nu}}~ \Big) && \mbox{Phase skew}
\\ \\
&\tfrac{2mc}{\hbar}~\bar{\psi}\gamma^\mu\gamma^5\psi &=&-
&i\bar{\psi}\gamma^5\psi~~\Big(~\partial^\mu {\cal M} +~\partial_\alpha\,J^{\alpha\mu}~ \Big) && \mbox{Magnitude} \\
& &&+~~
& \bar{\psi}\overset{\hspace{-9pt}\star}{\sigma^{\mu\nu}}\psi~~\Big(~\partial_\nu\,\phi~\,-~\partial_\alpha\, \overset{\hspace{-6pt}\star}{{\cal P}^\alpha_{~\nu}}~\, \Big) && \mbox{Phase}  \\
& &&+~~
&\bar{\psi}\sigma^{\mu\nu}\psi~~\Big(~\partial_\nu\,\vartheta~\,+~\partial_\alpha\, {\cal P}^\alpha_{~\nu} ~\, \Big) && \mbox{Balance}  \\
& &&-~~
&\bar{\psi}\psi~~\Big(~\partial^\mu\,{\cal P}~-~\partial_\alpha\, \overset{\hspace{-6pt}\star}{J^{\alpha\mu}}~ \Big) && \mbox{Phase skew}
\\ \\
&\tfrac{2mc}{\hbar}~i\bar{\psi}\gamma^5\psi &=&+
&\bar{\psi}\gamma^\mu\gamma^5\psi~~\Big(~\partial_\mu {\cal M}\,+~\partial_\alpha\,J^{\alpha}_{~\mu}~~ \Big) && \mbox{Magnitude} \\
& &&-~~
&\bar{\psi}\gamma^\mu\psi~~\Big(~\partial_\mu\,\vartheta~\,+~\partial_\alpha\, {\cal P}^\alpha_{~\mu} ~\, \Big) && \mbox{Balance}

For a complete description of this Gordon decomposition and the contents of the components of this expression you can download the PDF.

                       Download the PDF                  Download the PDF from GOOGLE