The Unified Fermion Field presentation is available now:
Friday, September 28, 2018
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
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.
Understanding
Relativistic Quantum Field Theory
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
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
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
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
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
Papers
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 componentsWe 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 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
$\begin{aligned}
&\mbox{Magnitude:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~~~~\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Magnetization:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} \vec{\sigma}~~~~~~0 \\ 0~~~~~~\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&\mbox{Phase:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} i\sigma^o~~~0~~ \\ 0~~~~i\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Rotation:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~~~~i\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&\mbox{Balance:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~-\!\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Boost:}
&{\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{Polarization:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~-\!i\vec{\sigma} \end{array}\!\right)$}}
\end{aligned}$
&\mbox{Magnitude:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~~~~\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Magnetization:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} \vec{\sigma}~~~~~~0 \\ 0~~~~~~\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&\mbox{Phase:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} i\sigma^o~~~0~~ \\ 0~~~~i\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Rotation:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~~~~i\vec{\sigma} \end{array}\!\right)$}}
\\ \\
&\mbox{Balance:}
&{\mbox{ $\tfrac12\left(\!\begin{array}{r} \sigma^o~~~~0~~ \\ 0~~~-\!\sigma^o \end{array}\!\!\!\right)$}}~~~~~~~~
&\mbox{Boost:}
&{\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{Polarization:}
&{\mbox{ $-\tfrac12\left(\!\begin{array}{r} i\vec{\sigma}~~~~~0 \\ 0~-\!i\vec{\sigma} \end{array}\!\right)$}}
\end{aligned}$
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.
$\begin{aligned}
\psi_R^*\psi_L \longrightarrow ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\tilde{\sigma}^\nu \,\partial_\nu ~\psi_L]^*\psi_L ~~+~~
\psi_R^*[~i \,\sigma^\nu\,\partial_\nu ~\psi_R]~ \Big)
\\
\psi_L^*\psi_R \longrightarrow ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\sigma^\nu \,\partial_\nu ~\psi_R]^*\psi_R ~~+~~
\psi_L^*[~i \,\tilde{\sigma}^\nu\,\partial_\nu ~\psi_L]~ \Big)
\end{aligned}$
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 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.
$\begin{aligned}
\psi_R^*\psi_L \longrightarrow ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\tilde{\sigma}^\nu \,\partial_\nu ~\psi_L]^*\psi_L ~~+~~
\psi_R^*[~i \,\sigma^\nu\,\partial_\nu ~\psi_R]~ \Big)
\\
\psi_L^*\psi_R \longrightarrow ~~\tfrac12\,\tfrac{\hbar}{mc}
\Big( ~[~i \,\sigma^\nu \,\partial_\nu ~\psi_R]^*\psi_R ~~+~~
\psi_L^*[~i \,\tilde{\sigma}^\nu\,\partial_\nu ~\psi_L]~ \Big)
\end{aligned}$
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
$\begin{aligned}
&\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}
\end{aligned}$
&\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}
\end{aligned}$
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
Download the PDF Download the PDF from GOOGLE
Saturday, September 22, 2012
Real Symmetric representation of Dirac's equation
We describe how one can transform the standard representation of the Dirac equation into a new, spatially symmetric, real valued representation. It is shown that this new representation can be used as a direct replacement in QED and other standard model applications, it produces the same results and the notation is carefully chosen to be virtually identical.
The
spatial symmetry of the 8
field parameters.
One major consequence of the
spatial symmetry is that the
8 field parameters get a
spatial symmetric
interpretation as well. The
cube as seen in the
illustrations occurs if we
apply the so called
"current" operator to the
individual field parameters.
Each parameter becomes
associated with a
propagation (current) from the center of the cube
to one of the 8 vertices.
Each of the 4 central axis
of the cube has two of them,
one is left circular
polarized and the other is
right circular polarized. The eight parameters
together define the
direction of the electron's
current density as well as
the direction of the
electron's spin.
Note that 4
axis- orientations are used
to define directions instead
of the usual 3 spatial
coordinates. This means that
there is a one parameter
redundancy in the
definitions. This redundant
parameter turns out to be
the phase
of the wave function.
It's the phase of the wave
function which enables the
interference of matter
waves. We have found a way
in which nature can define a
phase without needing a
special preferred direction
of rotation.
The
720 degrees of spinor
rotation.
One of the most puzzling
statements a student gets to
hear is that "an electron has
to rotate over 720 degrees
before it returns back to its original position".
This statement is directly
related to another statement
which says that the electron
is a spin 1/2 particle. All kinds of strange and
often irrelevant
visualizations are used by
people who are trying to
make sense of this. It turns
out that we can find the
mathematics corresponding to
this spin 1/2 behavior
directly in the
representation itself.
We'll see that a more
appropriate form of the 720
degrees statement is that "the phase
difference is always 1/2
of the geometrical angle"
The
electron and positron
eigenstates.
We can calculate the
eigenstates of the electron
(and the positron) directly
from the equation of motion.
The particles oscillate with
a frequency proportional to
their mass. If we look at
the parameters along, for
instance, the x-axis (in
case of a spin in the x
direction) then we see
that there are 45 degrees
phase relations between
parameters that are 90
degrees rotated from each
other. The images above show 4
parameters with phase shifts
of 0, 45, 90 and 135
degrees. The other 4
parameters (in the positron
case) have phase shifts of
180, 225, 270 and 315
degrees. In the case of the
electron there are extra
signs. This is what
distinguishes the negative
charge from the positive
charge.
The
rotation generators of spin
1/2 particles.
We can find the
mathematical origin of this
spin 1/2 behavior (or 720
degrees behavior) directly
in the matrices used in the
representation. We can rotate the current
and spin direction of the
electron field with the use
of the x, y and z-rotation
matrices shown below.
Generally when we perform a
continuous rotation then we
operate on two (orthogonal)
components so that these get
a 90 degrees sine-cosine
relation. The result is then
a circular motion. In the
real, symmetric
representation however we
operate on two parameters
which are not orthogonal (90
degrees) but diagonally
positioned (180 degrees). The result is
that the phase
difference is 1/2 of the
geometrical angle as is typical for a spin 1/2
particle.
One can see that each
rotation matrix defines four
such connections, each time
between two parameters. This
is why there are four lines
shown in each of the images
(two X's) The signs in the
matrices are so that all
four rotations are defined
as being in the same
direction.
The
charge generators of spin 1/2
particles.
If we place the
electron in a potential
field then its phase change
rate will become higher if
the electron's energy
becomes higher and lower
when the energy becomes
lower. If we rotate an electron
then we will also change the
phase of the parameters (by
1/2 the rate). However if we
give the electron a
different phase change rate
by placing it in a constant
potential field then we
won't rotate the direction
of the spin or the current,
now, what is the difference
mathematically?
We see in the image below
that the individual
components of the charge
generator (which is used to
describe the influence of
the potential field) are
almost identical to the
rotation generators. The
difference is in the
signs. Two of the four
connections are now opposite
in sign. The result is that
the net rotation is zero.
The complete charge
generator shown at the
bottom is symmetric in x, y
and z: The electric
potential field has no
preferred direction in
space.
The
boost generators of spin 1/2
particles.
If we boost a particle
then we accelerate the
particle to a different
speed. The matrices used for
this operation are the boost
matrices.
These matrices are somewhat
more complex but from the
images we can see that they
correspond to the x, y and
z-directions. The most important
difference with the rotation
matrices is that they are
non-zero on the diagonals.
This causes some parameters
to become bigger and bigger
when the boost becomes
larger, while other
parameters become smaller
and smaller. the +1's found on the
diagonals correspond to the
increasing parameters and
the -1's correspond to
decreasing parameters. The parameters that become
bigger are drawn with
lighter colors while the
ones that become smaller are
darker. Now, each parameter
corresponds with a current
in its direction, from the
center of the cube outward.
If we boost the electron in
a certain direction then we
indeed expect the currents
in that direction to
increase while the currents
in the opposite direction
should decrease.
The
mass coupling of the chiral
states.
If we don't place the
electron in a potential
field then we still expect
that its phase changes
continuously and it should
do so with a frequency
proportional to its mass.
The matrix which is
responsible for this is the
"mass coupling" matrix. This
is the matrix which couples
the four left chiral
parameters with the four
right chiral parameters. It
is the only matrix which
does so.
If we compare the 3
individual components of the
mass coupling below with the
3 individual components of the charge
generator above, then we see
that they have the same 4x4
sub matrices but in different
quadrants. This means that the mass
coupling has the same effect
as the charge generator when
the four left chiral
parameters are equal to the
four right chiral
parameters, and the opposite
effect if there is a minus
sign between the two chiral
sets. This is the case in
the electron and positron
eigenstates respectively.
The combined mass coupling
matrix is symmetric in x, y
and z. Mass does not have a
preferred direction in space
just like charge doesn't
have a preferred direction.
About the PDF document
The PDF was written
over a period of about 3
years. It contains an
overview in chapter 1 plus a
complete step by step
derivation of the real
symmetric representation
from the standard asymmetric
complex one in chapter 2.
A lot of attention was spend
at the development of the
notation. The final form is
highly compatible with the
standard notation so that
most expressions will be
either the same or almost
identical.The reader will be
able to switch between the
representations without
effort. The real symmetric
representation is obtained by
going from the standard
complex 2x2 Pauli matrices
to their equivalent 4x4 real
versions. The extra degrees
of freedom obtained make it
possible to make the
representation symmetric.
The consequence of the 4x4
matrices is that SO(4)
becomes the group of unitary
generators of rotation and
charge. The non Abelian charges introduced are
discussed in chapter 3.
Further chapters are under development.
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