Sunday, September 23, 2012

The maximal Gordon decomposition

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
\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}

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 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}

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