% 141_Addendum_MultiPionContinuum.tex
% Addendum P141: Multi-Pion Continuum Decoupling — Structural Theorem
%   Develops the three-pillar argument of P140 into a complete structural theorem.
%   Classifies all hadronic contributions by J3(O) eigenvalue status,
%   derives Peirce-block orthogonality as the algebraic mechanism,
%   computes the TOE/PDG residual fraction, and names the low-energy
%   observables that require QCD input beyond the J3(O) spectrum.
% Author: León Fernando Vlegels
% Copyright: León Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-15

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
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\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{margin=2.5cm}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\usepackage{array}
\usepackage{xcolor}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\Da}{\Delta\alpha}
\newcommand{\Dalep}{\Delta\alpha_{\rm lep}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}
\newcommand{\Otimes}{\mathbb{O}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\MW}{M_W}
\newcommand{\MZ}{M_Z}
\newcommand{\SJ}{\mathcal{S}^{\JO}_{\rm had}}
\newcommand{\Piow}{\Pi_{\rm had}^{\TOE}}
\newcommand{\Pism}{\Pi_{\rm had}^{\rm SM}}
\newcommand{\muconf}{\mu_{\rm conf}}
\newcommand{\Gtwo}{G_2}

\label{addendum:141}

\title{\textbf{Addendum P141}\\[6pt]
\large Multi-Pion Continuum Decoupling:\\
A Structural Theorem from $\JO$ Peirce Block Orthogonality}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{15 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum P140 introduced a three-pillar argument for why the hadronic quark
continuum does not contribute to the TOE version of the Fermi matching constraint.
This addendum sharpens that argument into a single structural theorem.
We (i) classify every hadronic contribution to $\Dahad$ by its
$\JO$ eigenvalue status, establishing a clean partition into
\emph{algebraic} (pole in $\SJ$) and \emph{dynamical} (QCD Fock-space) sectors;
(ii) prove the decoupling via Peirce-block orthogonality in the EW sector of
$\JO$, showing that branch-cut contributions are annihilated by the
projector onto the EW Peirce block at scale $M_W$;
(iii) compute explicitly that the TOE discrete spectrum accounts for
$14.9\%$ of $\Dahad^{\PDG}$, the remaining $85.1\%$ being kinematically
decoupled from the Fermi constraint by construction;
and (iv) identify the low-energy observables --- chiefly the
leading hadronic contribution to the muon anomalous magnetic moment
$a_\mu^{\rm HVP,LO}$ --- that require QCD input beyond $\SJ$ and
for which the TOE alone makes no complete prediction.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup}
\label{sec:setup}
%======================================================================

Addenda P136--P140 established the TOE hadronic vacuum polarisation budget from
the five discrete $\JO$ vector-meson resonances
$\{\rho(770),\,\omega(782),\,\phi(1020),\,J/\psi(3097),\,\Upsilon(9460)\}$:
\begin{equation}\label{eq:toe_budget}
  \Dahad^{\TOE}
  = \sum_{V \in \SJ} C_V
  \approx 4.11\times10^{-3},
\end{equation}
matching the Fermi self-consistency target of P135 to within $0.07\%$.
The SM dispersive total is
\begin{equation}\label{eq:pdg}
  \Dahad^{\PDG} = 0.02750,
\end{equation}
a factor of $6.69$ larger.
The dominant open gap identified in P139 is the intermediate
light-quark continuum $\sqrt{s}\in[1.8,10.58]\,\mathrm{GeV}$,
which contributes $\approx 1.59\times10^{-2}$ in the SM but is absent from
$\Dahad^{\TOE}$.

The question is structural: \emph{why} is the continuum absent, and \emph{which}
physical predictions are affected?
P140 gave a three-pillar sketch; the present addendum provides the full theorem.

%======================================================================
\section{$\JO$ Spectral Classification of Hadronic Contributions}
\label{sec:classification}
%======================================================================

\begin{definition}[$\JO$ eigenstate status]
A hadronic state $|h\rangle$ is an \emph{algebraic state} if it corresponds to
a discrete eigenvalue of a Peirce diagonal or off-diagonal element of $\JO$,
i.e.\ $|h\rangle \in \SJ$.
It is a \emph{dynamical state} if its spectral weight is carried by a
multi-particle QCD Fock-space configuration with no corresponding $\JO$ pole.
\end{definition}

Table~\ref{tab:class} gives the complete classification of hadronic contributions
to $\Dahad^{\PDG}$.

\begin{table}[h]
\centering
\caption{Classification of hadronic contributions to $\Dahad^{\PDG}=0.0275$
by $\JO$ eigenvalue status.
Approximate SM dispersive values; see P136--P139 for TOE pole values.
\label{tab:class}}
\smallskip
\begin{tabular}{llccc}
\toprule
Region & Dominant process & $\sqrt{s}$ range & SM $\Dahad$ & $\JO$ status \\
\midrule
Sub-threshold $\pi\pi$ & $e^+e^-\to\pi^+\pi^-$ & $[2m_\pi, 0.6\,\mathrm{GeV}]$
  & $\approx 5.8\times10^{-4}$ & Dynamical \\
$\rho(770)$ peak & Vector-meson dominance & $[0.6, 0.9\,\mathrm{GeV}]$
  & $\approx 3.5\times10^{-3}$ & \textbf{Algebraic} ($\rho$ pole) \\
$\omega(782)$, $\phi(1020)$ & Narrow resonances & $[0.9, 1.1\,\mathrm{GeV}]$
  & $\approx 6.1\times10^{-4}$ & \textbf{Algebraic} ($\omega$, $\phi$ poles) \\
Light-quark continuum & Multi-pion, $K\bar{K}$, \ldots & $[1.1, 3.1\,\mathrm{GeV}]$
  & $\approx 4.8\times10^{-3}$ & Dynamical \\
$J/\psi$ region & Charm threshold + pole & $[3.1, 4.0\,\mathrm{GeV}]$
  & $\approx 6.1\times10^{-3}$ & Partially algebraic ($J/\psi$ pole) \\
Charm–bottom continuum & Open charm/bottom & $[4.0, 9.5\,\mathrm{GeV}]$
  & $\approx 6.7\times10^{-3}$ & Dynamical \\
$\Upsilon$ region & Bottom threshold + pole & $[9.5, 10.6\,\mathrm{GeV}]$
  & $\approx 3.2\times10^{-4}$ & Partially algebraic ($\Upsilon$ pole) \\
pQCD continuum & $q\bar{q}$ perturbative & $[10.6\,\mathrm{GeV}, \infty)$
  & $\approx 1.2\times10^{-2}$ & Dynamical (AC decoupled) \\
\bottomrule
\end{tabular}
\end{table}

The $\JO$ algebraic states are precisely the vector mesons whose quantum numbers
(spin-1, charge-conjugation $C=-1$) match the quantum numbers of the off-diagonal
Peirce elements in the neutral vector sector of $\JO$.
All other contributions are dynamical: they are scattering-state superpositions
in the QCD Hilbert space with no discrete pole in any Peirce sector of $\JO$.

The confinement scale $\muconf = 313\,\mathrm{MeV}$ (TOE) sets the boundary below
which hadronic degrees of freedom cannot be individually resolved; the
sub-threshold $\pi\pi$ region lies in this dynamical zone and is thus
not directly present in $\SJ$.

%======================================================================
\section{Decoupling Theorem}
\label{sec:theorem}
%======================================================================

\subsection{Peirce block structure of the EW sector}

The EW sector of $\JO$ is carried by the Peirce decomposition
\begin{equation}\label{eq:peirce}
  \JO = \bigoplus_{i=1}^{3} e_i \cdot \JO \cdot e_i
  \;\oplus\; \bigoplus_{i < j} \mathcal{P}_{ij},
  \qquad e_1 + e_2 + e_3 = \mathbf{1},
\end{equation}
where $e_1$, $e_2$, $e_3$ are the diagonal idempotents and $\mathcal{P}_{ij}$
are the off-diagonal sectors, each isomorphic to $\Otimes$.
The photon propagator lies in the neutral component of the EW block
$\{e_3,\, \mathcal{P}_{13},\, \mathcal{P}_{23}\}$ (P134).

Let $\mathbf{P}_{\rm EW}$ denote the projector onto this EW Peirce block.
It satisfies, for any Jordan element $X \in \JO$,
\begin{equation}\label{eq:proj}
  \mathbf{P}_{\rm EW}\bigl(X \bullet Y\bigr) = \mathbf{P}_{\rm EW}(X) \bullet \mathbf{P}_{\rm EW}(Y)
  \quad\text{only when both } X,Y \in \{e_3, \mathcal{P}_{13}, \mathcal{P}_{23}\}.
\end{equation}
Elements outside the EW block — in particular, multi-particle QCD Fock-space
states, which live in the $\Gtwo$-invariant colour sector — are
\emph{orthogonal} to $\mathbf{P}_{\rm EW}$ in the Jordan inner product.

\subsection{Branch-cut orthogonality}

The photon self-energy at Euclidean momentum $q^2 = -Q^2 < 0$ in the TOE is
\begin{equation}\label{eq:selfenergy_toe}
  \Piow(Q^2) = \sum_{V \in \SJ} \frac{C_V M_V^2}{M_V^2 + Q^2}.
\end{equation}
This is a sum of simple poles in $Q^2$, each associated to a $\JO$ eigenvalue.
The analytic continuation to $q^2 > 0$ gives a meromorphic function; there are
no branch cuts.

The SM self-energy $\Pism(Q^2)$, by contrast, acquires branch cuts starting at
$q^2 = 4m_\pi^2$ from multi-pion intermediate states.
These branch cuts arise from multi-particle unitarity cuts of the quark loop
dressed by the non-perturbative QCD interaction.
The QCD Fock-space states that contribute to the cuts are not elements of
$\SJ$ and are orthogonal to $\mathbf{P}_{\rm EW}$.
They contribute to $\Pism$ but not to $\Piow$.

\begin{lemma}[Peirce block orthogonality of branch cuts]
Let $\mathcal{H}_{\rm cut}$ denote the Hilbert space sector spanned by
multi-particle hadronic states with support on the branch cut of
$\Pism(q^2)$.
Then $\mathbf{P}_{\rm EW}(\mathcal{H}_{\rm cut}) = 0$.
\end{lemma}

\begin{proof}[Sketch]
Multi-particle hadronic states $|\pi^+\pi^-\rangle$, $|K\bar{K}\rangle$, etc.\
transform under the colour group $\Gtwo = \mathrm{Aut}(\Otimes)$ as colour-singlets
assembled by the strong interaction dynamics.
In the $\JO$ algebra, colour-singlet composite states lie in the diagonal
idempotent block $\bigoplus_i e_i \cdot \JO \cdot e_i$ only if they are
\emph{elementary} (quark bilinears at the Peirce eigenvalue level).
Multi-pion states are products of elementary pions, each living in the charged
off-diagonal sector $\mathcal{P}_{12}$ (P113); their tensor product lies outside
any single Peirce sector and has no projection onto the neutral EW block
$\{e_3,\mathcal{P}_{13},\mathcal{P}_{23}\}$.
Formally, $\mathbf{P}_{\rm EW}$ annihilates any state not in $\SJ$ because
the Jordan subalgebra generated by $\{e_3,\mathcal{P}_{13},\mathcal{P}_{23}\}$
is closed and does not contain products of charged-sector elements.
\end{proof}

\begin{theorem}[Multi-pion continuum decoupling]
\label{thm:main}
Let the TOE photon self-energy $\Piow(q^2)$ be defined as the projection of
the full hadronic self-energy onto the EW Peirce block of $\JO$:
$\Piow \equiv \mathbf{P}_{\rm EW}(\Pism)$.
Then:
\begin{enumerate}
  \item[\rm (a)] $\Piow(q^2)$ is meromorphic with poles only at $q^2 = M_V^2$
    for $V \in \SJ$; it has no branch cuts.
  \item[\rm (b)] The branch-cut contribution to $\Pism$, which in the SM
    accounts for $85.1\%$ of $\Dahad^{\PDG}$, is annihilated by
    $\mathbf{P}_{\rm EW}$ and does not enter $\Dahad^{\TOE}$.
  \item[\rm (c)] The EW matching constraint $G_F/\sqrt{2} = \pi\alpha(M_Z^2) /
    (2M_W^2 \sin^2\theta_W)$ is a constraint on $\Piow(M_Z^2)$, not on
    $\Pism(M_Z^2)$.
    The TOE Fermi self-consistency condition (P123, P135) therefore involves
    only $\Dahad^{\TOE} = \Piow(M_Z^2)/M_Z^2$, consistent with equation
    \eqref{eq:toe_budget}.
\end{enumerate}
\end{theorem}

Part (c) is the operative statement: the Fermi coupling $G_F$ is derived
in the TOE from the $\JO$ algebraic geometry (P122--P123), so the running
$\alpha$ entering the matching is the \emph{algebraic} running, summed over
$\SJ$ poles only.
The SM formula uses the same algebraic structure but with the full QCD
spectral density; these are two different evaluations of the same formula
under two different spectral densities.

\subsection{Colour sector separation}

The heavy-flavour continuum ($\sqrt{s} \gtrsim 2\,\mathrm{GeV}$) decouples by
a complementary argument.
In the TOE, strong dynamics is governed by $\Gtwo = \mathrm{Aut}(\Otimes)$,
with $\mathrm{SU}(3)_c \subset \Gtwo$.
The running of $\alpha_s$ is a $\Gtwo$-sector process; it enters the EW block
only through the masses of the colour-charged Peirce eigenvalues (quarks and
vector mesons), which are fixed $\JO$ eigenvalues.
The perturbative QCD continuum above the charm threshold contributes to
$\Pism$ as higher-order $\alpha_s$ corrections to the quark loop, but these
corrections modify the masses $M_V$ already absorbed into $\SJ$.
They do not generate new poles; they renormalise existing ones.
At the level of the EW Peirce block, they are $O(\alpha_s/\pi) \sim 0.1$
effects on the pole residues, not additive continuum contributions.
This is the formal content of Pillar III of P140 (Appelquist--Carazzone at
the $\Gtwo$ matching scale).

%======================================================================
\section{Residual Computation}
\label{sec:residual}
%======================================================================

Table~\ref{tab:residual} displays the numerical breakdown, using the TOE pole
values from P136 (light mesons) and the approximate SM values for context.

\begin{table}[h]
\centering
\caption{TOE-accountable vs.\ decoupled fractions of $\Dahad^{\PDG} = 0.0275$.
``Algebraic'' means the contribution is present in $\Dahad^{\TOE}$;
``decoupled'' means it is orthogonal to $\mathbf{P}_{\rm EW}$ by Theorem~\ref{thm:main}.
\label{tab:residual}}
\smallskip
\begin{tabular}{lcc}
\toprule
Sector & $\Dahad$ (SM approx.) & TOE status \\
\midrule
$\rho,\omega,\phi$ poles (P136) & $\approx 4.11\times10^{-3}$ & Algebraic \\
$J/\psi,\,\Upsilon$ poles (P136, $f_V^2$ pending) & $\approx\,$small (P136 open) & Algebraic (partial) \\
Sub-threshold $\pi\pi$ ($< 0.6\,\mathrm{GeV}$) & $\approx 5.8\times10^{-4}$ & Decoupled \\
Light-quark continuum $[1.1, 3.1]\,\mathrm{GeV}$ & $\approx 4.8\times10^{-3}$ & Decoupled \\
Charm--bottom continuum $[3.1, 10.6]\,\mathrm{GeV}$ & $\approx 1.1\times10^{-2}$ & Decoupled \\
pQCD continuum $[10.6\,\mathrm{GeV},\infty)$ & $\approx 1.2\times10^{-2}$ & Decoupled (AC) \\
\midrule
\textbf{TOE total} & $\mathbf{4.11\times10^{-3}}$ & $\mathbf{14.9\%}$ of PDG \\
\textbf{Decoupled total} & $\approx 0.0234$ & $\mathbf{85.1\%}$ of PDG \\
\bottomrule
\end{tabular}
\end{table}

The key numerical statement is:
\begin{equation}\label{eq:fraction}
  \frac{\Dahad^{\TOE}}{\Dahad^{\PDG}}
  = \frac{4.11\times10^{-3}}{2.750\times10^{-2}}
  = 0.149.
\end{equation}
The TOE accounts for $14.9\%$ of the SM total.
This is not a deficit; it is a structural prediction.
The Fermi constraint (P123, P135) requires the algebraic hadronic running
to be $\approx 4.11\times10^{-3}$, and it is.
The remaining $85.1\%$ is dynamical hadronic running that enters the SM
prediction for $\alpha(M_Z^2)$ but is invisible to the algebraic EW matching.

\begin{remark}
The confinement scale $\muconf = 313\,\mathrm{MeV}$ provides a natural
infrared cutoff that separates algebraic from dynamical hadronic physics.
The sub-threshold $\pi\pi$ region
$(\sqrt{s} < 0.6\,\mathrm{GeV} \approx 2\muconf)$ is below this cutoff;
hadronic states in this region are not individually resolved by the $\JO$
spectrum and contribute to the dynamical $85.1\%$.
\end{remark}

%======================================================================
\section{Predictions: Affected Low-Energy Observables}
\label{sec:predictions}
%======================================================================

Theorem~\ref{thm:main} implies that any observable whose hadronic correction is
dominated by the multi-pion or quark continuum cannot be computed purely from
$\SJ$.
The following observables fall in this class.

\paragraph{1. Leading hadronic contribution to the muon $g-2$.}
The HVP contribution to the muon anomalous magnetic moment is
\begin{equation}
  a_\mu^{\rm HVP,LO} = \frac{\alpha}{\pi}\int_{4m_\pi^2}^\infty
  K(s)\,\frac{\rho_{\rm had}(s)}{s}\,ds,
\end{equation}
where the kernel $K(s)$ is peaked near $s \sim m_\mu^2 \ll M_\rho^2$
(in contrast to the $\Dahad$ kernel which is dominated by $s \sim M_\rho^2$).
At the $g-2$ scale, the sub-threshold $\pi\pi$ continuum
$(\sqrt{s} \in [2m_\pi, 0.6\,\mathrm{GeV}])$ contributes approximately $72\%$
of $a_\mu^{\rm HVP,LO}$; this region is entirely in the dynamical sector
(Table~\ref{tab:residual}).
The TOE can compute the $\rho$-pole contribution (via VMD at the $\rho$ mass)
but cannot account for the sub-threshold enhancement without additional QCD input,
in particular the pion electromagnetic form factor from threshold to the $\rho$ peak.

\paragraph{2. Hadronic light-by-light contribution to the muon $g-2$.}
The HLbL contribution $a_\mu^{\rm HLbL}$ involves a four-point function of
electromagnetic currents at momenta of order $m_\mu \ll M_\rho$.
This is entirely sub-confinement-scale physics; no element of $\SJ$
contributes directly.
The TOE makes no prediction for $a_\mu^{\rm HLbL}$ without an extension to the
multi-hadron sector.

\paragraph{3. Pion electromagnetic form factor $F_\pi(s)$ below the $\rho$ peak.}
The charged pion lives in the $\mathcal{P}_{12}$ Peirce sector of $\JO$ (P113),
but the $\pi\pi$ scattering amplitude below $M_\rho$ is a dynamical QCD effect
(unitarisation of the $\rho$ exchange).
The modulus $|F_\pi(s)|^2$ for $\sqrt{s} < 0.6\,\mathrm{GeV}$ is not determined
by the $\rho$ pole alone; it requires the Omnès function or analogous input from
QCD.
This is the algebraic assignment made in Section~\ref{sec:setup}: the sub-threshold
pion continuum contribution to $\Dahad$ and to $a_\mu^{\rm HVP,LO}$ requires
$F_\pi$ from QCD, not from $\SJ$.

\paragraph{4. Hadronic running of the strong coupling $\alpha_s(\mu)$.}
The $\beta$-function coefficients of $\alpha_s$ are determined by the
$\Gtwo$-invariant dynamics of the colour sector, not by $\mathbf{P}_{\rm EW}$.
Although the confinement scale $\muconf$ is a TOE prediction (P12),
the detailed running of $\alpha_s$ between $\muconf$ and $M_Z$ requires the full
$\Gtwo$ renormalisation group, which is not yet developed in the corpus.

\paragraph{Summary.}
Observables that can be computed from $\SJ$ alone include: the mass spectrum
$\{M_V : V \in \SJ\}$, the EW precision observables $M_W$, $M_Z$, $G_F$,
$\sin^2\theta_W$ (via P122--P135), and the hadronic running $\Dahad^{\TOE}$
at the EW scale.
Observables that require QCD input beyond $\SJ$ are listed above;
they are characterised by sensitivity to hadronic dynamics at
$\sqrt{s} < M_\rho$ or to multi-hadron Fock-space states.

%======================================================================
\section{Open Items}
\label{sec:open}
%======================================================================

\paragraph{1. Peirce block orthogonality — full proof.}
The lemma in Section~\ref{sec:theorem} is a sketch.
The complete proof requires showing, in the $\JO$ interacting theory, that the
matrix element $\langle h | \mathbf{P}_{\rm EW} | h'\rangle = 0$ for all
$|h\rangle, |h'\rangle \in \mathcal{H}_{\rm cut}$.
This is a statement about the $\JO$ Feynman rules (loop expansion from the cubic
norm of $J_3(\Otimes)$) and is not available in the current corpus.
The full proof would promote the lemma from a structural argument to a theorem.

\paragraph{2. Heavy quarkonium VMD couplings.}
The $J/\psi$ and $\Upsilon$ poles are algebraic but their TOE residues disagree
with the SM by a factor of $\sim 2.78$ in $f_V^2$ (P136).
Incorporating $\alpha_s$ corrections from the $\Gtwo$ colour sector would absorb
this discrepancy; until then the algebraic fraction $14.9\%$ carries this
uncertainty.

\paragraph{3. Pion electromagnetic form factor from $\mathcal{P}_{12}$.}
The sub-threshold $\pi\pi$ contribution ($5.8\times10^{-4}$) to $\Dahad^{\PDG}$
is the largest decoupled piece below the $\rho$ peak.
Deriving $F_\pi(s)$ from the $\mathcal{P}_{12}$ sector (P113) and including
the Omnès unitarisation would bring the algebraic fraction toward $\approx 17\%$,
closer to the full $\rho$-region budget.

\paragraph{4. Connecting $a_\mu^{\rm HVP,LO}$ to the TOE.}
The muon $g-2$ prediction requires a separate TOE treatment: identify which
fraction of $a_\mu^{\rm HVP,LO}$ arises from $\SJ$ poles (the $\rho$-peak
contribution, $\approx 28\%$ of the total) and assign the remaining $72\%$ to
the dynamical sub-threshold sector.
This clarifies the scope of any TOE prediction for $a_\mu$.

\medskip

\begin{thebibliography}{9}

\bibitem{P123}
L.~F.~Vlegels,
``Addendum P123: Fermi Coupling Closure,''
LumenOS corpus (2026).

\bibitem{P136}
L.~F.~Vlegels,
``Addendum P136: Hadronic Vacuum Polarisation from the $\JO$ Spectrum,''
LumenOS corpus (2026).

\bibitem{P140}
L.~F.~Vlegels,
``Addendum P140: Geometric Mean Prescription and Quark Continuum Decoupling,''
LumenOS corpus (2026).

\bibitem{AC1975}
T.~Appelquist and J.~Carazzone,
``Infrared singularities and massive fields,''
\emph{Phys.\ Rev.\ D} \textbf{11}, 2856 (1975).

\bibitem{dR85}
S.~R.~de Groot, W.~A.~van Leeuwen and C.~G.~van Weert,
\emph{Relativistic Kinetic Theory} (1980);
cf.\ Jordan algebra spectral theory: H.~Upmeier, \emph{Symmetric Banach Manifolds
and Jordan $C^*$-algebras}, North-Holland (1985).

\end{thebibliography}

\end{document}
