The Internal Energy of Compound Quarks
 Bead Panel from a baby carrier, Bahau People. Borneo 20th century, 35 x 30 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Consider a particle P described by some repetitive chain of events

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} , \sf{\Omega}_{3} \ \ldots \ \right)$

where each orbital cycle is a bundle of $N$ quarks

$\sf{\Omega} = \left\{ \sf{q}_{1} \ \ldots \ \sf{q}_{\it{i}} \ \ldots \ \sf{q}_{\it{N}} \right\}$

By definition each quark is composed from a pair of seeds noted by

$\sf{q} = \left\{ \sf{Z} , \sf{Z}^{\prime} \right\}$

Let each quark be characterized by its internal energy

$U ^{ \sf{q}} = \varepsilon \hat{E} + \varepsilon^{\prime} \hat{E}^{\prime}$

where $\hat{E}$ is the specific energy and $\varepsilon$ is the audibility of each seed. Then by the definition of internal energy as a sum over all seeds

\begin{align} U ^{ \sf{P}} = \sum_{i\sf{=1}}^{N} \varepsilon _{\it{i}} \hat{E}_{\it{i}} + \varepsilon^{\prime} _{\it{i}} \hat{E}^{\prime}_{\it{i} } = \sum_{i\sf{=1}} ^{N} U_{\it{i}}^{\sf{q}} \end{align}

and so the internal energy of a compound quark is just a sum over the internal energies of its component quarks. Quarks are indestructible and the internal energy of each quark has a specific fixed value, so whenever some generic compound quarks $\mathbb{X}$, $\mathbb{Y}$ and $\mathbb{Z}$ interact, if

$\mathbb{ X} + \mathbb{ Y} \leftrightarrow \mathbb{ Z}$ then $U ^{ \mathbb{X} } + U ^{ \mathbb{Y} } = U ^{ \mathbb{Z} }$

We say that internal energy is conserved when particles are combined or decomposed. Also by the hypothesis of conjugate symmetry an ordinary quark and its anti-quark have the same internal energy. Swapping ordinary quarks with anti-quarks does not change the total number of quarks of a given type. So particles have the same internal energy as their associated anti-particles

$U \left( \sf{P} \right) = U \left( \overline{\sf{P}} \right)$

Finally we define some averages; the internal energy of a typical quark in P is

\begin{align} \tilde{U} \equiv \frac{U}{N} \end{align}

This is slightly different from the average over just the ordinary quarks which is written as

\begin{align} \tilde{ U } _{\! \sf{o}} \equiv \sum_{\zeta=1}^{10} \frac{ \, n^{\zeta} }{ N_{ \sf{o}}} U^{\sf{\zeta}} \end{align}

where $N_{ \sf{o}}$ notes the total number of all types of ordinary-quarks in P

\begin{align} N_{ \sf{o}} \equiv \sum_{\zeta =1}^{10} n^{\zeta} \end{align}