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Work

Consider a particle P characterized by some repetitive chain of events \Psi  = \left( \mathsf{\Omega}_{1} , \mathsf{\Omega}_{2}  \;   \ldots  \;       \right) where each orbital cycle is a bundle of quarks written as \mathsf{\Omega} = \left( \mathsf{q}_{1}, \mathsf{q}_{2} \; \ldots   \;  \mathsf{q}_{N} \right). Let these quarks be assembled into a model of P. The work required to bring these quarks together to build the model is defined as

 W \equiv k_{\mathsf{F}}  \left\| \,   \overline{\rho} \,  \right\|

where  \left\| \,   \overline{\rho} \,  \right\| is the norm of the radius vector of P. We consider that W might be an imaginary number because the norm may be imaginary under some circumstances. Recall that the constant  k_{\mathsf{F}} was introduced earlier to relate the internal energy of quarks to their radii. So W is just another, slightly different representation for the internal energy of the quarks in P.

Particles and anti-particles have opposing radius vectors, that is, \overline{\rho} \left( \mathsf{P} \right) = - \overline{\rho} \left( \mathsf{\overline{P}} \right). But they both have the same norm. So the work required to assemble any particle is the same as the work done to build its corresponding anti-particle

W  \left( \mathsf{P} \right) = W \left(  \mathsf{\overline{P}} \right)

If extra quarks are absorbed or emitted by P, then  \mathsf{\Omega} is replaced by a new bundle  \mathsf{\Omega}^{\prime} and  W changes to  W^{\prime}. The quantity  \Delta W \equiv W^{\prime} - W may be used to describe the change. Particle radii may also vary, and then we say that the interaction has done work on the particle by changing its shape.

The norm can be written as

    \begin{equation*}   \left\| \,   \overline{\rho} \,  \right\|  =  \left(  \begin{split} &    \; k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  &  + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}    \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right)^{\frac{1}{2}} \end{equation*}

where the constants  k_{\alpha \beta} are components of the quark metric, and  \rho_{\alpha} are components of P’s radius vector. So the square of the work can be written as

    \begin{equation*}   W^{2}  =  k^{2}_{\mathsf{F}}  \left(  \begin{split} &    \; k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  &  + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}   \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right) \end{equation*}

And since we explicitly consider that the work may be imaginary, then W^{2} may be negative. The foregoing expression is key for calculating the mass of P. And experimental observations of mass are possibly the most important data for understanding the mechanics of particles. So we will refer back to the work, but next we discuss clocks.

The work required to construct this 19th century tampan from Sumatra is suggested by extensive and layered chromatic patterns.
Tampan, Paminggir people. Lampung region of Sumatra 19th century, 70 x 70 cm. Photograph by D Dunlop.