Tampan, Paminggir people. Sumatra 19th century, 70 x 70 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop. |

Consider a particle P characterized 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 some bundle of quarks

$\sf{\Omega} = \left\{ \sf{q}_{1}, \sf{q}_{2}, \sf{q}_{3} \ \ldots \ \right\}$

Let these quarks be assembled into a quark model of P that uses the radius vector $\overline{\rho}$ to describe P in quark space. Definition: The **work** done to build the model is the number

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

where $\left\| \, \overline{\rho} \, \right\| \,$ is the norm of the radius vector. So constructing models of centered particles requires the least amount of work. Recall that the constant $k_{\sf{F}}$ was introduced earlier to relate the internal energy of quarks to their radii. So $W$ is just another, slightly different representation of the internal energy of the quarks in P. The square of the work done may be expressed in terms of the radial components as

$W^{2} = \left( \vphantom{\sum^{\, 2}} \rho_{m}^{ 2} k_{mm} + \rho_{e}^{2} k_{ee} + \rho_{z}^{2} k_{zz} + 2 \rho_{e} \rho_{m} k_{em} + 2 \rho_{e} \rho_{z} k_{ez} + 2 \rho_{m} \rho_{z} k_{mz} \right) k^{2}_{\sf{F}}$

and we explicitly consider that this quantity may be negative. If extra quarks are absorbed or emitted by P, then $\sf{\Omega}$ is replaced by a new bundle $\sf{\Omega}^{\prime}$ 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. Furthermore, since all of the quarks in a description can in principle be thought of as a single particle, an overall value of $\Delta W$ may be relevent if quarks are introduced or removed *anywhere* in a description, not just in P. Theorem: Particles and anti-particles have opposing radius vectors $\overline{\rho} \left( \sf{P} \right) = - \overline{\rho} \left( \sf{\overline{P}} \right)$ but their norms are the same, so

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

and the work required to assemble any particle is the same as the work done to build its corresponding anti-particle.

Summary |

Noun | Definition | |

Work | $W \equiv \, \parallel \overline{\rho} \parallel k_{\sf{F}}$ | 5-19 |