Let particle P be characterized by its enthalpy $H$ and the work $W$ required to bring together its component quarks. Definition: the **rest mass** of P is given by

$\begin{align} m \equiv \frac{\sqrt{ \: H^{2} - W^{2} \ \vphantom{ {\Sigma^{2}}^{2} } }}{ c^{2} } \end{align}$

where $c$ is a constant. This definition distinguishes several types of particles by their mass. If $m$ is positive then P is a material particle; an ordinary particle of matter, like a coin or a bullet. There is an important special case for material particles when the work required to make them is negligible compared to their enthalpy, then we say they are *heavy* particles. If the mass is zero then P is ethereal, so called for *Αἰθήρ* the mythical personification of the breath of Gods. Finally if $m^{2} \! < \! 0$ then P has an imaginary^{1} mass. Roughly speaking, the rest mass describes how much internal energy is leftover after the work of assembling a particle has been completed. When a particle is in an excited state we may use the mass to describe how hard or *dense* it is. Recall that $\widetilde{\rho}$ notes the mean radius of P's phase-components. Definition: the **density** of P is

Particle Type | Definition |

heavy |
$W^{2} \, \ll \, H^{2}$ |

material |
$W^{2} \, < \, H^{2}$ |

ethereal |
$W^{2} \, = \, H^{2}$ |

imaginary |
$W^{2} \, > \, H^{2}$ |

$\begin{align} \varrho \equiv \frac{ m c^{2} }{ \widetilde{\rho} } \end{align}$

Theorem: particles and anti-particles have the same mass as each other. We have already seen how $H \left( \sf{P} \right) = - H \left( \overline{\sf{P}} \right)$ and $W \left( \sf{P} \right) = W \left( \sf{\overline{P}} \right)$ when conjugate symmetry is assumed. But the mass depends on these quantities squared. So

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

Theorem: photons are ethereal because by definition, any photon $\gamma$ is a phase anti-symmetric particle. So $\rho ( \gamma) =0$ and no work is required to assemble the quarks in a photon. Phase anti-symmetry also guarantees that the net number of quarks must be nil for any type of quark. Substituting this $\Delta n = 0$ condition into the definition of enthalpy shows that $H ( \gamma) =0$ as well. Then the definition of mass given above implies that

$m \left( \large{\gamma} \right) =0$

Sensory Interpretation: Enthalpy characterizes the magnitude of all classes of sensation, whereas the work represents just somatic and visual sensations. The mass is established by their difference, which is mostly due to thermal sensation. For heavy particles, thermal perceptions are more important than visual sensations. And for particles with an imaginary mass, audio-visual sensations dominate awareness.Summary |

Adjective | Definition | |

Mass | $\begin{align} m \equiv \frac{1}{ c^{2} } \sqrt{ \: H^{2} - W^{2} \ \vphantom{ {\Sigma^{2}}^{2} } } \end{align}$ | 8-2 |

Adjective | Definition | |

Density | $\begin{align} \varrho \equiv \frac{ m c^{2} }{ \widetilde{\rho} } \end{align}$ | 8-3 |