Tampan, Paminggir people. Sumatra circa 1900, 43 x 46 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop. |

Consider a particle P described by a repetitive chain of space time events

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

where each orbital cycle is parsed into eight components

$\sf{\Omega} ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3}, \sf{P}_{4}, \sf{P}_{5}, \sf{P}_{6}, \sf{P}_{7}, \sf{P}_{8} \right)$

as required to establish a three-dimensional description. Definition: **quark-level homogeneity** is an assumption that for almost any particle in a description, these eight sub-orbital events have many characteristics the same as each other. Whole particles may differ from each other, but for perfect spatial homogeneity, every particle is presumed to have an all-around similarity between its eight sub-orbital components. Specifically we require that each octant has the same lepton number, baryon number and charge $q$. So this is a lot like spatial isotropy but it is even more stringent because it requires that sub-orbital events are similar all to each other, not just similar in pairs. For example, a presumption of spatial homogeneity constrains the charge as

$q\,\left(\sf{P}_{1} \right) =q\,\left(\sf{P}_{2} \right) =q\,\left(\sf{P}_{3} \right) =q\,\left(\sf{P}_{4} \right) =q\,\left(\sf{P}_{5} \right) =q\,\left(\sf{P}_{6} \right) =q\,\left(\sf{P}_{7}\right) =q\,\left(\sf{P}_{8} \right)$

Then recall that charge is conserved so the total charge on P is eight times the charge of one sub-orbital event

$\begin{align} q^{\sf{P}} = \sum _{\it{k} \mathrm{=1}}^{8} q\left(\sf{P}_{\it{k}}\right) =8 \, q\left(\sf{P}_{1}\right) \end{align}$

But the charge is defined by multiples of the fraction one-eighth as $\, q \equiv \frac{ 1 }{8} \left( {\Delta}n^{\mathsf{T}}-{\Delta}n^{\mathsf{B}}+{\Delta}n^{\mathsf{C}}-{\Delta}n^{\mathsf{S}} \right)$ where ${\Delta}n$ notes a particle's quark coefficients. So in a perfectly homogenous space

$\begin{align} q ^{\sf{P}} = {\Delta}n^{\mathsf{T}}\left(\sf{P}_{1}\right)-{\Delta}n^{\mathsf{B}}\left(\sf{P}_{1}\right)+{\Delta}n^{\mathsf{C}}\left(\sf{P}_{1}\right)-{\Delta}n^{\mathsf{S}}\left(\sf{P}_{1}\right) \end{align}$

The quark coefficients of $\sf{P}_{1}$ are always integers so $\, q^{\sf{P}}$ is determined by a sum or difference of integers. Therefore quark-level homogeneity constrains the charge of P as $\, q = 0, \, \pm1, \, \pm2, \, \pm3 \ \ldots$ Similar reasoning leads to the same result for lepton and baryon numbers

$L = 0, \, \pm1, \, \pm2, \, \pm3 \ \ldots$ and $B = 0, \, \pm1, \, \pm2, \, \pm3 \ \ldots$

The notion of homogeneity depends on the extent or scale of a description. By the foregoing definition, a space must be filled with particles that are at least as large as atoms to achieve quark-level homogeneity. But almost any ordinary space *is* full of atoms and molecules, and $q$, $L$ and $B$ are invariably observed to be integers. At the atomic level, particles are described using Cartesian coordinates and similarity conditions are expressed in terms of $x$, $y$ and $z$. Definition: **atomic-level homogeneity** requires that particles are distributed so that their shapes are not oriented in any special direction. The variation in their radii is presumably similar in all directions so that $\delta \rho _{x} = \delta \rho_{y} = \delta \rho_{z}$. And at the macroscopic level, further conditions like *well-stirred mixture* assumptions may have to be included to make a complete specification of spatial homogeneity.

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Spatial Homogeneity |