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Spatial Homogeneity

Consider some particle P that is described by a repetitive chain of events written as

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

And let each orbital cycle  \mathsf{\Omega} be parsed into eight sub-orbital events as required to establish a three-dimensional spatial description

\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2} \, \ldots \, \mathsf{P}_{k} \, \ldots \, \mathsf{P}_{8} \right)

Then we define quark-level homogeneity as an assumption that for almost any particle in a description, these eight sub-orbital events have characteristics that are 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  L, baryon number  B 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(\mathsf{P}_{1} \right) =q \! \left(\mathsf{P}_{2} \right) = q \! \left(\mathsf{P}_{3} \right) = \, \ldots \, = q \! \left(\mathsf{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

\displaystyle q^{\mathsf{P}} = \sum_{\it{k} \mathrm{=1}}^{8} q \! \left( \mathsf{P}_{k} \right) = 8 \, q \! \left(\mathsf{P}_{1}\right)

But 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

q^{\mathsf{P}} =  \Delta n^{\mathsf{T}} \! \left(\mathsf{P}_{1}\right) - \Delta n^{\mathsf{B}} \! \left(\mathsf{P}_{1}\right) + \Delta n^{\mathsf{C}} \! \left(\mathsf{P}_{1}\right) - \Delta n^{\mathsf{S}} \! \left(\mathsf{P}_{1}\right)

The quark coefficients of  \mathsf{P}_{1} are always integers so  q^{\mathsf{P}} is determined by a sum or difference of integers. Therefore quark-level homogeneity constrains the charge on P as

q^{\mathsf{P}} = 0, \, \pm1, \,  \pm2, \, \pm3 \, \ldots

Similar reasoning leads to the same result for lepton and baryon numbers

L = 0, \, \pm1, \,  \pm2 \, \ldots

and

B = 0, \, \pm1, \,  \pm2 \, \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. Then 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.

Spatial homogeneity is suggested by the uniform distribution of woven motifs in this Indonesian textile.
Tampan, Paminggir people. Sumatra circa 1900, 43 x 46 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.