A Space for Quark Models
WikiMechanics begins with the premise that we can understand ordinary space by describing sensation. And, we also need some room to build quark models. So let's start by defining an algebraic vector spaceXlink.png made from the radius vectors $\, \overline{\rho}$ of some collection of particles $\, \mathbf{P}$1, $\mathbf{P}$2, $\mathbf{P}$3$\mathbf{P}$N. This mathematical construction is generically written as

$\mathbb{Q} = \left\{ \overline{\rho}^{ 1 }, \ \overline{\rho}^{2}, \ \overline{\rho}^{3} \ldots \ \overline{\rho}^{\, i} \ldots \ \overline{\rho}^{\, N} \right\}$

Radius vectors are defined by describing sense perceptions, so ultimately $\mathbb{Q}$ is defined by sensation too. We say $\mathbb{Q}$ is three-dimensional because the three components of a radius vector represent three distinct classes of sensation which may change independently of each other. These three components are not the Cartesian coordinates that we usually use in geometry because instead of being associated with lengths, they are defined by counting quarks. Moreover, this space is explicitly constructed to keep track of quarks, so $\mathbb{Q}$ is called a quark space. The following basis vectorsXlink.png are used to make general descriptions; the magnetic axis is defined by $\hat{m} \equiv (1, 0, 0)$, the electric axis from $\hat{e} \equiv (0, 1, 0)$ and the polar axis by $\hat{z} \equiv (0, 0, 1)$. Then, any radius vector in $\mathbb{Q}$ can be expressed in terms of its components as

$\overline{\rho} = \rho_{m} \hat{m} + \rho_{e} \hat{e} + \rho_{ z} \hat{z}$

The norm of a radius vector is written without an overline as

$\rho \equiv \left\| \, \overline{\rho} \, \right\| = \sqrt{ \vphantom{\sum^{\, 2}} \ \rho_{m}^{ 2} k_{mm}+ \rho_{e}^{2} k_{ee} + \rho_{z}^{2} k_{zz} + 2 \rho_{e} \rho_{m} k_{em}+ 2 \rho_{e} \rho_{z} k_{ez} + 2 \rho_{m} \rho_{z} k_{mz} \ }$

metric.png

This function compresses all three components of a radius vector into a single quantity that depends on six numbers known collectively as the quark metric. Definition: if $\rho = 0$ then P is centered with respect to the electric, magnetic and polar axes. Theorem: particles and anti-particles have symmetrically opposed radius vectors $\overline{\rho} \left( \sf{P} \right) = - \overline{\rho} \left( \sf{\overline{P}} \right)$ so their norms are the same

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

Quark space is a poor approximation to ordinary space, it is coarse and grainy because quark coefficients are always integers. And $\mathbb{Q}$ is squashed in a funny way because metric components are not Euclidean. But these messy details are handled mathematically, and in the following articles we make idealized quark models using conventional graphics and freely available software.

Right.png Next step: the work required to build a quark model.
Summary
Noun Definition
Quark Space $\mathbb{Q} = \left\{ \overline{\rho}^{ 1 }, \ \overline{\rho}^{2}, \ \overline{\rho}^{3} \ldots \ \overline{\rho}^{\, k} \ldots \ \overline{\rho}^{\, N} \right\}$ 5-6
Noun Definition
Magnetic Axis $\hat{m} \equiv (1, 0, 0)$ 5-7
Noun Definition
Electric Axis $\hat{e} \equiv (0, 1, 0)$ 5-8
Noun Definition
Polar Axis $\hat{z} \equiv (0, 0, 1)$ 5-9
Noun Definition
Centered $\rho = 0$ 5-18
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