Ergonomic Quark Models – EthnoPhysics

Outline

To summarize, so far we have defined the elementary particles of EthnoPhysics by objectifying some common sensations as seeds. Then we considered pairs of seeds and called them quarks. We talked about how quarks are counted and conserved. And we characterized them by their internal energy and temperature.

Over the next few articles EthnoPhysics makes models of larger particles from collections of these quarks. Everything from photons, nuclei and atoms, to clocks and even reference-frames are introduced as compound quarks. We focus on models that are minimalist, intuitive and specifically designed for ease of use: Ergonomic quark models.

On this page we look at the shape of a compound quark. This leads to a discussion of radii, metrics and more generally to quark-space as a venue for the presentation and analysis of quark models. The assessment of shape ultimately leads to a definition of the work required to assemble a compound quark. But before all that, we start with a more differential notion of size, the enthalpy.

Enthalpy

Enthalpy represents the net difference in size between sensations felt on the left side versus the right side. The comparison may be considered over various classes of sensation. To be more specific, let a quark model of particle P be characterized by its quark coefficients $n$ and their associated internal energies $U \! .$ Recall that $\zeta$ is an index that notes quark-type. The enthalpy of P is defined as

$\displaystyle H \equiv \sum_{\zeta=1}^{16} \Delta n^{\zeta} U^{\zeta}$

We also make use of a partial sum over just the chemical quarks that is written as

$\displaystyle H_{\! chem} \equiv \sum_{\zeta=11}^{16} \Delta n^{\zeta} U^{\zeta}$

Enthalpy is conserved when compound quarks are formed or decomposed because it is defined by sums over quarks, and quarks are conserved.

The assumption of conjugate symmetry requires that the internal energy of ordinary-quarks and anti-quarks are the same as each other. Also, the net number of quarks $\Delta n$ , in particle $\mathsf{P}$ and its conjugate twin $\overline{\mathsf{P}}$ , are related by

$\Delta n^{\zeta} \mathsf{ ( P ) } = - \Delta n^{\zeta} \mathsf{ ( \overline{P} ) }$

So the enthalpy of a particle and its conjugate twin are related as

$H ( \mathsf{P} ) = - H ( \mathsf{\overline{P}} )$

The internal energy $U$ is defined from the specific energy $\widehat{E}$ of a particle. This specific energy represents the size of a perception. And quarks are objectified from thermal, visual, lateral and oral sensations. So a sensory interpretation of enthalpy is an awareness of size for all these kinds of sensations, net left-side from right. Next we use this notion of size to define the radius of a composite quark.

Enthalpy represents an extended notion of size somewhat like the radiating pattern of this Indonesian tampan. — Tampan, Paminggir people. Lampung region of Sumatra, circa 1900, 74 x 90 cm. From the collection of Vice-President Adam Malik, Jakarta. Photograph by D Dunlop.

Radii

Radii and radius-vectors are used to express spatial concepts like extension and containment. More exactly, consider a quark model of particle P that is characterized by some repetitive chain of events

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

where each orbital cycle is a bundle of quarks

$\mathsf{\Omega} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2} \; \ldots \; \mathsf{q}_{N} \right)$

that are described by their quark coefficients $n$ and internal energies $U \! .$ Recall that $H_{chem}$ notes the enthalpy of the chemical quarks in P. And let a constant $k_{\mathsf{F}}$ have a positive value, with units of a force. These quantities are used to describe the shape and extent of P as follows.

The core radius of particle P is given by

$\mathcal{R}_{core} \equiv \dfrac{H_{\! chem}}{k_{\mathsf{F}}}$

The magnetic radius of P is

$\rho_{m} \equiv \dfrac{ \Delta n^{\mathsf{A}} U^{\mathsf{A}} - \Delta n^{\mathsf{M}} U^{\mathsf{M}} }{ k_{\mathsf{F}} }$

the electric radius is defined by

$\rho_{e} \equiv \dfrac{ \Delta n^{\mathsf{G}} U^{\mathsf{G}} - \Delta n^{\mathsf{E}} U^{\mathsf{E}} }{ k_{\mathsf{F}} }$

and the central radius of P is given as

$\rho_{z} \equiv \, \mathcal{R}_{core} + \dfrac{ \Delta n^{\mathsf{U}} U^{\mathsf{U}} - \Delta n^{\mathsf{D}} U^{\mathsf{D}} }{ k_{\mathsf{F}} }$

The numbers $\rho_{m} \, ,$ $\rho_{e}$ and $\rho_{z}$ are called radial components of the radius vector of P which is defined as

$\overline{\rho} \equiv \left( \rho_{m}, \, \rho_{e}, \, \rho_{z} \right)$

We also define a radial-direction vector, which is a lot like a unit-vector

$\widehat{\rho} \equiv \begin{cases} \hspace{5px} (0,0,0) &\textrm{if} \hspace{5px} \overline{\rho} = (0,0,0) \\ \hspace{7px} \overline{\rho} \, / \, \| \overline{\rho} \| &\textrm{if} \hspace{5px} \overline{\rho} \ne (0,0,0) \end{cases}$

Radii are Conserved

The radius-vector $\overline{\rho}$ is conserved when particles are formed or decomposed. To see this consider, for example, just the magnetic-radius which was defined above as $\rho_{m} \! \equiv \! \left( \Delta n^{\mathsf{A}} U^{\mathsf{A}} \! - \! \Delta n^{\mathsf{M}} U^{\mathsf{M}} \right) \hspace{-3px} / k_{\mathsf{F}} .$ It can be rearranged as

$\rho_{m} = \left(\frac{U^{\mathsf{A}}}{k_{\mathsf{F}}}\right) n^{\overline{\mathsf{a}}} + \left(\frac{-U^{\mathsf{A}}}{k_{\mathsf{F}}}\right)n^{\mathsf{a}} + \left(\frac{-U^{\mathsf{M}}}{k_{\mathsf{F}}}\right) n^{\overline{\mathsf{m}}} + \left(\frac{U^{\mathsf{M}}}{k_{\mathsf{F}}}\right) n^{\mathsf{m}}$

where $U^{\mathsf{A}}$ , $U^{\mathsf{M}}$ and $k_{\mathsf{F}}$ are all constants. This expression shows that the magnetic-radius can be written as a sum of products between quark-coefficients and constants. This form is known as a linear equation . Therefore we may deploy all the usual techniques of linear algebra in its analysis. This applies to the other radial components as well. So in general, the radius-vector and its offshoots are treated as plain algebraic vectors in the following discussion.

Quarks are conserved when particles are formed or when they decay. So for any sort of quark noted by $\mathsf{q} \hspace{1px} ,$ and for any generic particles $\mathbb{X} ,$ $\mathbb{Y}$ and $\mathbb{Z} ,$ if

$\left\{ \mathbb{ X } , \mathbb{ Y } \rule{0px}{10px} \right\} \equiv \mathbb{ Z }$

then

$n^{\mathsf{q}} (\mathbb{X}) + n^{\mathsf{q}} (\mathbb{Y}) = n^{\mathsf{q}} (\mathbb{Z})$

These relationships can be applied to the magnetic-radius using the associative properties of addition to obtain

$\rho_{m} (\mathbb{X}) + \rho_{m} (\mathbb{Y}) = \rho_{m} (\mathbb{Z})$

Thus the magnetic-radius is conserved. And this holds for other radial-components as well. Then straightforward vector-addition implies that the radius-vector is also conserved

$\overline{\rho}^{\mathbb{X}} + \overline{\rho}^{\mathbb{Y}} = \overline{\rho}^{\mathbb{Z}}$

Furthermore, the generic particles $\mathbb{X}$ and $\mathbb{Y}$ might just be individual quarks, so by repetition, we know the radius-vector for any particle $\mathbb{Z}$ is given by a sum over the radius-vectors of its component quarks. We write

$\overline{\rho} \hspace{1px} ( \mathbb{Z} ) \, = \! \! \displaystyle{ \sum_{\; \mathsf{q} \, \in \, \mathbb{Z} } \overline{\rho}^{\, \mathsf{q}} }$

Twin Radii are Opposed

We have already remarked that by the anti-commutative property of subtraction the quark-coefficients of conjugate twins are related as $\Delta n^{\mathsf{Z}} ( \mathsf{P} ) \! = \! - \Delta n^{\mathsf{Z}} ( \mathsf{\overline{P}} ) \hspace{2px} .$ And since $H ( \mathsf{P} ) \! = \! - H ( \mathsf{\overline{P}} ) \hspace{2px}$ we also know $H_{chem} ( \mathsf{P} ) \! = \! - H_{chem} ( \mathsf{\overline{P}} ) \hspace{2px} .$ Then by substitution, particles and their conjugate twins have symmetrically opposed radius vectors

$\overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} )$

And symmetrically opposed radial-direction vectors

$\widehat{\rho} ( \mathsf{P} ) = - \widehat{\rho} ( \mathsf{\overline{P}} )$

Radii and Shape

Going forward, we use these radius-vectors to develop a description for the shape of a compound quark. As a starting example; if P has the same radii every cycle, then we call it a rigid particle.

Some particles are not very solid or exactly located. Then an analysis of shape might not provide a useful description. So instead we may use the following radii to describe their containment or range. The inner radius is defined as

$\mathcal{R}_{in} \equiv \sqrt{\dfrac{hc}{8 k_{\mathsf{F}}} \rule{0px}{14px} } \; \dfrac{ \, \left| \Delta n^{\mathsf{D}} \rule{0px}{9px} \right| \, }{8\pi}$

And the outer radius of P is

$\mathcal{R}_{out} \equiv \sqrt{ \dfrac{hc}{8 k_{\mathsf{F}}} \rule{0px}{14px} } \; \dfrac{ \, N^{\mathsf{D}} \, }{8\pi}$

The numbers $\mathcal{R}_{in}$ and $\mathcal{R}_{out}$ are used to describe spheres. So the symbol $\mathcal{R}$ is generically called a spherical radius.

We say that a particle is free when the inner-radius is small, but not zero. And the outer-radius is huge. Or more exactly, when $\left| \Delta n^{\mathsf{D}} \right| \! \gtrsim 8 \!$ and $N^{\mathsf{D}} \! \! \to \! \infty .$ But if $\left| \Delta n^{\mathsf{D}} \right| \! < \! 8$ then we say that a particle is not free.

Here is a sensory explanation of the radius-vector. In these formulae $\Delta n$ means that contributions from sensations on the right side are cancelled by sensations felt on the left. The radius-vector depends on their net size. Coefficients of baryonic quarks do not appear in these definitions, only dynamic quarks. So the radius-vector does not depend on thermal sensation. Instead, it represents visual sensations. And the null value for energy is referred to down-quarks which are objectified from black sensations. So overall, the radius-vector is interpreted as a description of visual size, relative to black sensations, net right from left.

Quark models introduce spatial concepts like radii and containment, perhaps somewhat like this rattan basket from Borneo. — Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. Photograph by D Dunlop.

Quark Space

We need some room to build quark models. So let’s start by defining an algebraic vector space made from the radius vectors $\overline{\rho}$ of some finite collection of particles $\mathsf{P}_{1}, \; \mathsf{P}_{2}, \; \mathsf{P}_{3} \ldots \mathsf{P}_{N} .$ This mathematical construction is generically written as

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

Radius-vectors are defined by describing sense perceptions, so ultimately $\mathbb{Q}$ is defined by sensation as well. We say that $\mathbb{Q}$ is a three-dimensional space because radius-vectors have three distinct components. And these components are distinct because they are defined from three different classes of sensation which may vary independently of each other.

The components of $\overline{\rho}$ are not Cartesian coordinates because they are defined by counting quarks. And also $\mathbb{Q}$ is expressly constructed to illustrate quark arrangements. So the vector-space $\mathbb{Q}$ is called a quark space. The following basis vectors are used to make general descriptions in quark-space. The magnetic axis is established with $\widehat{m} \equiv (1, 0, 0) \, ,$ the electric axis from $\widehat{e} \equiv (0, 1, 0)$ and the central axis by $\widehat{z} \equiv (0, 0, 1) \, .$ Then, any vector in $\mathbb{Q}$ can be expressed in terms of its radial components $\rho_{m}$ , $\rho_{e}$ and $\rho_{z}$ as

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

Quark-space is grainy because quark coefficients are always integers. And $\mathbb{Q}$ is not exactly right-angled. But these messy details are handled mathematically and in the following articles we make idealized quark models using conventional graphics. Next, a bit more about metrics.

Metrics

EthnoPhysics begins with the premise that we can understand ordinary space by describing sensation. This is done by objectifying reference sensations as quarks and then considering spaces to be mathematical sets of quarks. Different kinds of space are defined from different distributions of quark types. And empty space is not defined. So overall, our understanding of a space is based on the particles that are in the space. Specifically, we assess the shape or radii of these particles.

Usually a radius is quantified by its length. But a full discussion of length requires some ideas that are initially quite vague. So to begin, we define a radius vector $\overline{\rho}$ from quark inventories. Using this vector, a quark space $\mathbb{Q}$ is defined for some finite collection of particles $\mathsf{P}_{1}, \ \mathsf{P}_{2}, \ \mathsf{P}_{3} \; \ldots \; \mathsf{P}_{N} \, .$ This space is generically written as

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

$\mathbb{Q}$ is then characterized using a statistical account of commonalities and variation in the shape of these particles. Shape is described by the radius vector $\overline{\rho} \equiv \left( \rho_{m}, \, \rho_{e}, \, \rho_{z} \right).$ So a generic radial component is written as $\rho_{\alpha}$ where $\alpha \in \{ m, \, e, \, z \}.$ The arithmetic mean of $\rho_{\alpha}$ taken over all the particles in $\mathbb{Q}$ is noted by

$\displaystyle \widetilde{\rho}_{\alpha} = \dfrac{1}{N} \sum_{i=1}^{N} \rho_{\alpha}^{\, i}$

The standard deviation of the variation in $\rho_{\alpha}$ is

$\sigma_{\alpha} = \sqrt{ \dfrac{1}{N} \sum_{i=1}^{N} \left( \rho_{\alpha}^{\, i} - \widetilde{\rho}_{\alpha} \right)^{2} \; }$

And the covariance between pairs of radial components is given by

$\displaystyle \mathrm{cov}_{\alpha \beta} = \dfrac{1}{N} \sum_{i=1}^{N} \left( \rho_{\alpha}^{\, i} - \widetilde{\rho}_{\alpha} \rule{0px}{11px} \right) \! \left( \rho_{\beta}^{\, i} - \widetilde{\rho}_{\beta} \right)$

where $\beta \in \{ m, \, e, \, z \} .$ Next we use these descriptive statistics to specify metrics for various quark-spaces.

A quark metric is a set of six numbers that are later used to express notions about distance or separation in quark-space. They are constants, so we note them using the letter $k$ with two subscripts. For a generic quark-space $\mathbb{Q}$ , a generic quark-metric is given by

$k_{\alpha \beta} \equiv \dfrac{ \; \mathrm{cov}_{\alpha \beta}^{\mathbb{Q}} \; }{ \mathrm{cov}_{zz}^{\mathbb{Q}} }$

These numbers are symmetric such that $k_{\alpha \beta} = k_{\beta \alpha} \hspace{0.4px} .$ And they are scaled relative to variations in the central radius. So there are no units, and $k_{zz}$ is always equal to one.

The Terrestrial Metric
central component	$k_{zz}$	1
electric component	$k_{ee}$	-0.0152286648
magnetic component	$k_{mm}$	+0.7453740340
electromagnetic component	$k_{em}$	-0.9292374609
electroweak component	$k_{ez}$	+1.5428187522
magnetoweak component	$k_{mz}$	-1.2742065050

For the important special case where every quark on planet Earth is included in $\mathbb{Q} ,$ we identify the terrestrial metric. Recall that touching the Earth is a reference sensation for EthnoPhysics. So we presume that the Earth is implicitly part of every description of human experience. Moreover, the Earth is very large and stable. So we assume that the law of large numbers guarantees that sensory covariances have fixed values that do not change over geological time-scales. Then a unique metric can be specified by

$k_{\alpha \beta} \! \left( \mathsf{terrestrial} \rule{0px}{9px} \right) \equiv \dfrac{ \; \mathrm{cov}_{\alpha \beta} \! \left( \mathsf{Earth} \rule{0px}{9px} \right) \; }{ \mathrm{cov}_{zz} \! \left( \mathsf{Earth} \rule{0px}{9px} \right) }$

The numbers shown in the accompanying table are called the terrestrial metric. The labels given to these numbers can be arbitrary, but the names shown in the table have been carefully chosen for their mnemonic value. They will smoothly fit into our traditional ways of discussing physics and be easy to remember.

The Euclidean Metric
$k_{zz} \equiv \mathsf{1}$	$k_{xy} = \mathsf{0}$
$k_{xx} = \mathsf{1}$	$k_{xz} = \mathsf{0}$
$k_{yy} = \mathsf{1}$	$k_{yz} = \mathsf{0}$

We also consider the possibility that a space may be defined from some set of particles that are restricted to certain shapes or sizes. Then a statistical analysis could yield a different metric for $\mathbb{Q} .$

For example, in our classrooms and laboratories we usually assume that space is filled with room-temperature atoms, not just any composite quark. An extended analysis of this sort of space is detailed later. But the overall result is easily summarized as the Euclidean metric shown in the accompanying table. This Euclidean metric is very different from the terrestrial metric because it does not have any negative numbers. A space that uses a terrestrial metric is not Euclidean.

Norms

Consider some particle P and its radius vector noted as $\overline{\rho} = \left( \rho_{m}, \rho_{e}, \rho_{z} \right)$ . These three numbers can be compressed into a single useful quantity, called the norm of the radius-vector, which is marked by $\left\| \, \overline{\rho} \, \right\|$ and defined as

$\left\| \, \overline{\rho} \, \right\| \equiv + \sqrt{ \, \left| \, \overline{\rho} \cdot \overline{\rho} \, \right| \; }$

The plus-sign indicates that we always take the positive root, so norms are never negative. The phrase $\overline{\rho} \cdot \overline{\rho}$ refers to a dot product. Evaluating this dot product depends on some choice for a metric. If the metric is Euclidean, then $\overline{\rho} \cdot \overline{\rho}$ is always positive, and taking the absolute-value is not necessary. But if the terrestrial metric is used, then $k_{\alpha \beta} = k_{\beta \alpha}$ and the norm can be written-out as

$\begin{equation*} \left\| \, \overline{\rho} \, \right\| = \left| \begin{split} & \; k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} \\ & + 2 k_{em} \rho_{e} \rho_{m} + 2k_{mz}\rho_{m} \rho_{z} \\ & \hspace{30px} + 2 k_{ez}\rho_{e} \rho_{z} \; \end{split} \, \right|^{\frac{1}{2}} \end{equation*}$

Recall that particles and their conjugate twins have opposing radius-vectors such that $\overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ) .$ And note that all radial components appear as paired factors in the foregoing formula. So sign differences between particles and their twins get cancelled, and both have the same norm. We can write

$\left\| \; \overline{\rho} ( \mathsf{P} ) \, \rule{0px}{12px} \right\| = \left\| \; \overline{\rho} ( \mathsf{\overline{P}} ) \rule{0px}{12px} \, \right\|$

Surfaces

Here is different way to describe a particle with just a single quantity. Let P have a radius-vector noted by $\overline{\rho} = \left( \rho_{m}, \rho_{e}, \rho_{z} \right) .$ Then another number $A,$ called the surface area of P, is defined by

$A \equiv 4\pi \left| \, \overline{\rho} \cdot \overline{\rho} \, \rule{0px}{10px} \right|$

This surface-area is related to the norm of the radius-vector as

$A = 4\pi \left\| \, \overline{\rho} \, \right\|^{2}$

So $A$ takes on the same value that it would have if P was a classical geometric sphere of radius $\left\| \, \overline{\rho} \, \right\|$ in an ordinary Euclidean space. This is true by definition regardless of P’s actual shape which may be quite distorted. If P is in a quark-space where the terrestrial metric is employed, then the surface area is given by

$\begin{equation*} A = 4\pi \left| \begin{split} & \; k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} \\ & + 2 k_{em} \rho_{e} \rho_{m} + 2k_{mz}\rho_{m} \rho_{z} \\ & \hspace{30px} + 2 k_{ez}\rho_{e} \rho_{z} \; \end{split} \, \right| \end{equation*}$

Recall that particles and their conjugate twins have opposing radius-vectors such that $\overline{\rho} ( \mathsf{P} ) \! = \! - \overline{\rho} ( \mathsf{\overline{P}} ) .$ And note that all radial components appear as paired factors in this expression for $A$ . So sign differences get cancelled and both particles have the same surface area; $A ( \mathsf{P} ) \! = \! A ( \mathsf{\overline{P}} ) .$

If some particle definitely has a surface, then $A \! \ne \! 0$ and we can consider more detail. For example, the surface-area quantifies a boundary between the inside and the outside of particle P. Any quarks associated with P are presumably squeezed together more tightly if P’s surface-area decreases. This inverse relationship is used to define the interior pressure of P by

$P \equiv 3k_{\mathsf{F}} / \! A$

We may also reckon that P’s surface has some kind of orientation by defining a surface direction as

$\epsilon_{\! A} \equiv \dfrac{ \, \overline{\rho} \cdot \overline{\rho} \, }{ \, \left| \, \overline{\rho} \cdot \overline{\rho} \, \right| \, } = \pm 1$

When $\epsilon_{\! A} \! = \! 1$ we say that the surface of P has a direction that is facing outside. For a Euclidean metric, the surface of P is always directed outward. Conversely, if $\epsilon_{\! A} = -1$ then we say that the surface of P is facing inside. In quark-space, some components of the terrestrial metric are negative numbers. So some well known particles do have inward facing surfaces. For example, look at this quark model for 𝝎, the omega meson.

The surface-direction is a binary quantity that could be used to express the difference between a pebble and a bubble. Or for a more visual example, $\epsilon_{\! A}$ might represent the difference between inside and outside surfaces of a woven basket¹Here are some examples of contrasting inside/outside weavings. which tend to look like photographic negatives of each other. In any case, since $\overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} )$ we obtain $\epsilon_{\! A} ( \mathsf{P} ) = \epsilon_{\! A} ( \mathsf{\overline{P}} ) .$

A Simple Particle Model

Here is a simple model to represent a generic particle in quark-space: We are going to depict particles as concentric spheres. But to begin, we have to ask what it even means to be a sphere in a grainy, non-Euclidean framework like quark-space?

Remember that for EthnoPhysics any particle P is characterized by some repetitive chain of events written as $\Psi = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \; \ldots \; \mathsf{\Omega}_{i}\; \ldots \; \right)$ . And also recall that each individual cycle $\mathsf{\Omega}_{i}$ is then described by a few different types of radii as discussed above. Specifically, each cycle has a radius-vector $\overline{\rho}_{i} \, .$ And these vectors point in different directions depending on the quarks in $\mathsf{\Omega}_{i} .$

Dozens of dots are arranged to show the surface of a geometric sphere. — A sphere in quark-space ?

So we visualize spheres in quark-space as compound events like $\Psi ,$ where all individual cycles share the same value for a spherical radius noted by $\mathcal{R} \hspace{0.5px} .$ Then a figure can be drawn where green dots that represent each individual cycle are pasted on the brown surface of an ordinary geometric sphere which has the same value of $\mathcal{R}$ for its radius.

This is a rough heuristic image. It might be better to say P is a spheroid since quark-space is not Euclidean. But complicated details are handled mathematically, and we use conventional graphics to display idealized forms. Thus spheres are pictured as sets of dots, even though many of these dots may be required for P to look like a ball, rather than say, a spherical cloud.

This image shows the three zones of a particle; core, field and outside. — Three zones around a particle.

Based on this notion of a sphere, we present a generic model of a physical particle as two concentric spheres. The model is illustrated in the adjoining diagram. The two spheres outline boundaries for three zones. These zones are called; the core of P, the field of P, and a remote exterior dubbed the outside of P. Next we make this simple prototype more exact by mathematically specifying these spheres and their radii.

Let us consider the core of P as an individual particle so that its repeated cycles $\mathsf{\Omega}_{i}$ have their radius-vectors noted by $\overline{\rho}_{core} \, .$ Then $\left\| \, \overline{\rho}_{\! core} \, \right\|_{i}$ marks the norm of the radius-vector of the $i$ ^th cycle.

Since the core of P is supposed to be spherical, all its cycles presumably have the same value for their radii. So we assume that $\left| \mathcal{R}_{core} \right| = \left\| \, \overline{\rho}_{\! core} \, \right\|_{i}$ for all $i .$ But $\mathcal{R}_{core}$ is defined by the ratio of $H_{\! chem}$ to $k_{\mathsf{F}} \, .$ So the simple particle model requires that radius-vectors are constrained by a condition that

Here is a cross-sectional diagram showing the regions and radii of a particle. — A particle’s cross-section showing its zones and radii.

$\left\| \, \overline{\rho}_{\! core} \, \right\|_{i} \, = \, \dfrac{ \, \left| H_{\! chem} \right| \, }{ k_{\mathsf{F}} }$ for all $i$

This is an equation for a spherical surface in quark-space. The surface area of the core is then determined as

$A_{core} = 4\pi \mathcal{R}_{core}^{2} = \dfrac{4\pi}{k_{\mathsf{F}}^{2}} H_{\! chem}^{2}$

Thus the core of P is positioned and quantified in the simple particle model. The same reasoning also applies to the outer sphere where

$\mathcal{R}_{out} \equiv \sqrt{ \dfrac{hc}{8 k_{\mathsf{F}}} } \dfrac{ \, N^{\mathsf{D}} \, }{8\pi}$

and

$A_{out} = 4\pi \mathcal{R}_{out}^{2} = \dfrac{ \, hc [ N^{\mathsf{D}} ]^{2} }{ 128\pi k_{\mathsf{F}}}$

On the inside of this outer sphere, as defined by the union of the core and field zones, we may also describe events using the inner-radius $\mathcal{R}_{in}$ discussed above. Recall that

$\mathcal{R}_{in} \equiv \sqrt{\dfrac{hc}{8 k_{\mathsf{F}}} } \; \dfrac{ \, \left| \Delta n^{\mathsf{D}} \right| \, }{8\pi}$

$A_{in} = 4\pi \mathcal{R}_{in}^{2} = \dfrac{ \, hc | \Delta n^{\mathsf{D}} |^{2} }{128\pi k_{\mathsf{F}}}$

The inner-radius is useful because ${\Delta}n^{\mathsf{D}} \equiv n^{\mathsf{\overline{d}}} - n^{\mathsf{d}}$ and $N^{\mathsf{D}} \equiv n^{\mathsf{\overline{d}}} + n^{\mathsf{d}} \, .$ These quark coefficients cannot be negative. So $0$ $\le \left| \Delta n^{\mathsf{D}} \right| \le N^{\mathsf{D}}$ and it is always true that

$0$ $\le \mathcal{R}_{in} \le \mathcal{R}_{out}$

Quark space presents a structured array of sensation, perhaps somewhat like the way this tampan from Sumatra displays nine dragons. — Nine Dragon Tampan, Paminggir people. Sumatra circa 1900, 41 x 43 cm. Photograph by D Dunlop.

Particle Volume

Consider some particle P described by its radius vector $\overline{\rho} .$ We may also characterize P using a number called the volume defined by

$V \equiv \dfrac{ 4 \pi }{3} \left\| \, \overline{\rho} \, \right\|^{3}$

This is true by definition, even if P is not a sphere, and even if the descriptive framework is not Euclidean. But if P happens to be a sphere in an ordinary classroom or laboratory, then $V$ is the same number as Archimedes’ value for a perfect sphere of radius $\left\| \, \overline{\rho} \, \right\|$ in classical Euclidean geometry.

Bound Particles

Now equipped with a simple model of a physical particle, we ask: What holds this particle together? How can some individual quarks become bound into a single unified object, yet others do not? A generic approach to questions like this involves energy analysis. And we start by remarking that the most important sort of energy associated with holding a particle together is marked by $\mathcal{B}$ and defined by

$\mathcal{B} \equiv -k_{\mathsf{F}} \rho_{z}$

where $k_{\mathsf{F}}$ is a positive constant introduced earlier and $\rho_{z} \hspace{1px}$ is the central radius. This quantity could be named something like “the stored potential energy due to a centripetal binding force”. But that is unwieldy, so we prefer to call $\mathcal{B}$ the central binding energy.

Recall that the central component of the terrestrial metric $k_{zz}$ is equal to one. So for any convention about the sign of $\mathcal{B}$ we can always write

$\mathcal{B}^{\, 2} = k_{\mathsf{F}}^{2} \, k_{zz} \rho_{z}^{2}$

By the foregoing definition, if the central-radius is greater than zero, then $\mathcal{B}$ is negative. Negative binding-energies imply that bound particles have less energy than their separately unbound constituents. And lower energies are associated with more common, long-lived particles. So the sign of $\rho_{z} \hspace{1px}$ is relevant for understanding particle stability. This central-radius has already been defined by

$\rho_{z} \equiv \, \mathcal{R}_{core} + \dfrac{ \Delta n^{\mathsf{U}} U^{\mathsf{U}} - \Delta n^{\mathsf{D}} U^{\mathsf{D}} }{ k_{\mathsf{F}} }$

where up-quark coefficients are noted by $\Delta n ^{\mathsf{U}} ,$ down-quarks by $\Delta n ^{\mathsf{D}} ,$ and their internal energies by $U^{\mathsf{U}}$ and $U^{\mathsf{D}} .$ Recall that $\mathcal{R}_{core}$ notes the core radius of P. It was defined by the ratio $H_{\! chem} / k_{\mathsf{F}}$ where $H_{chem}$ marks the enthalpy of any chemical quarks in P. So, $\mathcal{B}$ can be put in terms of quark coefficients as

$\mathcal{B} = \Delta n^{\mathsf{D}} U^{\mathsf{D}} - H_{\! chem} - \Delta n^{\mathsf{U}} U^{\mathsf{U}}$

The summands in this equation are enormously different from each other because the internal-energy of an up-quark is about sixteen orders-of-magnitude larger than a down-quark. The chemical quarks are somewhere in between. To evaluate the chemical term, we consider that

$\begin{equation*} H_{\! chem} \, \simeq \, \left( \begin{split} \text{\sf{number}} \hspace{5px} \\ \text{\sf{of}} \hspace{18px} \\ \, \text{\sf{molecules}} \\ \text{\sf{in}} \hspace{3px} \mathsf{P} \hspace{13px} \\ \end{split} \right) \, \times \, \left( \begin{split} \text{\sf{average}} \hspace{6px} \\ \; \text{\sf{chemical}} \; \\ \text{\sf{enthalpy}} \hspace{3px} \\ \text{\sf{per}} \hspace{17px} \\ \text{\sf{molecule}} \hspace{1px} \\ \end{split} \right) \end{equation*}$

For a rough estimate, we assume that all of the molecules in P are hydrogen gas molecules. Then the average chemical enthalpy per molecule is just equal to the chemical enthalpy of $\mathrm{H}_{\mathsf{2}} \, .$

We have a working quark model for diatomic hydrogen. It gives a bond dissociation energy for these molecules of $D_{\! \mathsf{o}} ( \mathrm{H}_{\mathsf{2}} ) = 4.4772 \; \mathsf{(eV)} .$ And for simple diatomic molecules, the dissociation-energy of their bond is the same as the chemical enthalpy of the molecule. So the average chemical enthalpy per molecule is approximated by $D_{\! \mathsf{o}} \hspace{0.5pt} .$

Also we remark that atoms are baryons. And there are two atoms in each diatomic molecule. So half the baryon number $B$ is a proxy for the number of molecules. Thus we obtain

$\mathcal{B} \, \simeq \, \Delta n^{\mathsf{D}} U^{\mathsf{D}} \! - \frac{1}{2} B D_{\! \mathsf{o}} - \Delta n^{\mathsf{U}} U^{\mathsf{U}}$

The foregoing relationship is just an approximation, but it is good enough to assess the sign of $\mathcal{B}$ . This sign is important for explaining the distinction between matter and anti-matter. So next we go into more detail about the central binding-energy and its sign. Here are four mutually exclusive categories that classify particles based on $\mathcal{B}$ .

Strong Bonding

We say that any particle containing unbalanced up-quarks is held together by strong bonds, or more exactly, by strong bonding along the central axis. Mathematically, the category is defined by a condition that $\Delta n^{\mathsf{U}} \! \ne 0 .$ Particles in this class are outstanding because the internal-energy of an up-quark is about a billion times larger than any chemical quark. And about $10^{16}$ times larger than a down-quark. So if there are any net up-quarks in a particle, they completely dominate the foregoing approximation such that

$\mathcal{B} \simeq -\Delta n^{\mathsf{U}} U^{\mathsf{U}} \; \; \; \; \sim \! 10^{9} \hspace{0.5px} \mathsf{(eV)}$

Then since $U^{\mathsf{U}}$ is a positive constant, the sign of $\mathcal{B}$ is given by

$\epsilon_{\! \mathcal{B}} \equiv \dfrac{\mathcal{B} }{ \, \left| \mathcal{B} \hspace{0.5pt} \right| \, } = \dfrac{ -\Delta n^{\mathsf{U}} U^{\mathsf{U}} }{ \, \left| -\Delta n^{\mathsf{U}} U^{\mathsf{U}} \right| \, } = - \dfrac{\Delta n^{\mathsf{U}} }{ \, \left| \Delta n^{\mathsf{U}} \right| \, }$

There are many particles that are joined together by this kind of bonding. Most noteworthy are the electron 𝙚 and the hydrogen atom 𝐇. But also including the other leptons 𝝁 𝞽 and 𝞶, the baryons 𝞚 𝞢 𝞝 and 𝞨, plus the mesons 𝞰 𝞺 𝙆^◦ 𝙛 𝘿 𝘽 𝙅 𝝍 𝟂 and 𝞆.

Chemical Bonding

We say that chemical bonding holds together any baryon or baryonic composite that has all its up-quarks balanced. Mathematically, the category is defined by conditions that $\Delta n^{\mathsf{U}} \! = 0$ and $B \! \ne 0$ where $B$ is the baryon number. So up-quarks are irrelevant, and the binding-energy is determined by down-quarks and chemical quarks. But the internal-energy of a down-quark is roughly a million times smaller than a chemical quark. So chemical-quarks are paramount and

$\mathcal{B} \, \simeq \, - \frac{1}{2} B D_{\! \mathsf{o}} \; \; \; \; \sim \! 10 \hspace{0.5px} \mathsf{(eV)}$

Then since $D_{\! \mathsf{o}}$ is always positive, the sign of $\mathcal{B}$ is exactly opposite to the sign of the baryon number

$\epsilon_{\! \mathcal{B}} \equiv \dfrac{\mathcal{B} }{ \, \left| \mathcal{B} \hspace{0.5pt} \right| \, } = \dfrac{ - \frac{1}{2} B D_{\! \mathsf{o}} }{ \, \left| - \frac{1}{2} B D_{\! \mathsf{o}} \right| \, } = - \dfrac{ B}{ \, \left| B \right| \, }$

Protons 𝙥, neutrons 𝙉 and the delta baryons 𝞓, are all tied together by this sort of bonding.

Kaonic Bonding

This category is defined by the conditions that $\Delta n^{\mathsf{U}} \! = \! 0$ and $B \! = \! 0 ,$ but that $\Delta n^{\mathsf{D}} \! \ne 0 .$ That is, non-baryons with extra down-quarks, but no net up-quarks. These particles are said to be held together by kaonic bonding. Under these conditions $\mathcal{B}$ can be approximated as

$\mathcal{B} \simeq \Delta n^{\mathsf{D}} U^{\mathsf{D}} \; \; \; \; \sim \! 10^{-5} \hspace{0.5px} \mathsf{(eV)}$

Then since $U^{\mathsf{D}}$ is a negative constant, the sign of $\mathcal{B}$ is given by

$\epsilon_{\! \mathcal{B}} \equiv \dfrac{\mathcal{B} }{ \, \left| \mathcal{B} \hspace{0.5pt} \right| \, } = \dfrac{\Delta n^{\mathsf{D}} U^{\mathsf{D}}}{ \, \left| \Delta n^{\mathsf{D}} U^{\mathsf{D}} \right| \, } = - \dfrac{ \Delta n^{\mathsf{D}} }{ \, \left| \Delta n^{\mathsf{D}} \right| \, }$

This sort of binding technically includes photons. But photons have no mass, and no well-defined position when considered as individuals. So solitary photons are not usually considered like other bound particles. In any case, the only material quanta in this eponymous category are the charged kaons 𝙆^±. There are about two dozen of them.

Thermoelectric Bonding

Finally we consider particles for which $\Delta n^{\mathsf{U}} \! = \! 0 \, ,$ $B \! = \! 0$ and $\Delta n^{\mathsf{D}} \! = \! 0 \, .$ The shape of these particles does not extend along the central axis, and $\rho_{z} \! =0 .$ So their central binding-energy is null

$\mathcal{B} = 0 \; \; \mathsf{(eV)}$

And

$\epsilon_{\! \mathcal{B}} \equiv \dfrac{\mathcal{B} }{ \, \left| \mathcal{B} \hspace{0.5pt} \right| \, } = \text{\sf{not defined}}$

Particles in this category are not bound by a strong central force. Instead, we say that thermal or electromagnetic effects hold them together with thermoelectric bonding. This category includes the mesons $\text{\textsf{\textbf{\textsl{\large{a}}}}}$ $\text{\textsf{\textbf{\textsl{b}}}}$ $\pi$ $\mathit{\phi}$ $\text{\textsf{\textbf{\textsl{h}}}}$ $\text{\textsf{\textbf{\textsl{X}}}}$ ϒ and also the big electromagnetic bosons 𝙒 𝙕 and 𝙃. The binding for most of these particles is described by their Coulomb energy. But the pions $\pi^{\mathsf{0}}$ and $\pi^{\pm}$ are thermally united. The meson $\text{\textsf{\textbf{\textsl{a}}}}_{\mathsf{0}} \mathsf{(1450)}$ is electrically tied. Whereas the meson $\mathit{\phi}_{\mathsf{3}} \mathsf{(1850)}$ and the meson $\text{\textsf{\textbf{\textsl{b}}}}_{\mathsf{1}} \mathsf{(1235)}$ are magnetically bonded.

$\rule{200px}{2px}$

The main flow of development here at EthnoPhysics is toward Newtonian mechanics. So next we are going to talk about work. But physics has many interesting branches. And there are some important particle characteristics related to rotation and electromagnetism that unfold from here. Details can be found on these pages.

Rotation

Achromatic visual sensation, or greyness, is identified as a key to understanding particle rotation. Spin angular momentum is discussed. Chirality and handedness are defined.

Electromagnetism

Chromatic visual sensations are analyzed. Rudimentary electromagnetic polarities, charges and potential-energies are defined.

Work

Consider some particle P described by the repetitive chain of events $\Psi = ( \, \mathsf{\Omega}_{1} , \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \; \ldots \; )$ . And let P have a radius vector written as $\overline{\rho} = ( \rho_{m}, \rho_{e}, \rho_{z} )$ . Then the work required to assemble the quarks in P is defined by

$W \equiv k_{\mathsf{F}} \sqrt{ \, \overline{\rho} \cdot \overline{\rho} \; }$

For particles that have surfaces, $A \! \ne \! 0$ and the surface direction $\epsilon_{\! A}$ is well defined. Then the work can also be expressed as

$W = k_{\mathsf{F}} \sqrt{ \epsilon_{\! A} \left| \, \overline{\rho} \cdot \overline{\rho} \, \right| \, } = \sqrt{ \epsilon_{\! A} \, } \; k_{\mathsf{F}} \! \left\| \, \overline{\rho} \, \right\| = \sqrt{ \epsilon_{\! A} \, } \; PV$

where $P$ is the interior pressure of particle P. And $V$ is its volume.

Particles and their anti-particles have opposing radius-vectors, that is $\overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ) .$ But any negative-signs in paired factors for the dot-product cancel each other. So the work required to assemble the quarks in a particle is the same as the work needed to build its conjugate twin; $W \! ( \mathsf{P} ) = W \! ( \mathsf{\overline{P}} ) .$

Recall that the constant $k_{\mathsf{F}}$ was introduced earlier to relate the internal energy of quarks to their radii. So $W$ is just another, slightly different representation for the internal-energy of the quarks in P. Nonetheless, $W$ has an essential role in our narratives about mechanics. We use it as an elementary verb like this: If any quarks are absorbed or emitted by P, then $\mathsf{\Omega}$ changes into another bundle $\mathsf{\Omega}^{\prime}$ and $W$ shifts to $W^{\prime} .$ If particle radii change, then the interaction is said to work on P by changing its shape. And this effort is quantified by $\Delta W \! \! = \! W^{\prime} \! \! - \! W .$ Thus the number $W$ is invested with extra physical relevance.

The foregoing formulae show that $W$ might be an imaginary number because sometimes the surface²The distinction between surface-directions may be expressed using toggle functions

$\delta_{out} \equiv \dfrac{\left| \left( \overline{\rho} \cdot \overline{\rho} \right) + \left| \, \overline{\rho} \cdot \overline{\rho} \, \right| \right|}{ 2\left| \, \overline{\rho} \cdot \overline{\rho} \, \right| }$

and

$\delta_{in} \equiv \dfrac{\left| \left( \overline{\rho} \cdot \overline{\rho} \right) - \left| \, \overline{\rho} \cdot \overline{\rho} \, \right| \right|}{ 2\left| \, \overline{\rho} \cdot \overline{\rho} \, \right| }$

to define

$W_{\! out} \equiv \delta_{out} \, PV$

and

$W_{\! in} \equiv \delta_{in} \, PV$

Here $W_{\! out}$ is called the work done from the outside of a particle. And $W_{\! in}$ is called the work done from the inside. Then, for $i^{2} = -1$ we can write an explicitly complex expression for the work as

$W = W_{\! out} + i \hspace{0.5pt} W_{\! in}$

of P may be facing inward. But $W^{2}$ is still important for determining the mass of a particle, so next we look at how the square of the work relates to binding and potential energies. Following immediately from definition we have

$W^{2} = \left( k_{\mathsf{F}} \sqrt{ \, \overline{\rho} \cdot \overline{\rho} \; } \right)^{2} = k^{2}_{\mathsf{F}} \left( \overline{\rho} \cdot \overline{\rho} \right)$

If the terrestrial metric is used, then the dot-product can be written-out to obtain

$\begin{equation*} W^{2} = k^{2}_{\mathsf{F}} \, \left( \begin{split} & \; k_{mm} \rho_{m}^{2} + k_{ee} \rho_{e}^{2} + k_{zz} \rho_{z}^{2} \\ & + 2 k_{em} \rho_{e} \rho_{m} + 2k_{mz}\rho_{m} \rho_{z} \\ & \hspace{30px} + 2 k_{ez}\rho_{e} \rho_{z} \; \end{split} \right) \end{equation*}$

This equation is the formula employed for numerical calculations in spreadsheets. It is direct and useful. But to go further toward a physical understanding of these terms, recall that $\mathcal{B}^{\, 2} \! = k_{\mathsf{F}}^{2} \, k_{zz} \rho_{z}^{2} \, .$ And note that this relationship can be substituted into the expression for $W^{2}$ given above. Furthermore, we can also substitute a variety of potential energies for the other terms. For example, the electric potential energy $\mathcal{U}_{e}$ can be squared to obtain $\mathcal{U}_{e}^{2} = k_{\mathsf{F}}^{2} \, k_{ee} \rho_{e}^{2} \, .$ So by substitution

$\begin{equation*} W^{2} = \left( \begin{split} & \; \mathcal{U}_{m}^{2} + \mathcal{U}_{e}^{2} + \mathcal{B}^{\, 2} \\ & \hspace{5px} + \mathcal{U}_{em}^{2} + \mathcal{U}_{mw}^{2} \\ & \hspace{22px} + \mathcal{U}_{ew}^{2} \; \end{split} \; \right) \end{equation*}$

But from discussions about the Coulomb energy $\mathcal{U}_{\! Coul}$ and the weak energy $\mathcal{U}_{\! weak}$ we have

$\mathcal{U}_{\! Coul}^{2} \! = \mathcal{U}_{e}^{2} + \mathcal{U}_{em}^{2} + \mathcal{U}_{m}^{2}$

and

$\mathcal{U}_{\! weak}^{2} \! = \mathcal{U}_{mw}^{2} + \mathcal{U}_{ew}^{2}$

And so, again by substitution, we can express the square of the work as

$W^{2} = \mathcal{B}^{2} + \mathcal{U}_{\! Coul}^{2} + \mathcal{U}_{\! weak}^{2}$

Thus the work required to assemble the quarks in a particle has three components that are associated with binding, Coulomb and weak energies. And later, we discuss various types of force that can be linked to each of these energies. So roughly speaking, we say that building a particle requires that work is done by these forces.

The work required to construct this 19th century tampan from Sumatra is suggested by extensive and layered chromatic patterns. — Tampan, Paminggir people. Lampung region of Sumatra 19th century, 70 x 70 cm. Photograph by D Dunlop.

Reliable Clocks

Clocks can be understood as collections of achromatic visual sensations. Historical order and solar clocks are discussed. Particle phase is defined.

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References
1	Here are some examples of contrasting inside/outside weavings.
2	The distinction between surface-directions may be expressed using toggle functions $\delta_{out} \equiv \dfrac{\left\| \left( \overline{\rho} \cdot \overline{\rho} \right) + \left\| \, \overline{\rho} \cdot \overline{\rho} \, \right\| \right\|}{ 2\left\| \, \overline{\rho} \cdot \overline{\rho} \, \right\| }$ and $\delta_{in} \equiv \dfrac{\left\| \left( \overline{\rho} \cdot \overline{\rho} \right) - \left\| \, \overline{\rho} \cdot \overline{\rho} \, \right\| \right\|}{ 2\left\| \, \overline{\rho} \cdot \overline{\rho} \, \right\| }$ to define $W_{\! out} \equiv \delta_{out} \, PV$ and $W_{\! in} \equiv \delta_{in} \, PV$ Here $W_{\! out}$ is called the work done from the outside of a particle. And $W_{\! in}$ is called the work done from the inside. Then, for $i^{2} = -1$ we can write an explicitly complex expression for the work as $W = W_{\! out} + i \hspace{0.5pt} W_{\! in}$