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Ergonomic Quark Models

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 observable 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 size and shape of a quark model. 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 spatial orientation of a model is defined. And related spatial concepts like angular momentum and charge symmetry are considered.

The discussion of shape ultimately leads to a definition of the work required to construct a quark model. But before all that, we start with a more collective 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. We write  U^{\zeta} = U^{\overline{\zeta}}. Also, the net number of quarks  \Delta n, in particle  \mathsf{P} and its anti-particle  \overline{\mathsf{P}}, are related by

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

So the enthalpy for a quark model of a particle and its anti-particle 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, somatic and taste 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 a net notion of size somewhat like the radiating patterns 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. For example, 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 chemical radius is

\rho_{chem} \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 polar radius of P is given as

\rho_{z} \equiv \, \rho_{chem} + \dfrac{ \Delta n^{\mathsf{U}} U^{\mathsf{U}} - \Delta n^{\mathsf{D}} U^{\mathsf{D}} }{ k_{\mathsf{F}} }

An ordered set of these radii defines the radius vector for a quark model of P as

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

The three components of  \overline{\rho} are conserved because they are defined from sums of quark coefficients, and quarks are conserved. So if some generic particles  \mathbb{X},  \mathbb{Y} and  \mathbb{Z} interact like \mathbb{ X } + \mathbb{ Y } \leftrightarrow \mathbb{ Z } then by the associative properties of addition, radii are related as

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

Also  \Delta n^{\mathsf{Z}} ( \mathsf{P} ) = - \Delta n^{\mathsf{Z}} ( \mathsf{\overline{P}} ) and  H ( \mathsf{P} ) = - H ( \mathsf{\overline{P}} ). So particles and anti-particles have symmetrically opposed radius vectors

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

Going forward, we use these radius-vectors to develop a description for the shape of a quark model. As a starting example; if P has the same radius vector for 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

\rho_{in} \equiv \dfrac{ \, \left| \Delta n^{\mathsf{D}} \rule{0px}{9px} \right| \, }{8}       \sqrt{  \dfrac{hc}{2\pi k_{\mathsf{F}}} \rule{0px}{14px} }

And the outer radius of P is

\rho_{out} \equiv \dfrac{ \, N^{\mathsf{D}} \, }{8} \sqrt{ \dfrac{hc}{2\pi k_{\mathsf{F}}} \rule{0px}{14px} }

We say that a particle is free when  \left| \Delta n^{\mathsf{D}} \right| \gtrsim 8 and  N^{\mathsf{D}} \to \infty . That is, when the inner radius is small and the outer radius is big. But if  \left| \Delta n^{\mathsf{D}}  \right| < 8 then we say that P is located at the core of a quark model.

XXX

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, only somatic and visual sensations.

Also 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 size for visual sensations, relative to black sensations, net right from left.

Quark models introduce spatial concepts like radii and containment, 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

EthnoPhysics 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 space made from the radius vectors  \overline{\rho}, for 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 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 quark models, so  \mathbb{Q} is called a quark space.

The following basis vectors 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 in quark space is given by

    \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_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}  \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right)^{\frac{1}{2}} \end{equation*}

This function compresses all three components of a radius vector into a single quantity that depends on six constant numbers noted by  k_{\alpha \beta} \; . These constants are known collectively as the terrestrial metric and they are discussed in detail later.

Particles and anti-particles have symmetrically opposed radius vectors, \overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ). So their norms are the same as each other

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

Quark space 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. These tangled details are handled mathematically, and in the following articles we make idealized quark models using conventional graphics. But first we say a bit more about metrics and norms.

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

Metrics

WikiMechanics 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 as collections 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 quark models. Traditionally, a radius is quantified by making a measurement of length. But a full discussion of length requires some ideas that are initially quite vague.

So to begin, we evaluate the shape of a quark model by counting its quarks. Then we define  \overline{\rho}, a radius vector, from quark inventories. Using this vector, an algebraic vector space called  \mathbb{S}, can be defined for some finite collection of particles  \mathsf{P}_{1}, \ \mathsf{P}_{2}, \ \mathsf{P}_{3} \; \ldots \; \mathsf{P}_{N}. Such a mathematical construction is generically written as

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

 \mathbb{S} is characterized using a statistical account of commonalities and variation in the shape of quark models for these particles. Averages and standard deviations are given by

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

and

\delta \! \rho_{\alpha} \equiv \sqrt{ \dfrac{1}{N}  \sum_{i=1}^{N} \left( \rho_{\alpha}^{i} - \widetilde{\rho}_{\alpha} \right)^{2} \; }

where \alpha \in \{ m, \, e, \, z \} notes different components of the radius vector \overline{\rho} = \rho_{m} \hat{m} + \rho_{e} \hat{e} + \rho_{ z} \hat{z}. Spaces are also described by correlations between these components using the coefficients

\chi_{\alpha \beta} \equiv \sqrt{ \frac{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 \}. Correlation coefficients may be combined to define some ratios, which are then used to systematically characterize \mathbb{S}. For the important special case where \mathbb{S} is the Earth, we have the terrestrial metric.

The Terrestrial Metric

Earthiness is illustrated by this planet icon for the sensation of touching the earth.

Recall that touching the Earth is a reference sensation for EthnoPhysics. So we presume that the Earth is implicitly part of every description. And since this celestial body is so big, the law of large numbers implies that correlation coefficients will have specific values that do not vary on geological time scales. So we can define five unique constants by

k_{\alpha \beta} \equiv \dfrac{ \chi_{\alpha \beta}^{\mathsf{Earth}} }{ \chi_{zz}^{\mathsf{Earth}} }

The Terrestrial Metric
centripetal 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

and use them to determine the norm of a radius vector. Note that by this definition k_{zz} is exactly one, and k_{\alpha \beta} = k_{\beta \alpha} \; . A set of numbers used to calculate a norm is called a metric, so we call this collection the terrestrial metric. Using it compresses a radius vector into a single quantity that depends on attributes of the Earth.

The labels given to components of the terrestrial metric are arbitrary, but the names listed in the table above have been chosen for their mnemonic value. They will smoothly fit into our traditional ways of discussing physics and be easy to remember. Later we use them to understand the forces involved when constructing quark models.

The Euclidean Metric

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

We also consider that \mathbb{S} may be a collection of quark models where membership in the set is restricted to certain shapes or other attributes. Then a statistical analysis of shape could yield a different metric.

For example, in our laboratories we usually assume that space is filled with room-temperature atoms, not just any composite quark. An extended analysis of this sort of terrestrial space is detailed later. But the overall result is easily summarized as the Euclidean metric shown in the accompanying table.

The Norm

Consider an ordered-set of three numbers  \rho_{m},  \rho_{e} and  \rho_{z} that constitute an algebraic vector written as \overline{\rho} \equiv \left( \, \rho_{m}, \rho_{e}, \rho_{z} \right). These numbers can be compressed into a single number called the norm of   \overline{\rho} which is written as  \left\| \, \overline{\rho} \, \right\|.

Evaluating the norm depends on several more numbers that are called components of a metric. The numerical values of these metric components are established by the context of a calculation, they are often implicit.

A complete mathematical discussion of metrics and norms can be quite extended, so for EthnoPhysics we focus on the specific case where  \rho_{m}, \rho_{e} and  \rho_{z} are the three radii used to describe the shape of a quark model. These radii are determined by counting quarks, so to assess their norm we use components of the terrestrial metric, written as  k_{\alpha \beta}. First we define a directed surface area by

    \begin{equation*} \begin{split}  \widehat{A} & \equiv  k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  & \hspace{20px} + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z} \\ & \hspace{40px} + 2 k_{ez}\rho_{e}  \rho_{z} \end{split} \end{equation*}

In general, radii may be positive or negative, so  \widehat{A} may be positive or negative too. If \widehat{A}>0 we say that the surface of P is outside facing. And if \widehat{A}<0, then the surface is facing inside. The norm of   \overline{\rho} is defined as

\left\| \, \overline{\rho} \, \right\| \equiv \sqrt{\widehat{A} \; \rule[-3px]{0px}{15px} }

This number may be imaginary if P’s surface is facing inward. Note that particles and anti-particles have opposing radius vectors, \overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ). And all radial components appear as paired factors in the expression for  \widehat{A}. So both vectors have the same norm, and we write

\left\| \rule[-3px]{0px}{12px} \, \overline{\rho} \, \right\|^{\mathsf{P}} = \left\| \, \rule[-3px]{0px}{12px} \overline{\rho} \,  \right\|^{\mathsf{\overline{P}}}

Here is similar way to distill two radius vectors into a single number. Let us call the vectors \overline{a} = \left( \, a_{m}, a_{e}, a_{z} \right) and \overline{b} = \left( \, b_{m}, b_{e}, b_{z} \right). Then the inner product is defined by

    \begin{equation*} \begin{split}  \overline{a} \cdot \overline{b}  & \equiv  k_{mm} a_{m} b_{m} + k_{ee} a_{e} b_{e} + k_{zz} a_{z} b_{z}  \\  & \hspace{20px} + 2 k_{em} a_{m} b_{e} + 2k_{mz} a_{m} b_{z} \\ & \hspace{40px} + 2 k_{ez} a_{e} b_{z} \end{split} \end{equation*}

We say that \overline{a} and \overline{b} are orthogonal if \overline{a} \cdot \overline{b} = 0.

The norm is used to organize spatial relationships, somewhat like this ajat basket from Borneo.
Ajat basket, Penan people. Borneo 20th century, 23 (cm) diameter by 36 (cm) height. Hornbill motif. Photograph by D Dunlop.

Spatial Orientation

The following quantities are defined from just the dynamic seeds in a quark model. They establish an orientation in quark space and are also used to describe displacements in ordinary space.

Magnetic Polarity

Redness is illustrated by this icon for visual sensations that are reddish or greenish.

Let a quark model for some particle P be characterized by N^{ \mathsf{M}} and N^{\mathsf{A}} the coefficients of its muonic seeds. These quantities are used to define another number \delta _{\widehat{m}} called the magnetic polarity of P as

\delta_{\widehat{m}} \equiv \begin{cases} +1 & \mathsf{\text{if}}  \; \; N^{\, \mathsf{M}} > N^{\, \mathsf{A}} \\ \; \; 0 & \mathsf{\text{if}} \; \; N^{\, \mathsf{M}} = N^{\, \mathsf{A}} \\ -1   & \mathsf{\text{if}} \; \; N^{\, \mathsf{M}} < N^{\, \mathsf{A}} \end{cases}

If \delta_{\widehat{m}}=+1 then northern seeds are more numerous than southern seeds and we say that P is oriented to the north. If P is part of a magnet, we might even call it a north pole. If southern seeds predominate then we say that P is directed to the south, or perhaps aligned in a southerly direction. And if \delta_{\widehat{m}}=0 then we say that P is not magnetically polarized.

Sensory interpretation: Muonic seeds are objectified from red and green sensations. So \delta_{\widehat{m}} is a binary description of whether a complicated visual sensation is more reddish or greenish. If \delta_{\widehat{m}}=0 then P is not remarkably red or green.

Electric Polarity

Yellowness is illustrated by this icon for visual sensations that are yellowish or bluish.

Let a quark model of P also be characterized by N^{ \mathsf{E}} and N^{\mathsf{G}} the coefficients of its electronic seeds. These numbers are used to define another quantity \delta_{\widehat{e}} called the electric polarity of P as

\delta_{\widehat{e}} \equiv \begin{cases} +1 & \mathsf{\text{if}}  \; \; N^{\, \mathsf{G}} > N^{\, \mathsf{E}} \\ \; \; 0 & \mathsf{\text{if}} \; \; N^{\, \mathsf{G}} = N^{\, \mathsf{E}} \\ -1 & \mathsf{\text{if}} \; \; N^{\, \mathsf{G}} < N^{\, \mathsf{E}} \end{cases}

If \delta_{\widehat{e}}=+1 then positive seeds are more numerous than negative seeds and we say that P is positive too. If P is part of a battery, we might even call it a positive electrode. If negative seeds predominate then we may say that P is oriented or aligned in a negative direction. If \delta_{\widehat{e}} = 0 then P is not electrically polarized. And, if both of \delta_{\widehat{e}} and \delta_{\widehat{m}} are zero, then we say that P is centered on the electric and magnetic axes.

Sensory interpretation: Electronic seeds are objectified from yellow and blue sensations. So \delta_{\widehat{e}} is a binary description of whether a complex visual sensation is more yellowish or bluish. If P is not clearly yellowish or bluish, then \delta_{\widehat{e}}=0. And if both of \delta_{\widehat{m}} and \delta_{\widehat{e}} are zero, then P is a colorless or achromatic sensation.

Helicity

Whiteness is illustrated by this icon for a binary description of grey visual sensations.

Finally, let a quark model of P be characterized by N^{ \mathsf{U}} and N^{\mathsf{D}} the coefficients of its rotating seeds. These numbers are used to define \delta_{z} as the helicity of P

\delta_{z} \equiv \begin{cases} +1 & \mathsf{\text{if}} \; \; N^{\, \mathsf{U}} > N^{\, \mathsf{D}} \\ \; \; 0 & \mathsf{\text{if}} \; \; N^{\, \mathsf{U}} = N^{\, \mathsf{D}} \\ -1 & \mathsf{\text{if}}\; \; N^{\, \mathsf{U}} < N^{\, \mathsf{D}} \end{cases}

If \delta_{z} > 0 then we say that P is a spin-up particle. Conversely, if \delta_{z} < 0 then P is called a spin-down particle. And if \delta_{z}=0 then we say that P is not rotating. Sensory interpretation: Rotating quarks are objectified from achromatic visual sensations. So \delta_{z} is a binary description of whether a complicated greyish vision is overall a light grey or a dark grey.

Angular Momentum

SpinHelicitySeeds
spin-up\delta_{z}=+1 N^{\mathsf{U}} > N^{\mathsf{D}}
non-rotating\delta_{z}=0 N^{\mathsf{U}} = N^{\mathsf{D}}
spin-down\delta_{z}=-1 N^{\mathsf{U}} < N^{\mathsf{D}}

Consider a quark model of P, that is described by the coefficients of its rotating seeds  N^{\mathsf{U}} and  N^{\mathsf{D}}. We say that P has a spin that is defined by these coefficients, as noted in the accompanying table.

We also use the helicity to make quantitative descriptions of P’s spatial orientation. And later, if P is also being used as a frame of reference, then  N^{\mathsf{U}} and  N^{\mathsf{D}} may be used to establish the phase of other particles. So rotating seeds have an important role in describing motion.

This task is expanded by considering the coefficients of the leptonic seeds  N^{\mathsf{A}},  N^{\mathsf{M}},  N^{\mathsf{E}} and  N^{\mathsf{G}} to define

\mathrm{J}_{m} \equiv \delta_{\hat{m}} \dfrac{h}{16\pi} \sqrt{ \left( N^{\mathsf{A}}-N^{\mathsf{M}} \right)^{2} + 8 \left| N^{\mathsf{A}}-N^{\mathsf{M}} \right| \; \rule{0px}{12px} }

\mathrm{J}_{e} \equiv \delta_{\hat{e}} \dfrac{h}{16\pi} \sqrt{ \left( N^{\mathsf{G}}-N^{\mathsf{E}} \right)^{2} + 8 \left| N^{\mathsf{G}}-N^{\mathsf{E}} \right| \; \rule{0px}{12px} }

\mathrm{J}_{z} \equiv \delta_{z} \dfrac{h}{16\pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \left| N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \rule{0px}{12px} }

where \delta_{\hat{e}} is the electric polarity and \delta_{\hat{m}} is the magnetic polarity. Then we specify the total angular momentum vector as  {\mathrm{\overline{J}}} \equiv \left( {\mathrm{J}}_{m} , \ {\mathrm{J}}_{e} , \ {\mathrm{J}}_{z} \right). Exchanging quarks for anti-quarks does not alter seed counts, so  {\mathrm{\overline{J}}} ( \mathsf{P} ) = {\mathrm{\overline{J}}} ( \mathsf{\overline{P}} ).

In general, the components  {\mathrm{J}}_{m},  {\mathrm{J}}_{e} and  {\mathrm{J}}_{z} have non-zero values, and P’s motion is complicated. But for a quark model that is not electrically or magnetically polarized we may construct a framework where P is centered on the electric and magnetic axes. Then it is easy to assess the norm of  \mathrm{\overline{J}} because N^{\mathsf{A}} = N^{\mathsf{M}} and N^{\mathsf{G}} = N^{\mathsf{E}}. The vector is aligned with the polar-axis, {\mathrm{\overline{J}}} = \left( 0, \ 0, \ {\mathrm{J}}_{z} \right) and so

\left\| \, \mathrm{\overline{J}} \, \right\| = \dfrac{h}{16 \pi}  \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left|    N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \rule{0px}{12px} }

This expression can be simplified because the total angular momentum quantum number \textsl{ \textsf{J} } is defined by

{\textsl{\textsf{J}}}  \equiv   \dfrac{ \, \left| \,  N^{\mathsf{U}}   - N^{\mathsf{D}} \, \right| \, }{8}

so that

\left\| \, {\mathrm{\overline{J}}} \, \right\| = \dfrac{h}{\rm{2} \pi} \sqrt{ \, {\textsl{\textsf{J}}} \, \left( {\textsl{\textsf{J}}} + 1 \right) \; \vphantom{X^{X}} }

Conservation of Angular Momentum

If {\textsl{\textsf{J}}} \ne 0 then the  z-component of the angular momentum vector can be expressed in terms of { \textsl{\textsf{J}}} as

\mathrm{J}_{z} = \delta_{z} \dfrac{{\textsl{\textsf{J}}} h}{2\pi} \sqrt{ \, 1 + \dfrac{1}{ {\textsl{\textsf{J}}} \, } \rule{0px}{13px} }

And if {\textsl{\textsf{J}}} \gg 1 then the radical is approximately one, and

\mathrm{J}_{z} \simeq \, \delta_{z} \dfrac{ \textsl{\textsf{J}} h}{2\pi} = \dfrac{h}{16 \pi} \delta_{z} \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| = \dfrac{h }{16 \pi} \left( N^{\mathsf{U}} - N^{\mathsf{D}} \right)

Similar results obtain for the other axes so that

\overline{\mathrm{J}} \, \simeq \dfrac{h}{16\pi} \left( N^{\mathsf{A}} - N^{\mathsf{M}}, \; \; N^{\mathsf{G}} - N^{\mathsf{E}}, \; \; N^{\mathsf{U}} - N^{\mathsf{D}} \right)

But seeds are conserved, so the quantity and character of the seeds in a description cannot change. Whenever some generic compound quarks  \mathbb{X},  \mathbb{Y} and  \mathbb{Z} interact, if \mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z} then the coefficients for any sort of seed Z are related as

N^{\mathsf{Z}} \left( \mathbb{X} \right) + N^{\mathsf{Z}} \left( \mathbb{Y} \right) = N^{\mathsf{Z}} \left( \mathbb{Z} \right)

Then by the associative properties of addition, the angular momentum must be approximately conserved too. For quark models of macroscopic particles, {\textsl{\textsf{J}}} is huge because h is so small, and the approximation is excellent.

The internal energy of down quarks is represented by the black background in this icon for achromatic visual sensation.

Sensory interpretation: Rotating seeds are objectified from achromatic visual sensations. So for quark models of spin-up particles, white sensations outnumber black sensations. Collectively they are bright. For spin-down particles, black sensations are more numerous than white sensations, they look dark. Quark models of non-rotating particles are in between, they are greyish. So \delta_{z} indicates if a complex achromatic visual sensation is brighter or darker than some medium grey visual experience. And {\textsl{\textsf{J}}} notes the size of the difference.

The angular momentum is used to model spatial relationships, somewhat like this ajat basket from Borneo.
Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Photograph by D Dunlop.

Charge Symmetry

Let some quark model for particle P be described by a chain of events where the quarks in each orbital cycle \mathsf{\Omega} can be parsed into two sets

\mathsf{\Omega}^{\mathsf{P}} = \left\{ \mathsf{P}_{\large{\Uparrow}} \, , \mathsf{P}_{\large{\Downarrow}} \right\}

that have opposite magnetic polarities

\delta_{\widehat{m}} \left( \mathsf{P}_{\large{\Uparrow}} \right) = - \, \delta_{\widehat{m}} \left( \mathsf{P}_{\large{\Downarrow}} \right) = \pm 1

Then \mathsf{P}_{\large{\Uparrow}} and \mathsf{P}_{\large{\Downarrow}} are called the northern and southern components of P. When these two components have the same charge  q then the outcome of any calculation using the charge is not affected by a change of polarity. The magnetic polarity is used to specify direction on the magnetic axis. So for quark models of P, the charge distribution along the magnetic axis is symmetric. Descriptions of phenomena associated with the charge of P are unaltered by any confusion or mix-up between north and south. This indifference is useful, so if

q \left( \mathsf{P}_{\large{\Uparrow}} \right) = q \left( \mathsf{P}_{\large{\Downarrow}} \right)

then we say that P has charge-symmetry on the magnetic-axis. See the quark model of atomic hydrogen for an example of this kind of symmetry.

Redness is illustrated by this icon for visual sensations that are reddish or greenish.

Sensory interpretation: Magnetic polarity can be interpreted as a description of if a visual sensation is more reddish or greenish. So for quark models with charge-symmetry on the magnetic-axis, the charge distribution does not depend on how the model is objectified from red and green sensations. This symmetry relieves us from having to pay very much attention to whether a sensation is red or green.

Charge symmetry on the magnetic axis is suggested by the red and green beads of this baby carrier from Borneo.
Baby carrier panel, Bahau people. Borneo 20th century, 29 x 26 cm. Photograph by D Dunlop.

Alternatively, let P be described by a chain of events where the quarks in each orbital cycle \mathsf{\Omega} can be parsed into two sets

\mathsf{\Omega}^{\mathsf{P}} = \left\{ \mathsf{P}_{\large{\oplus}},   \mathsf{P}_{\large{\ominus}} \right\}

that have opposite electric polarities

\delta_{\widehat{e}} \left( \mathsf{P}_{\large{\oplus}} \right) = - \, \delta_{\widehat{e}} \left( \mathsf{P}_{\large{\ominus}} \right) = \pm 1

Then  \mathsf{P}_{\large{\oplus}} and  \mathsf{P}_{\large{\ominus}} are called the positive and negative components of P. When these two components have the same charge  q then the outcome of any calculation using the charge is not affected by a change of polarity. The electric polarity is used to specify direction on the electric axis. So for quark models of P, the charge distribution along the electric axis is symmetric. Descriptions of phenomena associated with the charge of P are unaltered by any confusion or mix-up between positive and negative components. This indifference is useful, so if

q \left( \mathsf{P}_{\large{\oplus}} \right) = q \left( \mathsf{P}_{\large{\ominus}} \right)

then we say that P has charge-symmetry on the electric-axis. See this quark model of an electron for an example of electric-axis charge-symmetry.

Yellowness is illustrated by this icon for visual sensations that are yellowish or bluish.

Sensory interpretation: Electric polarity can be understood as a binary description of if a complex visual sensation is more yellowish or blueish. So for a quark model with charge-symmetry on the electric-axis, the charge distribution does not depend on how the model is objectified from yellow and blue sensations. This symmetry relieves us from having to pay very much attention to whether a sensation is yellow or blue.

Work

Consider a quark model of P that is characterized by some repetitive chain of events \Psi  = \left( \mathsf{\Omega}_{1} , \,  \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3}  \;   \ldots  \;       \right) where each orbital cycle is a bundle of quarks written as \mathsf{\Omega} = \left( \mathsf{q}_{1}, \,  \mathsf{q}_{2} \; \ldots   \;  \mathsf{q}_{N} \right) . Let these quarks be assembled into a model of P. The work required to bring these quarks together to build the model is defined as

 W \equiv k_{\mathsf{F}} \left\| \, \overline{\rho} \, \right\|

where  \left\| \,   \overline{\rho} \,  \right\| is the norm of the radius vector of P. We consider that  W might be an imaginary number because the norm may be imaginary under some circumstances. 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.

Models of particles and anti-particles have opposing radius vectors, that is, \overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ) . But they both have the same norm. So the work required to assemble the quark model of any particle is the same as the work done to build a model of its corresponding anti-particle

W \! ( \mathsf{P} ) = W \! ( \mathsf{\overline{P}} )

If extra quarks are absorbed or emitted by P, then  \mathsf{\Omega} is replaced by a new bundle  \mathsf{\Omega}^{\prime} and  W changes to  W^{\prime} . The quantity  \Delta W \equiv W^{\prime} - W may be used to describe the change. Particle radii may also vary, and then we say that the interaction has done work on the particle by changing its shape.

The norm can be written 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_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}    \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right)^{\frac{1}{2}} \end{equation*}

where the constants  k_{\alpha \beta} are components of the terrestrial metric, and  \rho_{\alpha} are components of P’s radius vector. So the square of the work can be written as

    \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_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z}   \\ & \hspace{30px} + 2 k_{ez}\rho_{e}  \rho_{z} \;  \end{split} \right) \end{equation*}

And since we explicitly consider that the work may be imaginary, then W^{2} may be negative. The foregoing expression is key for calculating the mass of P. And experimental observations of mass are possibly the most important data for understanding the mechanics of particles. So we will refer back to the work, but next we discuss EthnoPhysics faviconClocks.

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.