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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 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 size and 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 spatial orientation 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 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 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 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. 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 core radius of particle P is given by

\rho_{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 polar radius of P is given as

\rho_{z} \equiv \, \rho_{core} + \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 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

\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 smallish and the outer radius is huge. But if  \left| \Delta n^{\mathsf{D}}  \right| \lesssim 8 then P is not free.

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, 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

EthnoPhysics begins with the premise that we can understand the ordinary everyday space of our classrooms and laboratories 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} 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; the magnetic axis is established as \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 vector in  \mathbb{Q} can be expressed in terms of its components  \rho_{m} ,  \rho_{e} and  \rho_{z} as

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

Quark space is grainy because quark coefficients are always integers. And  \mathbb{Q} is not exactly rectilinear. 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.

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 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.

The Quark Metric

A quark metric is a set of six numbers that are later used to express notions about distance or separation in a 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 defined 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} . And they are scaled relative to variations in the polar radius. So there are no units, and k_{zz} is always equal to one.

The Terrestrial Metric

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

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. 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. Thus a unique metric can be specified by

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

k_{\alpha \beta} = \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 components 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

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 is restricted to certain shapes or other attributes. 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.

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 a norm depends on several supplementary numbers that are called metric components. These numbers are determined by the context of a calculation, they are often implicit. A full discussion of metrics and norms can be quite extended. So here we focus on the specific case where  \rho_{m}, \rho_{e} and  \rho_{z} are the three radii used to describe the shape of some compound quark P. The norm of P’s radius-vector is then found using the terrestrial metric which is written as  k_{\alpha \beta}. To assess  \left\| \, \overline{\rho} \, \right\| we first define the surface area of particle P as

    \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_{e}  \rho_{m} + 2k_{mz}\rho_{m}  \rho_{z} \\ & \hspace{40px} + 2 k_{ez}\rho_{e}  \rho_{z} \end{split} \end{equation*}

Then the the norm of   \overline{\rho} is defined by

\left\| \, \overline{\rho} \, \right\| \equiv \sqrt{ \, \left| \widehat{A} \, \right| \rule[-3px]{0px}{16px} }

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 facing outside. Conversely, if \widehat{A}<0 then P’s surface is facing inside. But the norm does not depend on these directions, just the modulus of the surface area. It can be written-out explicitly in terms of its components 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 anti-particles 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 expression for  \widehat{A}. So sign changes get cancelled and both particles have the same surface area and norm. We can write

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

Here is a related 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 their 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 to each other if \overline{a} \cdot \overline{b} = 0. We also use inner product to define the square of a vector as \overline{b}^{2} \equiv \overline{b} \cdot \overline{b} \, . Then the surface area of a particle can be stated as

\widehat{A} = \overline{\rho}^{2}

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.

Spatial Orientation

Next we talk about polarity and helicity. These quantities are innate particle characteristics. They depend on quark content, not the reference frame. So we can use them to establish the orientation of a particle in quark space as follows.

Magnetic Polarity

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}} \ne 0 then we say that P is magnetically polarized.

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

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

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. And, if \delta_{\widehat{e}} \ne 0, then we say that P is electrically polarized.

Yellowness is illustrated by this image of golden coins.

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. When both \delta_{\widehat{e}} and \delta_{\widehat{m}} are zero, then P represents a colorless or achromatic sensation.

If P is not electrically or magnetically polarized, it said to be centered around the polar axis.

Helicity

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}, the helicity of P by

\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} = +1 then we say that P is a spin-up particle. Conversely, if \delta_{z} = -1 then P is called a spin-down particle. And if \delta_{z} = 0 then we say that P is not rotating. The helicity of a particle depends on the coefficients of its rotating seeds  N^{\mathsf{U}} and  N^{\mathsf{D}} . So the spin also depends on these coefficients as shown in the accompanying table.

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}}

If P is being used as a frame of reference then  N^{\mathsf{U}} and  N^{\mathsf{D}} also determine the phase of other particles. So these rotating seeds have an important role in describing motion.

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 \delta_{z} = +1 white sensations outnumber black sensations. Collectively they are bright. For \delta_{z} = -1 black sensations are more numerous than white sensations, they look dark. Particles with \delta_{z}=0 are greyish. So  \delta_{z} indicates if a complex visual experience is overall brighter or darker than some dull grey sensation.

The norm is used to describe spatial relationships, perhaps somewhat as suggested by the patterns in 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.

Angular Momentum

The coefficients of dynamic seeds may be used to specify some angular momenta. Recall that \delta_{\hat{e}} is the electric polarity and \delta_{\hat{m}} is the magnetic polarity. We 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} }

These three numbers are components of the total angular momentum vector which is written as  {\mathrm{\overline{J}}} \equiv \left( {\mathrm{J}}_{m} , \ {\mathrm{J}}_{e} , \ {\mathrm{J}}_{z} \right) . Exchanging quarks for anti-quarks switches conjugate seeds but does not change dynamic seeds, 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 particle that is not electrically or magnetically polarized we may construct a framework where P is centered around the polar axis. 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 angular momentum vector is aligned with the polar-axis. We can write  {\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 further simplified because the total angular momentum quantum number \textsl{ \textsf{J} } is given 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}} }

The Handedness

Sweetness is illustrated by this icon for taste sensations that are honey flavoured.

Let a quark model of P be characterized by N^{ \mathsf{U}} and N^{\mathsf{D}} the coefficients of its rotating seeds. And let N^{\mathsf{U}} be different from N^{\mathsf{D}} so that the angular-momentum quantum number  \textsl{ \textsf{J} } is not zero. Then P is rotating and it has a well-defined angular momentum vector. Furthermore, let P also be described by N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} and N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{D}}}}} the coefficients of its stereochemical seeds. That is, N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} marks the total number of levo quarks and N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{D}}}}} notes the quantity of all dextro quarks in P. Then the particle’s rotation is characterized by its handedness which is written as \delta_{\! R} and defined by

\delta_{\! R} \equiv \begin{cases} +1 & \mathsf{\text{if}}  \; \; N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} > N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{D}}}}} \\ \; \; 0 & \mathsf{\text{if}} \; \; N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} = N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{D}}}}} \\ -1   & \mathsf{\text{if}} \; \; N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{L}}}}} < N^{\textcircled{\raisebox{.5pt}{\sf{\tiny{D}}}}} \end{cases}

If \delta_{\! R} = +1 then we say that the particle is levorotatory. That is, its rotation is turning with a left-handed disposition compared to the angular-momentum vector. But if \delta_{\! R} =-1 then P is called dextrorotatory or right-handed. These rotational characteristics are later associated with geometric isomerism in chemical molecules.

An image of a left-hand associated with levorotatory spiral motion, and a right-hand together with dextrorotatory spiral motion.
Handedness and two kinds of rotatory motion.

For spacetime descriptions, the handedness of a rotating particle may be depicted using stripes on its angular-momentum vector, like a barber’s pole. But the stereochemical quarks are millions of times smaller than leptonic and baryonic quarks. So for simplicity, stripes and stereochemical-quarks are just not included in many illustrations. Then the handedness of a particle may be noted in extra text, or perhaps completely set aside as an implicit assumption.

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

\overline{\mathrm{J}}  \left( \mathbb{X} \right) + \overline{\mathrm{J}}  \left( \mathbb{Y} \right)  \simeq \overline{\mathrm{J}}  \left( \mathbb{Z} \right)

For macroscopic particles, {\textsl{\textsf{J}}} is huge because h is so small, and the approximation is excellent.

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 some particle P that is characterized by a 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 brought together to form a quark-model of P. The work required to assemble the model is defined by

 W \equiv k_{\mathsf{F}} \sqrt{ \widehat{A} \rule[-3px]{0px}{15px} \; }

where  \widehat{A} is the surface area of P. We consider that  W might be an imaginary number because the surface area may be negative or facing inward 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.

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 surface area. So the work required to assemble the quark model of any particle is the same as the work done to build 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 P by changing its shape.

The square of the work can be written as

    \begin{equation*}   W^{2}  =    \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) k^{2}_{\mathsf{F}} \end{equation*}

where the constants  k_{\alpha \beta} are components of the terrestrial metric. This expression is important for calculating the mass of P.

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.
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EthnoPhysics faviconReliable Clocks

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