Accurate Quark Counting

Outline

Counting is a primordial part of science. From counting days to counting stars, a systematic quantitative approach to observation is crucial for objectifying the description of human experience. Counting is closely related to making measurements. And reproducible measurements are what distinguish physical particles from other narrative devices like genies or ghosts. So in this article we tackle questions about physical particles by counting quarks.

Physical Particles

To summarize developments so far, we have defined seeds by objectifying some common sensations. Seeds are the elementary components of EthnoPhysics. All other objects are subsequently defined by aggregations of seeds. This approach is not new, it is influenced by the ancient philosophy of Anaxagoras.

Then we considered pairs of seeds and called them quarks. Quarks are discussed in more detail over the next few pages, but we can already use them to make this rudimentary definition: Physical particles are compound quarks. But quarks are defined by sensation. So, as guided by David Hume, we ultimately understand physical particles to be bundles of sensation.

EthnoPhysics then expands on Hume’s idea in an effort to understand particle mechanics without resorting to mysteriously received notions about length, mass and time. First we remarked that experiencing a sensation is itself an event. Then we organized a way of mathematically describing events using ordered-sets called chains of events generically noted by

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

Recognizing patterns of sensation, and identifying particles, requires some repetition within a stream of consciousness. So physical particles are mathematically represented using orbital chains of events where some bundle of sensation $\mathsf{\Omega}$ is experienced over and over again. Then, if we speak informally of the quarks in a particle, we mean the quarks in one bundle. For instance we may say that particle $\mathsf{P}$ contains the quarks $\mathsf{u},$ $\overline{\mathsf{s}}$ and $\mathsf{d},$ or we may write phrases like

$\mathsf{u} + \overline{\mathsf{s}} + \mathsf{d} \rightarrow \mathsf{P}$

as an abbreviation for writing out the full expression for the chain

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

where the convention that seeds are conserved requires that

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

From these general considerations, different sorts of particles can be obtained from rules that specify new quark combinations. For example, the conjugate twin of any particle $\mathsf{P}$ is noted by $\overline{\mathsf{P}}$ and defined by exchanging ordinary-quarks and anti-quarks of the same type. So for example if $\mathsf{P}$ contains $\mathsf{u},$ $\overline{\mathsf{s}}$ and $\mathsf{d},$ then $\overline{\mathsf{P}}$ is composed from $\overline{\mathsf{u}},$ $\mathsf{s}$ and $\overline{\mathsf{d}}.$ And here is a quick introduction to some more compound quarks.

Frames of Reference are compound quarks where the total number of quarks is usually enormous. They provide a descriptive context for other particles.

Clocks are defined from compound quarks that have a fixed relationship with events on Earth.

Nuclear Particles are compound quarks that are very symmetric so that they are stable enough to be measured.

Photons and Gravitons are compound quarks that have no mass. They are mostly known by their effect on other particles.

Newtonian Particles are compound quarks that are dense enough so that they can absorb a few photons or gravitons without decaying.

Spaces and Fields are described by mathematical sets of quarks too. Different kinds of fields are defined from different distributions of quark types.

So in brief, EthnoPhysics employs sensations, seeds and quarks to define all physical things.

A Quark Index

Index	Seed	Quark
$\zeta$	Z	z
1	U	u
2	D	d
3	E	e
4	G	g
5	M	m
6	A	a
7	T	t
8	B	b
9	S	s
10	C	c
11	Ⓐ	a
12	Ⓑ	b
13	Ⓘ	i
14	Ⓦ	w
15	Ⓓ	d
16	Ⓛ	l

Next we specify a quark index because when considering quarks it is often more convenient to identify them by number instead of a letter. We start with a non-conjugate seed noted by Z that is used to define an ordinary quark $\mathsf{z} \equiv \{ \mathsf{Z}, \, \mathsf{O} \}$ and its associated anti-quark $\overline{\mathsf{z}} \equiv \{ \mathsf{Z}, \, \overline{\mathsf{O}} \} .$ These quarks are occasionally referred to using $\zeta,$ the Greek letter zeta, as shown in the adjoining table. When used like this, zeta is called a quark index. The two particles $\mathsf{z}$ and $\overline{\mathsf{z}}$ are sometimes collectively called $\zeta$ -type or Z-type quarks. This notation is especially helpful when using summation notation.

Counting Quarks

Counting quarks is a way to scientifically describe sensation. More exactly, we start by counting seeds. Let P be a generic particle composed of some aggregation of seeds. A simple way to make a mathematical description of P is just to sort-out the number of different types of seeds in P. To satisfy Anaxagorean narrative conventions, Cantor’s definition of a set, and Pauli’s exclusion principle, we require that seeds are perfectly distinct. Therefore seed counts always report a positive integer or zero, never fractions or negative numbers.

If all seeds are paired in quarks, then P can also be represented as a set of quarks and mathematically described by counting quarks. We note the results of such an inventory using the letter $n$ in a serified italic font. These numbers are called quark coefficients because they can be interpreted as factors in a nuclear reaction that yields P. For example if $\mathsf{s}+2\mathsf{c} \to \mathsf{P}$ then the quark coefficients of P are $n^{\mathsf{s}} = 1$ and $n^{\mathsf{c}} = 2$ . Quark coefficients are always non-negative integers because quarks are defined from perfectly distinct pairs of perfectly distinct seeds.

Quark Coefficients
coefficients of ordinary quarks	$n^{\mathsf{z}} \hspace{0.4cm} n^{\zeta}$
coefficients of anti-quarks	$n^{\mathsf{\overline{z}}} \hspace{0.4cm} n^{\mathsf{\overline{\zeta}}}$
the total number of Z-type quarks	$N^{\mathsf{Z}} \equiv n^{\mathsf{\overline{z}}} + n^{\mathsf{z}}$
the net number of Z-type quarks	${\Delta}n^{\mathsf{Z}} \equiv n^{\mathsf{\overline{z}}} - n^{\mathsf{z}}$
the total number of all types of quarks	$\displaystyle N_{\! \mathsf{q}} \equiv \sum_{\zeta =1}^{16} \; n^{ \overline{\zeta}} + n^{\zeta}$

The Roman letter Z, or the Greek letter $\zeta$ are used to indicate quark-type. In general, we use the symbols $n^{\mathsf{z}}$ or $n^{\zeta}$ to note the coefficients of ordinary quarks. Coefficients of anti-quarks are written with an overline. A few other numbers used for describing particles are defined in the accompanying table.

For thermodynamic and chemical quarks, the total number of quarks in some particle, $N^{\mathsf{Z}} ,$ is also the total number of Z-type seeds in the particle. This is because by definition there is just one thermodynamic or chemical seed in any quark. So we often refer to $N^{\mathsf{Z}}$ as a seed coefficient.

Counting Anti-Quarks

Particles that are conjugate twins to each other are specified by exchanging quarks with anti-quarks. So the quark coefficients for some particle $\mathsf{P}$ and its conjugate twin $\overline{\mathsf{P}}$ are related as

$n^{\mathsf{z}} (\mathsf{P}) = n^{\mathsf{\overline{z}}} (\overline{\mathsf{P}})$

and

$n^{\mathsf{\overline{z}}} (\mathsf{P}) = n^{\mathsf{z}} (\overline{\mathsf{P}})$

Then $\Delta n$ can be written as

$\begin{equation*} \begin{split} \Delta n^{\mathsf{Z}} (\mathsf{P}) & \equiv n^{\mathsf{\overline{z}}} (\mathsf{P}) - n^{\mathsf{z}} (\mathsf{P}) \\ & = - \left[ n^{\mathsf{z}} (\mathsf{P}) -n^{\mathsf{\overline{z}}} (\mathsf{P}) \rule[-3px]{0px}{16px} \right] \\ & = - \left[ n^{\mathsf{\overline{z}}} (\overline{\mathsf{P}}) -n^{\mathsf{z}} (\overline{\mathsf{P}}) \rule[-3px]{0px}{16px} \right] \; \text{\sf{by substitution}} \\ & = - \Delta n^{\mathsf{Z}} (\overline{\mathsf{P}}) \end{split} \end{equation*}$

This is just some simple arithmetic known as the anti-commutative property of subtraction . But when applied to quark coefficients, it expresses a fundamental relationship between material particles and their conjugate twins.

Quarks are Conserved

Quarks are conserved because physics depends on mathematics. Recall that a logical style of description that uses counting and mathematics requires that seeds are conserved. For the same reasons, when we shift the description to counting quarks, then quarks must be conserved too. The overall quantity and quality of the quarks in a description cannot change. As a narrative convention, we say that quarks are indestructible. Whenever some compound quarks $\mathbb{X}$ , $\mathbb{Y}$ and $\mathbb{Z}$ are combined or decomposed, if

$\mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}$

then the coefficients of any specific type of quark $n^{\mathsf{q}}$ are related as

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

And counting out a sum over all types of quarks $N_{\mathsf{q}}$ is constrained as

$N_{\mathsf{q}}^{\mathbb{X}} + N_{\mathsf{q}}^{\mathbb{Y}} = N_{\mathsf{q}}^{\mathbb{Z}}$

Quark Coefficients are Integers

To satisfy Anaxagorean narrative conventions, Cantor’s definition of a set, and Pauli’s exclusion principle, we require that every seed Z is perfectly distinct. Seeds are always distinguished from each other, either by their innate characteristics, or by their association with other particles in nested sets. Therefore when counting seeds we always report a positive integer or zero, never fractions or negative numbers. So

$N^{\mathsf{Z}} = 0, \, 1, \, 2, \, 3 \ \ldots \ \ \forall \, \mathsf{Z}$

For the same reasons, when we define quarks from seeds, and shift the description to counting quarks, then quark coefficients must be integers too. That is, $n^{\mathsf{q}}$ is always a non-negative integer because quarks are defined from perfectly distinct pairs of perfectly distinct seeds. We write

$n^{\mathsf{q}} = 0, \, 1, \, 2, \, 3 \ \ldots \ \ \forall \, \mathsf{q}$

The foregoing relationships are the logical basis for a variety of conservation laws that are found throughout physics. We often refer back to them. But next we look at how counting quarks leads to particle classification using quantum numbers.

Counting quarks is like counting the beads in this baby carrier panel from Borneo. — Baby Carrier Panel, Basap people. Borneo 19th century, 39 x 20 cm. Photograph by D Dunlop.

Quantum Numbers

Quantum numbers are used to identify and classify particles. They are employed in both atomic and nuclear physics. The atomic quantum numbers are discussed later. We start here by specifying the nuclear numbers from quark coefficients noted by $n$ and $N.$ The total angular momentum quantum number is defined by

$\textsl{ \textsf{J} } \equiv \dfrac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \rule{0px}{10px} \right| \, }{8}$

The charge quantum number is

$q \equiv \dfrac{ \Delta n^{\mathsf{T}} - \Delta n^{\mathsf{B}} + \Delta n^{\mathsf{C}} - \Delta n^{\mathsf{S}} }{8}$

The lepton number is defined as

$L \equiv \dfrac{ \Delta n^{\mathsf{G}} - \Delta n^{\mathsf{E}} + \Delta n^{\mathsf{M}} - \Delta n^{\mathsf{A}} }{8}$

The baryon number is given by

$B \equiv \dfrac{ \Delta n^{\mathsf{T}} - \Delta n^{\mathsf{B}} - \Delta n^{\mathsf{C}} + \Delta n^{\mathsf{S}} }{8}$

And finally the strangeness quantum number is defined as

$S \equiv \dfrac{ \Delta n^{\mathsf{D}} - \Delta n^{\mathsf{U}} - \left| n^{\mathsf{u}} - n^{\overline{\mathsf{d}}} \right| + \left| n^{\mathsf{d}} - n^{\mathsf{\overline{u}}} \right| \, }{8}$

Particle Types
a charged particle	$q \neq 0$
a neutral particle	$q=0$
a lepton	$B = 0 \textsf{ and } L \neq 0$
a baryon	$B \neq 0 \textsf{ and } L = 0$
a meson	$B=0 \textsf{ and } L=0$
a strange particle	$S \neq 0$

Recall that quark coefficients are always integers. So these formulae all yield integer multiples of one-eighth. But later, if we assume spatial homogenity, then $q ,$ $S ,$ $L$ and $B$ are also restricted to being integers.

Particles may be classified by their quantum numbers into a few different¹By definition the strangeness quantum-number describes how a particle is rotating. It is not necessarily related to the presence of strange-quarks. But some well known particles like kaons, contain lots of strange-quarks and also have $S \! \ne \! 0 .$ Moreover, strange-quarks themselves have $S \! = \! 0 .$ So the label strange can be misleading if used without context. Perhaps, if confusion is persistent then the $\mathsf{s}$ -quarks could just be renamed, for example, as silly quarks. Or maybe we could say that a particle is shiny if $S \! \ne \! 0 .$ types as shown in the accompanying table. In general, particle identities and attributes are quantized because EthnoPhysics is fundamentally based on a finite categorical scheme of binary distinctions. Any characteristic defined using a quark coefficient is necessarily quantized because quark coefficients are always integers.

Now considering conjugate twins, recall that for any seed Z, the net number of quarks in particle $\mathsf{P}$ and its twin $\overline{\mathsf{P}}$ are related as

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

This relationship implies that the charge, strangeness, lepton-number and baryon-number of particles and their conjugate twins have the same absolute value, but opposite signs. That is,

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

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

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

and

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

But swapping quarks with anti-quarks just changes their conjugate seeds. The rotating seeds that determine $\textsl{\textsf{J}}$ are not affected. So $\textsl{\textsf{J}} \, ( \mathsf{P} ) = \textsl{\textsf{J}} \, ( \mathsf{\overline{P}} ) .$

Quarks are conserved. So the overall quantity of each quark-type in any given description may not 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 quark $\mathsf{q}$ are related as

$n^{\sf{q}} \left( \mathbb{X} \right) + n^{\sf{q}} \left( \mathbb{Y} \right) = n^{\sf{q}} \left( \mathbb{Z} \right)$

But the lepton number, for example, is defined above from sums of quark coefficients. So by the associative properties of addition we have

$L^{\mathbb{X}} + L^{\mathbb{Y}} = L^{\mathbb{Z}}$

Thus the lepton number is conserved too. This reasoning also applies to the baryon number and charge, so we write

$B^{\mathbb{X}} + B^{\mathbb{Y}} = B^{\mathbb{Z}}$

and

$q^{\mathbb{X}} + q^{\mathbb{Y}} = q^{\mathbb{Z}}$

However, the strangeness and angular momentum quantum-numbers are defined using absolute-value functions which are not generally associative. So $\textsl{\textsf{J}}$ and $S$ are not always conserved when compound quarks are formed or decomposed.

Quantum numbers are related to counting quarks, which is like counting the beads in this woven panel from Borneo. — Baby Carrier Panel, Basap people. Borneo 19th century, 30 x 21 cm. Photograph by D Dunlop.

Quirks and Quarks

Quirks are key details that regulate how quarks are combined to form larger particles. Some quarks are bigger than others. And some are hot. There are three quirks to master quarks.

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References
1	By definition the strangeness quantum-number describes how a particle is rotating. It is not necessarily related to the presence of strange-quarks. But some well known particles like kaons, contain lots of strange-quarks and also have $S \! \ne \! 0 .$ Moreover, strange-quarks themselves have $S \! = \! 0 .$ So the label strange can be misleading if used without context. Perhaps, if confusion is persistent then the $\mathsf{s}$ -quarks could just be renamed, for example, as silly quarks. Or maybe we could say that a particle is shiny if $S \! \ne \! 0 .$