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Accurate Quark Counting

Outline

Physical Particles

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. So in this article we tackle questions about physical particles by counting quarks.

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 formally make this simple definition: Physical particles are compound quarks. So together with David Hume we understand 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. Starting with anti-particles: The anti-particle 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 and thereby described objectively.
  • Photons and Gravitons are compound quarks that have almost no character and are mostly used to explain changes in other particles.
  • Newtonian Particles are compound quarks that are dense enough so that they can absorb a few photons or gravitons without changing very much.
  • 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.

Physical particles as aggregations of quarks are illustrated in this bead panel from Borneo.
Bead Panel, Bahau people. Borneo 20th century, diameter 38 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.

A Quark Index

Next we define a quark index because when counting quarks it is often more convenient to use a number instead of a letter to represent different kinds of quarks. So consider a seed noted by Z where

Z ∈ { U, D, E, G, M, A, T, B, S, C, Ⓐ, Ⓑ, Ⓘ, Ⓦ, Ⓓ, Ⓛ }

IndexSeedQuark
 \zetaZz
1Uu
2Dd
3Ee
4Gg
5Mm
6Aa
7Tt
8Bb
9Ss
10Cc
11a
12b
13i
14w
15d
16l

The seed Z can be used to define an ordinary quark written as \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 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 in formulae.

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 integers because quarks are defined by pairs of perfectly distinct seeds.

CharacteristicDefinition
the total number of Z-type quarksN^{\mathsf{Z}} = n^{\mathsf{\overline{z}}} + n^{\mathsf{z}}
the net number of Z-type quarks{\Delta}n^{\mathsf{Z}} = n^{\mathsf{\overline{z}}} - n^{\mathsf{z}}
the total number of all types of quarks\displaystyle N_{\mathsf{q}}  = \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 as   n^{\overline{\mathsf{z}}} or   n^{\overline{\zeta}} . A few other numbers used for describing particles are defined in the accompanying table. The letter  \Sigma indicates the use of summation notation.

Note that if Z represents a thermodynamic or chemical quark, then  N^{\mathsf{Z}} also gives the number of these sorts of seeds in P. This is because there is just one Z-type seed for each quark, and all quarks are named after their non-conjugate seed. So we may call   N^{\mathsf{Z}} a seed coefficient when discussing thermodynamic seeds.

Counting Anti-Quarks

By the foregoing definitions, the net number of quarks in particle  \mathsf{P} and its anti-particle  \overline{\mathsf{P}} are related as

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

This is just an arithmetic fact, little more than a tautology. It follows directly from the anti-commutative properties of subtraction. However, when we apply it to counting quarks it expresses a fundamental physical symmetry between matter and anti-matter. So we will come back to it later. But next we look at why quarks are forever.

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.

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 reason, 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. Therefore when counting seeds we always report a positive integer or zero, not fractions or negative numbers

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

For the same reason, when we define quarks from seeds, and shift the description to counting quarks, then the coefficient of any quark  n^{\mathsf{q}} must always be a non-negative integer as well

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 and quark-coefficients are related to quantum numbers.

Quarks are conserved and so are the beads shown in this 19th century weaving from Borneo.
Baby Carrier Panel, Basap people. Borneo 19th century, 27 x 19 cm. Photograph by D Dunlop.

Quantum Numbers

Quantum numbers are used to identify and classify particles. They are utilized in both atomic and nuclear physics. We start with the nuclear numbers which are determined from quark coefficients noted by  n and  N . Recall that these quark-coefficients are determined by counting quarks so they are always integers. Therefore the following quantum numbers are all integer multiples of one eighth. They are quantized, thus their name.

The total angular momentum quantum number is defined by

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

Nuclear particles can be classified by these quantum numbers into a few general categories as noted in the accompanying table.

Nuclear Particle Types
a boson \textsl{\textsf{J}} = \textsf{an integer}
a fermion  \textsl{\textsf{J}} = \textsf{an integer} + \frac{1}{2}
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 neutral particle q=0
a charged particle q \neq 0
a strange particle S \neq 0

In general, attributes and identities 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.

Recall that for any seed Z, the net number of quarks in particle  \mathsf{P } and its anti-particle  \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 anti-particles have the same absolute value, but opposite signs.

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

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

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

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

But exchanging quarks for anti-quarks does not alter thermodynamic seed counts, so for the angular momentum quantum number  \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}}

Therefore the lepton number is conserved. By the same reasoning the baryon number and charge are conserved too, so

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

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

But 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.
Next

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