The Semiotic Flora of Elementary Particles

© This paper is not for reproduction without the express permission of the author.

Abstract

This paper
refers (but adds nothing) to the standard model of elementary particles,but
presents many of these particles in a "botanical" way, like the
flowers in a Flora.The vacuum-background for the particles is treated with
special emphasis on the zero-point-energy and its measurable effect — the
Casimir effect. The special importanceof the number 3 in the standard model
leads to the idea that classification may bebased on C.S. Peirce's *triadic*
philosophy of signs — his *Semiotic.* A slightly abbreviated Danish version
of this article will appear in the collection: Thellefsen andDinesen (Eds.) *Semiotiske*
*Unders0gelser,* Gyldendal, 2003.

Thanks are due to Bent C. Jorgensen for suggesting the botanical metaphor and to Edwina Taborsky for inspiring applications of Peirce's semiotic to physics.

1 Introduction

Most natural sciences
start out with a *deictic* ontology (Poli 2001: 1-5), a view that builds
on the distinguishability of objects through nomenclature and placing in a
system of classification.Thus, a natural *science* like biology builds on
a natural *history,* like botany that through the classification of Linné
allows the naming of plants using a well defined system of indexing — a *Flora.*
The physics of elementary particles is long past the state of natural history
by theuse of a strong, but heavy mathematical apparatus in Quantum Field Theory
and group-representations. As the particles by and by have become as numerous
as flowers we can still use a "Flora" for naming and schematically
surveying them. A suitable system for this can be found in Peirce's semiotic.
This makes it possible to find a shorter way through the mathematical jungle,
and certain regularities that still appear enigmatic in the mathematical theory,
seem more understandable in the semiotic perspective.

2 The Wild Vacuum

The physical concept
of a *particle* *—* a point with mass — is, semiotically speaking, an
*icon* *—* a sign whose object is *potential* or *virtual.*
The particle as the physical object the icon refers to has definite properties,
but not necessarily *existence.* A virtual particle is just a possibility
for excitation of the physical vacuum — the empty space. That space is *empty*
does not mean that it is without properties. It has three types of properties,
viz. *optical,* *topological,* and *metrical* properties. The *optical*
properties[1]
entail that space has three dimensions and is *seen* as delimited by a *heavenly*
*sphere* which has no physical existence. Two parallel lines (light rays)
are seen as in the painter's perspective (Peirce CP 6.26) intersecting each
other in two diametrically opposite points and all possible points of infinity
make up "a line in the infinite" i.e., a great circle on the heavenly
sphere, called the horizon.[2]
The topological properties are described by Peirce with four integers, the so
called *Listing* *numbers'* *chorisis,* *cyclosis,* *periphraxis,*
and *immensity* that characterize every three-dimensional object: *Chorisis*
*is* *the* *number* *of* *separate* *pieces* *that*
*make* *up* *the* *object.* Cyclosis is the number of
through-going *holes* or *singularities* *with* *axial* *symmetry*
(like vortices). *Periphraxis* is the number of internal,
three-dimensional holes, and *Immensity* is a number is only different
from zero for an unlimited body. Looking at the whole universe it will have chorisis
and immensity equal to one, while its cyclosis and periphraxis are unknown
quantities reflecting singularities in the *metric* of space. The field equations
of General Relativity that combine the metrical properties with the field of
gravitation show that there are possible singularities corresponding to both
types: *Cosmic* *Strings* add to the *Cyclosis* of space and *Black*
*Holes* add to its *Periphraxis.* How many there are of such objects
in the visible universe is not known, but observations indicate that both types
exist. Within the normally accessible scales of length and energy the physical
vacuum appears completely without structure. It is, though, not without
properties, but hides itself under three fundamental constants of nature, viz:

1.
*c=*3·10^{8} m/s; the velocity of light in a
vacuum

1.
*ħ* = h/2π=10^{-34} J·s; Dirac's
quantum of action, *(h* is Planck's constant)

2.
G*=*6.67·10^{-13}
N·m^{2}/kg^{2}; Newton's constant
of gravitation

Expressed as here in normal (SI) units the numerical values of these constants are either very big or very small, but that just means that the SI-units (length in meters (m), time in seconds (s), and mass in kilograms (k)) are "human measures", far away from the world of elementary particles. However, it is possible to choose units of length, time, and mass, such that the three constants of nature, mentioned above, all get the value of unity in these new units, the so called Planck-units.

·
the Planck-length
is then: L_{p}=_{}*=*4·10^{-36} m

·
the Planck-time
is: t_{p }= L_{p}/c = 10^{-44} s

·
and the
Planck-mass is M_{p }= _{}= 5·10^{-7} kg

A natural starting point for pictures of elementary particles is then a sphere with radius one Planck-length and mass one Planck-mass. Compared to ordinary elementary particles (like electrons) the Planck-particle is of very small extension, but very heavy (ca 0.5 mg).

The force of
gravity on the surface of such a particle will be so strong, that the particle
"swallows itself” and becomes a mini-black hole. This has never been observed.and
will probably never be, since the Planck-energy M_{p}∙c^{2}
= 10^{18} GeV is far beyond the
range of even the largest accelerators. Perhaps there have been many of them
when the universe was only one Planck-time old, but as
"mini-black-holes" quickly evaporate by a process called Hawking-radiation,
they have all disappeared long ago. If we could view the physical vacuum
through a microscope with a resolution of one Planck-length we would likely see
that space on these scales is not without structure, but has both *cyclosis*
(from superstrings) and *periphraxis* (from mini-black-holes). Topology
(and hence also metric) is *chaotic* on the Planck-scale, both in space
and time.

3 Zero point energy

In the holistic "New Age Philosophy's" critique of physical reductionism (as expressed, e.g. by David Bohm) one often sees the assertion that the physical vacuum contains infinite amounts of energy (Wilber 1982). Even the smallest volume, like a cubic millimeter should, according to this conception, contain enough of energy to sustain the whole world for many years.[3]

We shall see how such an idea can arise from a — basically correct — application of physical principles and why it is, despite of this, altogether wrong.

Let us consider
a small part of space delimited by two parallel metal plates separated by a
distance *L.* *Between* *such* *plates* there can be a
series of electromagnetic oscillation-modes that are standing waves whose *half*
*wavelength* is a whole fraction of the distance *L.* An example
would be an oscillating string or a closed organ-pipe where we can distnguish between
a *ground-tone* with the wavelength *2L* and an infinite series of *overtones,*
*where* *the* nth overtone has the wavelength *2L/(n+1).* The
ground-tone has *n=0* and the overtones have *n* from I to ∞.
The frequency of oscillation of each such mode is found by dividing the
wavelength up into the velocity of light *c.* *Thus,* *the* *ground-tone*
*has* *the* *frequency* v*=c/2L.* Every mode can be
considered as a *harmonic* *oscillator,* and according to Quantum
Mechanics it can only have the discrete energy-values

_{}

where *m* is a positive integer or zero. We see that the energy is *quantized*
with the quantum *hv*.

Such
field-quanta can be regarded as particles, and when it, like here, are quanta
of an electromagnetic "light-field" we call the particles *photons.*
Likewise, we speak of *phonons* when it is a sound-field like the
oscillations on a string that are quantized, (c should then be the velocity of *sound).*
If the *n*th mode is excited to the *m*th level we say that there are
*m* photons (phonons) in the *state* *n.* Thus, the ground state
of vacuum is the one where *m=0* for all the states. From the above
formula for the energy-values we see that the energy of each mode in its
ground-state is not zero, but carries the zero-point-energy *hv/2*. As
there are infinitely many modes in the cavity, the total zero-point-energy is
infinite. This, however, is a purely formal consideration that does not
consider the semantic purport in the concept of energy, namely *ability* *to*
*perform* *work.* If an oscillator is excited to level *m* it
can perform work by delivering a quantum hv to the surroundings whereby the
oscillator itself makes a transition to level *m-*1*.* This, however,
is impossible, if the oscillator is in the ground-state *m=0,* because
there are no lower levels. So, the infinite vacuum-energy turns out to be a fiction,
and a "perpetuum mobile of the third kind" is an impossibility like
all other kinds of perpetuum mobile.

One should not,
however, entirely disregard thg zero-point-energy as being unreal, because it
shows itself in other ways than the ability to perform work, namely by *the*
*pressure* it exerts on the surroundings. The so called *Casimir-effect*
is an experimental demonstration of this pressure.[4]

The zero-point-energy
has physical actions and is therefore, according to Peirce's pragmatic
criterion of meaning, real. This assertion leads naturally to the question
"From where did it come?" This is a mischievous question that leads
to the mischievous answer: "We made it ourselves!" There is, namely,
a concept-logical connection between *localizing* a particle (to ensure
that it is situated in a certain, limited region of space) and to *transfer*
*energy* *to* *it.* This connection is expressed in *Heisenberg*’*s*
*uncertainty* *relation*

∆*x*∙∆*p>h*

where ∆x is the
uncertainty of spatial location and ∆p the uncertainty of momentum (mass
times velocity). If we try to localize the particle strongly, i.e., make
∆x very small, then ∆p will be, correspondingly, greater. The
particle will not rest qiuetly when we keep it in a narrow cage, and therefore
we have to perform work by narrowing its limits — a work that adds to the
kinetic energy of the particle. This argument is also valid when there is no
particle. For example there are *no* *photons* when all the
oscillatory modes are in their ground-state. The zero-point-energy of the photon-field's
ground state is, according to the previous derivation *hc/4L,* i.e., it increases
when we diminish *L* and the increase comes from the work we do by the compression.

4 The Vacuum Press

Let us perform a
thought-experiment wherein we compress the vacuum by means of the apparatus
shown in figure I. The cavity-length *L* is here the distance between the
piston and the bottom of the box.

**Figure
1:** The Vacuum Press

When we press down the
piston we change the wavelength of the ground-mode and thereby increase the
zero-point-energy. For sufficiently small values of *L* the
zero-point-energy will be greater than the relativistic rest-energy *mc²*
of a particle of mass *m.* This, however, is not sufficient to create the
particle, because, if it emerges within the box it will have a "localization-kinetic-energy"
according to Heisenberg's uncertainty relation, and this energy increases
faster (inversely proportional to *L²)* when *L* decreases and therefore
there will never be enough of zero-point-energy in the photon-field to create a
particle with mass. If there are holes in the box potentially existing
particles may escape and then have no localization-energy. There will then be
enough of energy in the photon-field to create an electron when *L*
becomes smaller than the *Compton* *wavelength* of the electron *λc=*
*ħ/mc ≈ *3∙10^{-13 }m, where *m **≈ *9∙10^{-31} kg is the mass of the electron.

When we try to
press the piston to the bottom various particles will sprout from the holes
like seeds of an orange when *L* passes below their respective Compton wavelengths.

The Compton wavelength puts a natural limit to how narrowly a particle may be localized.
If we think of the particle as a small hard sphere, we can think of the Compton wavelength as the radius of the sphere. The radius of the electron is
then ca 1000 times as small as the radius of a hydrogen atom and ca 2000 times
as big as the radius of the atomic nucleus (the proton). In the Planck system
of units (where *ħ*=l and c=l) the radius of the particle is simply
the reciprocal of its mass. A particle of one Planck-mass (a mini-black-hole) will
have radius one Planck-length — the smallest distance that can be connected
with classical conceptions of space-time.

It may seem contradictory when we claim that the zero-point-energy cannot perform work but is yet able to produce particles. The explanation is, again, that the holes in the box, that allow the particles to escape also makes it possible for the zero-point-oscillation to yield, i.e., decrease its frequency and thereby its energy. Still, we maintain that the work comes from the compression of the piston and the zero-point mode is only an intermediate storage-medium for the energy.

The most
efficient method of compressing space consists in providing two massive particles
with a high velocity in an *accelerator* and then arranging a *collision*
between these particles. In CERN's (newly abolished) LEP (Large-
Electron-Positron-Collider) the collision- energies reached about 100 GeV, and
that is not quite sufficient to produce the currently most interesting particles (as the Higgs-boson).[5] A new accelerator LHC
(Large Hadron Collider will, within a few years yield significantly higher
collision energy by using hadrons (like protons) that are about 200 times more
massive than electrons (and thereby also more compressed beforehand).

5 Renormalisation — just smart, or a bit too smart?

The previous
discussion of the vacuum press and the Casimir effect (the pressure on the
piston) is incomplete, because it only takes into account the ground-mode of
the photon- field. Naturally we must also regard the infinity of overtones, but
that leads to the problem that the total zero-point-energy (and thereby also
the pressure) becomes infinite. The zero- point energy of the *n*th mode
is:

_{}

It is therefore
clear that the complete zero-point-energy includes a factor that is the sum of
all positive integers from I to ∞, and this factor must, for a normal
consideration, be infinitely great This we could, perhaps, learn to accept,
for, as we have seen, the zero-point- energy cannot perform work, so we could
disregard it as being non-energy. But it's not so easy. Every single mode gives
rise to an upward-directed force on the piston that is *K _{n}* = -
d

Casimir's
calculations, as well as Spaarnay's experiment even show that the pressure is
negative, i.e., the force on the piston is directed downwards. We are,
therefore, forced to "explain away" or *renormalize* this
infinity. A way to do this is by using a mathematical technique called *analytic*
*continuation.* A very important function in Mathematics is *Riemann*’s
*xetafunction* ζ(z) that is defined for *complex* *numbers*
*z=x+*i*y* in the following way:

_{}

This definition
is entirely clear for all *z* whose real part, *x,* is greater than
1, because then the series converges to a finite value. However, the function
has a unique *analytical* *continuation* to the whole complex plane,
including negative real values of *z,* where the series is divergent.
Formally, we can put *z=*-1*,* whereby the infinite sum becomes the
previously mentioned sum of all positive integers, and we can assign it a value
given by the analytical continuation of the zetafunction to *z=*-1*.*
In this way we get at the renormalized value ζ(-1) = -1/12^{4} ,
i.e., not only have we transmutated the infinite factor to something finite we
have even given it the correct sign! In a similar way we can "prove"
other absurdities, e.g. that ∞ = -½, for if we put *z*=0 in the
above formula we get a sum of infinitely many 1s, i.e., ∞, and the
analytical continuation ζ(0) has the value -½.

Such a
mathematical renormalization-technique appears "a bit too smart"
because it may lead to screaming absurdities, but the method should not be
entirely rejected, as it is, in fact, applied and often leads to results that
are completely correct. An example is the so called *factorial* *function*
n! = l∙2∙2∙∙∙∙n, i.e., the number of
permutations of *n* objects, that is defined for positive integers *n.*
*An* *analytical* *continuation* *employing* *the* *so*
*called* Gammafunction allows us to define (-l/2! =√π), a
result that no mathematician or physicist will cast in doubt.[6] We shall not "throw
out the baby with the bathing water" by prohibiting renormalization by analytical
continuation, but still, I want to go through a physical argument of
argumentation reflecting Casimir's calculation and, hopefully, making it a
little less suspect.. I shall give a short outline of the argument here, while
the details can be found in the appendix.

Hitherto, we
have only considered the electromagnetic modes in the cavity *below* the piston
in figure 1. This infinity of modes, all have wavelengths less than 2*L.*
However, they exist also *above* the piston, where each of them gives rise
to a *downwardly* directed force that precisely cancels the *upwardly*
directed force from the corresponding mode below the piston. In this way we
remove the infinity, so what is left?

There are all
the modes whose wavelength is *greater* than 2*L,* and these modes
are only found *above* the piston. By adding the forces from these modes
one finds that the resulting force on the piston is *downwardly* directed
with the finite value

_{}

Curiously enough
the previous "bit too smart" renormalization argument gives almost the
same, viz._{}, i.e., the correct sign and
only a factor 6 smaller than the right numerical value.

The Casimir
"pressure" is thus a "suction" (because *K _{total}*
is negative) but it can only be felt when

If the vacuum
press in figure 1 shall be able to squeeze particles out of vacuum, the pressure
must be positive. The calculation above, giving a negative pressure can
therefore only be valid for distances *L* larger than the Compton wavelengths of the virtual particles.

The reason why
I've given a relatively lengthy discussion of vacuum is to avoid that the
following enumeration of particles and their properties should be regarded as
reduc- tionistic: We need a holistic conception: The whole is more than the sum
of its parts — the elementary particles don't have properties that are
independent of their context. A particle is *a field-quantum* and
interacts with virtual fields in vacuum. We cannot calculate the properties of
one single free particle, e.g. its mass, for its properties reflect the wild
vacuum, although it is not quite as wild as certain "quantum holists"
claim (infinite energy density, etc.)

6 The mysterious number 3

In a popular "nature-historic" account of the elementary particles (Petersen 1991) the author seems puzzled about "two mysterious 3-numbers" that have emerged in later years in particle-physics:

2.
There are three *generations*
of elementary particles, and

3.
The heavy nuclear
particles — *baryons* (as the proton and the neutron) consist of *three*
*quarks.*

We can easily expand
the list over the fundamental roles of the number 3: Space has three dimensions
and three types of properties (chapter 1). There are three types of units
(length, mass, and time) and three fundamental constants of nature (*ħ,c,*
and *G).* The quarks have three "colours" (red, green, and blue)
and strange electric charges that are not built of the electron's charge *e*
as the quantum of charge, but of l/3e.

Previously one
got used to *dichotomies,* or *two-partitions:* There are *Fermions*
(as electrons and quarks with *half-integer* spin, usually ½) and *bosons*
with integer-spin (0 for *π* mesons, 1 for photons, and 2 for gravitons).
There *are* *particles* and *antiparticles,* and there *are*
*positively* and *negatively* charged particles with *even* or *odd*
*parity.* These *dichotomies* can be understood from the concept of *mirroring.*
The mirror-image of a *particle* with *negative* charge and *even*
parity will be an *antiparticle* with *positive* charge and *odd*
parity, but the newly discovered *trichotomies* are not connected to
mirroring and therefore appear strange. I shall not attempt to seek a
mathematical justification of the trichotomies but instead take departure from
a philosophy that is built on trichotomies and thereby make the
"mysterious" number 3s appear as something natural and inevitable,
namely *Peirce's* *Semiotic.*

7 Semiotics of the particle-concept

A previous article by the author (Christiansen 1997) explained how Peirce's triadic doctrine of categories implies that signs can be divided in classes that can be put up in triangular schemes.

On the most
elementary level of description a sign is something that mediates between an *object*
and an *interpretant* schematically I---O where the line stands for the
sign-vehicle — a physical signal or link between O and I. By using Peirce's
categories we can distinguish between three types of links or mediating
processes:

3.
A *potential*
link that only exists as a possibility. The sign is then called an *icon.*

4.
An *actual*
link from an existing object is an *index.*

5.
A *general*
link referring to a general class of objects defines a *symbol.*

**Figure
2:** Sign classes for the I-link relation

On the next
level of description the sign-vehicle or the *Representamen* R is objectified
and gets two links to O and I after the scheme I---R---0. These two links
represent the elementary quantum processes *Preparation* (R---O) and *detection*
(I---R) and each of them can be classified by the three categories, though only
such that the category of the detection- link cannot exceed that of the
preparation-link. In this way we arrive at *the* *six* *quantum-*
*semiotic* *sign* *classes* shown in figure 3:

**Figure
3:** The 6 quantum-semiotic sign classes for the
two-link relation of I---R

As the
interpretant I is to be regarded as a sign-vehicle (representamen for a new interpretant
J, we are led on to consider a three-link relation J---I---R---O consisting of
the two two-link relations 1:I---R---I, and 2:J---I---R, where we see that the
I of the first relation appears as representamen in the second relation, and R
in the first relation appears as object in the second relation. When we, as
before assign the links categories that cannot increase when we move left in
the diagram we find Peirce's 10 classes of signs (CP2.256) arranged in a *Pythagorean*
*Tetraktys* (figure 4)

**Figure
4:** The 10 sign classes for the three-link-relation.
The six classes from figure 3 are the ones,that have 1 in the first place:
(111)-(133).

We shall mainly
employ the 6 quantum-semiotic sign-classes on figure 3. *A.* *particle*
*is* a *quantum* *of* *a* *field* and therefore has
two "handles" or links corresponding to the two fundamental quantum
processes *preparation* and *detection.*

There is a clear
line of development in the scheme of figure 3, starting with the qualisign 11
and continuing with successive *actualizations* of potential links (1 to
2) and *generalizations* of actual links (2 to 3) without skipping any
intermediate stages on the way to the symbol 33. I shall briefly sketch a
"nature-historic" interpretation of the sign classes in the right
order with hints to particle physics.

**11**- The qualisign is the empty space,
only containing *virtual* fields and particles, like the electromagnetic
modes, that can be occupied by photons, but otherwise are in the ground state.

**12 **- The hypoicon can be thought of as a *superstring*
*—* en string i vacuum having certain field-and particle-properties,
reflecting how the string is wound up in other dimensions than the one it is
stretching in. The theory of superstrings operates with 8 hidden spatial
dimensions and is *supersymmetrical* with regard to Fermion- or
Boson-properties of the strings.

**22 **- Sub-index: the string closes upon
itself to a ring with *area.*

**13 **- Icon: the ring moves in space and
thereby establishes a three-dimensional container-like region. Thus appears the
three-dimensional *continuum* *of* *quality* that is the premiss
for a sign to refer to an object by *likeness* as an icon does. If the
ring is creased there will be bumps in the container, corresponding to a
non-euclidean metric that reflects gravitation.

**23 **- Index: The iconic particle may
collide with another and produce a lot of unspecified particles through the
action of the vacuum press. This is *indexical* *semiosis,* not yet
generalized to lawlike behavior.

**33 **- Symbol: As we learn about the
properties of particles we become able to predict results of collisions between
known particles and interpret their traces. On this stage both preparation and
detection are generalized and we have reached the symbolic level of description.

This account of the evolution of signs is inspired by Edwina Taborsky (2002).

8 Contents of the botany-box.

When you go out to
botanize, identifying the names of flowers in a *Flora* there are certain
concepts you need in your head and certain tools to keep in your box, besides
the flora and the lunch-packet, e.g. a magnifying glass for counting petals and
stamens. Similarly, the elementary particles are classified by their *internal*
*properties* and their *interactions* with other particles and
fields.

·
The internal
properties are *spin,* *mass,* *and* *charge.*

·
The interactions
are *strong,* *electromagnetic,* *weak,* *and gravitational,*
here listed after decreasing strength.

All particles,
also the massless, as the photon, interact gravitationally. This universal force
is described in Einstein's General Theory of Relativity, but because it is so
weak and hitherto has evaded a quantum mechanical treatment it has mostly been
ignored by elementary particle- (or high energy-) physics. It is known, though,
that the field quantum of gravitation, the *graviton* is massless and has
spin 2.

All *charged*
particles (including quarks and "heavy leptons") interact
electromagneti- cally. The strength of this interaction is determined by the
electron's charge *e* which was long believed to be the universal quantum
of charge, until it was discovered that the charges of quarks are 2/3 or -1/3 e.

All *hadrons*
(quarks) and *baryons* (nuclear particles consisting of three quarks) interact
*strongly* *and* *weakly.* "The strong interaction was
earlier described as mediated by medium- heavy bosons, called *mesons* *(π*
and K), but now we have learned that mesons too are compounds (by a quark and
an antiquark). So what is left of the strong interaction is the force between
quarks, whose field-quanta are called *gluons* (8 kinds with spin 1)

Quarks and
leptons interact with each other through the *weak* *interaction*
whose mediating field quanta are the *intermediate* *vector-bosons*
(3 kinds with spin 1). The uncharged light leptons, the *neutrinos* only
interact weakly (and gravitationally).

After this short
account of *interactions* now follows a survey of the most important *internal*
*properties* of the particles:

The
(rest-) *mass* *m* and the *Energy* *E* are connected
through Einstein's relation

*E* = *mc ^{2}*

In the Planck
system of units, where *c=*1 they are identical. The most commonly used unit
for this quantity is the *electronvolt* *eV,* that is the energy an
electron gets by accelerating through a voltage-drop of I volt (I *eV* ==
1.6-10'9 J). Besides, we have the multiple units k*eV* (kilo=1000), M*eV
*(Mega = 10^{6}) and G*eV* (Giga = 10^{9}). The electron's
mass is ca 500 k*eV* = *½* M*eV,* *corresponding* *to*
*m*=9.1∙10^{-31} kg. The photon and the graviton are
massless and can therefore only move with the velocity of light. Earlier it was
believed that also the neutrinos are massless (they were observed almost simultaneously
with the light from the supernova in the big Magellanic Cloud in 1987), but
recent experiments in Japan have shown that at least one of the three
neutrinos has a mass about 10*eV* *about* *one* *hundredthousandth*
*of* *the* *electron-mass.* *As* *a* *rule* *of*
*thumb* *one* *may* *assume* *that* *the* *greater*
*mass,* *the* *later* *is* *the* *discovery* *of*
*the* *particle,* *because* *the* *accessible* *accelerator-energies*
*have* *increased* *gradually* *from* *the* k*eV*
to the Ge*V *range.

*Spin* is an internal *angular* *momentum*
(length times momentum) and is quantized in units of *ħ.* Earlier it
was believed that particles are small hard spheres and that the spin expressed
the sphere's rotation about its own axis, but it has turned out that only *integral*
spin- values can be interpreted in this way. Now we will say that spin is
concerned with how the symbolic representation of the field is changed by a
rotation of the coordinate-system used in the description. Spin can be integral
or half-integral as the particles are, respectively *bosons* or fermions.
Fermions are *exclusive',* there can only be one fermion of a given type
in a given quantum-state. Bosons, on the contrary, are "social". They
are prone to go together in the same state and form a *condensate* as
known from superfluid systems and laser-light (a condensate of photons). The
exclusivity of fermions makes it tempting to regard them as the most evident fundamental
bricks in an atomic description of matter. Boson-condensates are more apt for describing
classical fields. Every fermion has an anti-particle that is different from
itself, whereas a boson's anti-particle often (not always) is the same
particle. The theory of superstrings (ref. 8) — the newest candidate for a
uniting theory of particles and fields is *supersymmetrical* (hence the
name), i.e., it postulates that every boson has a fermion-partner, and vice
versa. The photon's supersymmetric partner is called the* photino*; it has
not yet been seen.

Spin ½ particles
(like quarks and leptons can have the spin pointing either *forwards *or *backwards*
in the direction of movement. This means that they possess *parity* or *helicity,
*i.e., they are different from their mirror images. Neutrinos are lefthanded
*(spin* *against* *movement),* *and* *antineutrinos* *are*
*righthanded.* An important theoretical result, *the* *CPT-theorem*
establishes that the theory must be invariant for the combination of three
mirror-operations *C,* *P,* and *T*, i.e., changes of sign for, respectively
charge (*C*), parity (*P*), and the direction of time *(T).*

9 Semiotic classification of elementary particles.

We must distinguish
between proper *elementary* *particles* and *compound* *particles*
that are built of elementary particles. Elementary particles are, for
historical reasons, divided in *three* *generations,* 1, 2, and 3,
whose order corresponds to the order of their discovery, which is connected with
the circumstance that the mass (and thereby the necessary accelerator-energy for
the production of them) increases from generation 1 to generation 3.

Each generation
consists of *two* *leptons* *and* *two* *quarks.* So,
altogether we have 6 leptons and 6 quarks, which makes it tempting to place
them in the semiotic classification with 6 sign-classes in figure 3. All these
12 particles are spin* *½ fermions, each having their own anti-particle.
So, there are really 24 different particles, but in the following we shall
disregard the
anti-particles. I cannot, at
present, give a proper semiotic reason for the placing of every single particle
in the scheme, so I'll just use a heuristic rule that uses the evolutionary sequence
of the 6 sign-classes and assume that this order reflects an increasing mass of
the particles. Thus, we get the schematic placement of the 6 leptons:

**Figure
5:** The leptons

**Generation** **1** consists of the electron *e *and
its neutrino ν_{e}

**Generation** 2 consists of the muon μ and its neutrino ν_{μ}.

**Generation** **3** consists of the tauon τ and its
neutrino ν_{τ}

The three
"ons" all have the same (negative) charge as the electron -e, while
the neutrinos are uncharged. As mentioned it is now shown that at least one of
the neutrinos has a restmass about 10 *eV,* which doesn't give a clue for
ordering after increasing mass, so I've just assumed that their mass increases
from generation 1 to generation 3 and ordered them accordingly in the lowest
row of the scheme.

Considering now
the *quarks* that are charged spin ½ fermions, we have the following distribution
on the generations: For each of the 6 quarks is noted its mass, measured in *MeV.*
*Their* *charges* *are* *given* *in* *the* *left*
*column:*

**generation**** I generation 2
generation 3**

+2/3e: *u* *(up)* *(5)*
*c* *(charm)* (350) t (top) (>80)

-l/3e: *d* *(down)* *(9)* *s*
*(strange)* *(160)* b (bottom) (4800)

** **

**Figure** **6:** The three
generations of quarks

We note that for
both leptons and quarks the *charge-difference* between the two particles
in a generation is always one electron-charge. The placement of quarks in the
semiotic scheme then looks like this:

** t**

** b
c**

** s d
u**

** **

**Figure**** 7**: The quarks

The top-quark **t**
is here placed highest in the scheme, because it is the latest discovered,
although its mass is not known very precisely.

As earlier
mentioned the quarks participate, both in the strong, the electromagnetic, and
the weak interactions. Weak interactions are mediated by a *vector-field*
analogous to the electromagnetic, whose quanta are spin-l-bosons, as the
photon. However, they are not massless as the photon, but, on the contrary,
very heavy (90-100 *GeV),* which has been difficult to understand, but is
now explained by assuming that they interact with an uncharged, spin 0 field (a
condensate) of so called *Higgs-bosons,* that are not yet seen with
certainty in experiments.

The weak
interactions make it possible that the heavy quarks may decay to the lighter, e.g.
the process d→u + W^{-} will be possible, when W^{-} has
a single negative charge

*-e* like the electron. Correspondingly, we have the process c→d + W^{+} where the positive W^{+}
is the anti-particle to W^{-}. Finally we have the possibility s→d
+ Z where Z is uncharged, as s and d both have the charge -⅓ *e.*
All these three vector-bosons are discovered at CERN. Because Z_{0} is
the heaviest of the three, the s-decay runs slower than the d- and c-decays.

10 Compound particles.

The greatest merit of
the quark-theory is that it can explain the properties of the heavy fermions —
the *baryons* *and* *the* *intermediately* *heavy* *bosons*
*—* the mesons. Earlier it was believed that mesons (especially *the
pions* π^{+}, π^{-}, π^{0}) were
elementary quanta of the field that mediates the strong interactions between
baryons, but now we know that all these particles are compounds:

·
A *baryon*
(spin ½ or 3/2) consists of three quarks

·
A *meson*
(spin 0 or 1) consists of a quark and an anti-quark.

Both quarks and
anti-quarks have spin ½*,* so a meson may have spin 0 or spin 1.

Traditionally
known strong interactions, mediated by π-mesons involve only generation 1
quarks u and d. For spin 0 mesons we then have the three possibilities (anti- quarks
are denoted with a stroke above the quark-symbol): (u,_{}) = π^{+},
(d,_{})
= π^{-} and π^{0} or η or (d,_{})[8] i.e. two charged and two
uncharged particles The diagrams show immediately that π^{+} and π^{-}
are each other's anti-particles, while π^{0} (or η) are its
own anti-particle.

Spin 1 mesons
are very unstable and decay rapidly to spin 0 mesons. This family contains,
besides π-mesons, also combinations that contain s quarks, namely the K-mesons
(kaons) K^{+} = (u,_{}), K^{-} = (s,_{}), and K^{0}
= (s,_{}).
It is seen that K+ and K' are each other's antiparticles, while K° is different
from anti-K°.

For *baryons*
it is valid, as mentioned, that each of them consists of *three* *quarks*
and, thus, may have spin ½ or 3/2. For spin 3/2 baryons all combinations of the
three lightest quarks (u, d, and s) are possible. We may then assign the three
quarks the Peircean categories: u=l, d=2, and s=3 and apply the scheme of
3-link sign-relations (figure 4) to the following figure 8:

**sss**

**dss uss**

**dds uds uus**

**ddd udd uud uuu**

**Figure** **8:** The
spin-3/2 baryon-decuplet

This figure is
called the spin 3/2 *decuplet* (because it contains 10 particles) is one
of the quark-theory's great successes, because it predicts the existence of a
hitherto unknown particle, the topmost in the scheme, (sss), called the Ω'-particle.
It carries a negative unit of charge, (as s has the charge -1/3) and is rather
stable (long-living) because it only decays via the weak interaction with the
heavy Z° particle.

Spin ½ baryons
cannot contain three identical quarks. This rule is coming from Pauli's
exclusion principle, that forbids two fermions to occupy the same quantum
state. Using again figure 4 (or 8) as our starting point we have to omit the
three corners of the triangle, so there are only 7 spin ½* * baryons, of
which the most important is the *proton* (uud) (charge *+e)* and the *neutron*
(udd) (charge 0). Spin 3/2 baryons (except Ω') decay rapidly to spin ½ baryons
via the strong interactions.

It was a great
obstacle for the early quark-theory, that one never sees a single (free) quark.
It seems that they only exist three or two at a time, confined in the prison of
baryons or mesons. In order to explain why the 3 always were together it was
decided to assign them a *colour* or colour-charge, r, g, or b (red,
green, or blue). The rule then is, that only "white" particles can
appear as free, namely the baryons (rgb) or the mesons (colour+complementary colour,
e.g. blue+yellow). In the modem theory of strong interactions, called
quantum-chromo- dynamics (due to the colours) it is assumed that the quarks
attract each other with forces that are weak at short distances, but strong at
large distances. These forces, which "glue" the quarks together in
"white" bundles, are mediated by field-quanta that are called *gluons,*
which like photons are massless spin-1-particles. As a force between two quarks
act between 3-3 colour- combinations, one should think there would be 9
different gluons, but it turns out that the photon is hiding among these
combinations, so there are only 8 gluons. With the high accelerator-energies
that are available today, it is possible to tear quarks out from baryons and mesons.
They are then seen as "jets", i.e., long stripes, consisting of
quarks, anti-quarks, gluons and a lot of other particles that are created when
one with brutal force tears the quarks loose from their attraction.

11 Conclusion.

The particles that are
here classified by means of semiotic schemes are all described in the so called
*standard* *model* of elementary particles, we have looked at 63
particles (24 leptons and quarks with anti-particles, 10 spin 3/2 baryons, 7
spin ½ baryons, 4 spin 0 mesons, 6 spin I mesons, 3 vector bosons, 8 gluons,
and I Higgs boson). The Higgs-boson is, as mentioned, not yet found with
certainty, and that is regarded as a problem for the standard model, because it
plays an important role for the understanding of why certain particles (as vector-bosons)
have mass. The theory indicates that the Higgs-boson is related to the heavy b-quark
and therefore only is produced at very high energies (over 100 *GeV).*

The semiotic
approach is a *schematization,* not a physical theory, like the standard model,
that has its own difficulties to fight against, notably the lack of
supersymmetry. Superstring theory solves this problem and also includes
gravitation, which the standard model has avoided. But, probably, there will
still be some use for a "nature-historic" account, as the semiotic,
that "steal around" the heavy mathematical apparatus that the
theories require.

12 Appendix

**Casimir-renormalization**

As we have seen, a
cavity between two plates separated by a distance *L* may contain an
infinity of standing waves whose wavelengths are all smaller than *IL.*
The zero point energy of these modes would give rise to an infinite positive
(upwards directed) force on the upper plate if it wasn't for the fact that the
same modes exist *above* the plate and each of the modes above provide a
negative, downwards directed force that precisely cancels the force from below.
There are still modes above the plate, not yet taken into account, namely all
the modes with wavelengths greater than 2*L.* So, let us imagine a second
plate with a distance *L'* from the first plate. The *n*th mode in
the cavity between *L* and *L"* will have the wavelength

_{}

the frequency of this mode is

_{}

and its zero-point energy is

_{}

which gives rise to the force

_{}

The force of the mode as a function of its wavelength X is then

_{}

In order to sum all
these forces we have to find the density of modes. First, the number of modes *dn*
in the frequency-interval *dv* is

_{}

So, the number of modes per wavelength interval is

_{}

The total force from all the modes with wavelengths greater than 21 can the be calculated as

_{}

with the result

_{}

It is seen that the
total force is downwards (negative) when *L<L'/2,* *and* *in*
*the* *limit,* *when* *the* *external* *cut-off,*
*L' *goes to infinity we get the earlier quoted renormalized result

_{}

references (Danish references are marked with a (D))

Abramowitz,
M. and I.A. Stegun. (Eds.) 1965*. Handbook of Mathematical Functions.* New York: Dover.

**(D)** Christiansen, Peder Voetmann, Fysik og semiotik
— semiotikken i fysikken, *Anvendt* *semiotik,* red. Keld Gall Jorgensen,
Gyldendal, 1997.

**(D)** Jensen, Michael Agermose 1998. ‘Article’.
Journal of the Niels Bohr Institute, Copenhagen:
Gamma, No. 110. June.

Peirce, Ch. S. 1976. (Ed. C. Eisele). *The New Elements of Mathematics*.

Peirce,
Ch. S. 1931-35. (Eds. C. Hartshorne, P. Weiss and A. Burks). *Collected
Papers. *Cambridge, MA: Harvard UP. References are by volume and paragraph.

(D) Petersen, Jens Lyng: Elementarpartikier — om felter, kvarker og kvanter. Gyldendals Intro-serie, 1991.

(D) Serensen, Bent. 1987. Superstrenge — en teori om alt og intet. Munksgaard, Nysyn, 1987.

Poli, R.
2001. Foreword, *Axiomathes,* vol.12: 1-5. Dordrecht: Kluwer.

Taborsky, E. 2002. ‘Energy and Evolutionary Semiosis’. Sign Systems Studies 30.1. University of Tartu.

Wilber,
K. (Ed.). 1982. *The Holographic Paradigm and other Paradoxes: Interview
with D. Bohm*. Boston: Shambala. Ch. 5.

[1] Pierce uses the name *optic* for the discipline that is
now called *Projective Geometry*. Topology he calls *topic*. Pierce
claims that *optic* and *topic* should precede *metric*.

[2] Every plane bundle of parallel directions of view has a horizon, and all horizons together make up the heavenly sphere.

[3] A hypothetical engine that can extract the vacuum energy is called a “Perpetuum Mobile of the third kind.”

[4] This effect, predicted by the Dutch physicist H.G.B. Casimir in 1948 was verified experimentally by M.J. Spaarnay in 1958.

[5] The most interesting particles are those predicted theoretically but not yet found with certainy experimentally.

[6] !

[7] (D) See, e.g., the article by Michael Agermose Jensen in the journal of the Niels Bohr Institute, Copenhapen, Gamma, nr. 110 from June 1998.

[8] The two uncharged particles π_{0} and η
are both quantum mechanical mixtures (superpositions) of (u,_{}) and the heavier (d_{})