### Beyond textbook physics – searching for the origin of colours

The present,
grey-coloured pages propose a unified description of
fundamental physics – including quantum gravity – called the
*strand tangle model*
and compare its
consequences and predictions with experiments.
Strands describe all of nature with a single principle:

General relativity and the standard model are due to fluctuating strands of Planck radius, for whicheachchange from an overpass to an underpass yields Planck's quantum of action ℏ.

Strands fill all of nature, matter and vacuum. Strands push each other. Pushing strands describe and explain all change and all movement in nature. Only strands explain the origin of each gauge interaction, its gauge groups and other properties, the origin of each elementary particle and its quantum numbers, and the origin and uniqueness of each fundamental constant: particle masses, the fine structure constant, the nuclear coupling constants, and the mixing angles. However, their exact values still need more precise calculations. In the research literature, only strands explain the origin of wave functions; and only strands allow deducing, without any possible alternative, both the Lagrangian of general relativity and the Lagrangian of the standard model.

The strand tangle model is daring in its ideas, complete in its coverage, but specific and restrictive in its experimental predictions.

A simple introduction to strands in 12 one-minute steps is found on this page.

Strands provide the *only* unified description of motion, as shown in
this text.

A pedagogical introduction to wave functions, forces and particles is
here.

A compact introduction to the origin of the gauge groups is
here.

Talk slides with an introduction to
the strand conjecture are here.
They provide
a complete description of motion, with animations
and experimental predictions, explaining the origin of the
fine structure constant 1/137.03, the electron mass, and thus all
colours.

A motivating introduction for scientists and philosophers is
this essay.

For a detailed comparison of the strand conjecture with experiments, read here.

For commented scientific publications and preprints about strands, read further.

Spinning electron.
Evolving photon.

● Step 1: Maximum force– implies general relativity

● Step 2: From maximum force to
unification– requires strands

● Step 3: From strands to black holes, general relativity, and quantum gravity

● Step 4: The strand
conjecture about wave functions and field theory

● Step 5a: The strand tangle
model yields the standard model
of particle physics

● Step 5b: Quantum electrodynamics deduced from strands

● Step 5c: Quantum chromodynamics deduced from strands

● Step 6: Cosmology deduced
from strands– about dark matter and dark energy

● Outreach

● Acknowledgements

● Volume VI of Motion Mountain

● Strands in other languages

● Publication list

### Summary: what happens beyond the standard model?

1. **All observations and all equations of fundamental
physics** follow from crossing switches of fluctuating strands with
Planck radius that reach the cosmological horizon.

2. **All fundamental constants** – the number of dimensions,
the coupling constants, the particle masses and the mixing angles
– are unique and follow from the strand tangle model. They **can be
calculated.**

3. **Strands predict** – like textbook physics
does and like Bronshtein's physics cube
does – that there is **no physics beyond the standard model**
with massive Dirac neutrinos and **no physics beyond general
relativity.** All experimental predictions deduced from strands so far
– listed here – agree with all
experiments. **No observation remains unexplained** in fundamental
physics, including the principle of least action. Finally, strands predict
that **no other, inequivalent model yields these results.**

4. **Particles** are rational tangles of strands, **wave
functions** are blurred strand crossing densities, **black hole
horizons** are blurred weaves of strands, **space** is made of blurred
crisscrossing strands, **curvature and gravity** are due to
inhomogeneous strands, and the three **gauge interactions** are due to
the three Reidemeister moves of strands.

5. **Properties:** One principle. Based on Lorentz invariance.
Implies 3 dimensions. Deduces the 3 gauge interactions. Derives the 3
generations and the known elementary particles. Reproduces spinors and
perturbative quantum field theory. Reproduces curvature and general
relativity. Reproduces black hole entropy. Unifies general relativity and
quantum theory. Makes testable predictions: no new
physics, no supersymmetry, no science fiction.

**All derives from the fundamental principle:**

The fundamental principle animated by Ben Kilgore (just click):

The fundamental principle states that each strand crossing switch produces a quantum of action ℏ. The principle allows deducing the complete standard model of particle physics and full general relativity. This includes deducing spinor wave functions, antiparticles, Dirac's equation, the gauge groups U(1), SU(3) and broken SU(2), electromagnetic fields, Maxwell's equations, QED, the nuclear interactions, QCD, the particle spectrum with its three generations, massive Dirac neutrinos, PMNS mixing, the Higgs mechanism, weak CP violation but strong CP conservation, and the quark model, but also curvature, the metric, Einstein's field equations, the Hilbert Lagrangian, and cosmology.

### Step 1: Maximum force

The story started with the discovery, in the years 2000 to 2003, of
the maximum force value c^{4}/4G in nature, by Gibbons and, independently, by
the present author. In the past decades, a few papers tried to refute the
result; over twenty papers by various groups across the world have
confirmed it and corrected the apparent refutations. The author is also
the discoverer of the *principle* of maximum force, i.e., of the
result that general relativity can be completely deduced from nature’s
limit c^{4}/4G. The last papers by the author on the topic are

A. Kenath, C. Schiller and C. Sivaram, From
maximum force to the field equations of general relativity - and
implications,
*International Journal of Modern Physics D* 31
(2022) 2242019, 10.1142/S0218271822420196.
This paper won an honourable mention in the 2022
Competition of the Gravity Research Foundation.
Download
the pdf here.

C. Schiller, Tests for maximum force and maximum power, Physical Review D 104 (2021) 124079. Preprint here.

C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104 (2021) 068501 10.1103/PhysRevD.104.068501. Free preprint here.

C. Schiller, From maximum force via the hoop conjecture
to inverse square gravity, *Gravitation and Cosmology* 28 (2022)
305–307, 10.1134/S0202289322030082.
Download the
pdf here.

All deduced results on maximum force c^{4}/4G – or on the
equivalent maximum power c^{5}/4G, maximum mass flow rate
c^{3}/4G or maximum mass to length ratio c^{2}/4G –
agree with all observations so far, including those of the LIGO and Virgo
collaborations, and of all neutron star and pulsar measurements. All these
limits are equivalent; each one defines general relativity.
For details, see the dedicated web page on maximum force and
maximum power.

### Step 2: From maximum force to unification with strands

The necessity to use strands to achieve unification is deduced in simple terms in C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity, published in Zeitschrift für Naturforschung A (2022) https://doi.org/10.1515/zna-2022-0243.

A deeper dive towards unification with strands, still accessible to every physicist, is C. Schiller, From the Bronshtein cube of limits to the degrees of freedom of relativistic quantum gravity.

A popular account has been published online in Essentia.

These texts make clear predictions on how to pursue unification, and deduce the lack of possible progress in most other directions. In particular, they summarize all experiments ever made and show that unification takes place in three spatial dimensions, that unification must take into consideration the smallest length in nature, the double Planck length, and that the fundamental, common constituents of space and particles must be filiform, fluctuating and of Planck radius. In short, the texts show that the fundamental constituents of nature must be strands.

### Step 3: From strands to black holes, general relativity, and quantum gravity

The description of nature with strands reproduces the field equations and quantum gravity. This is shown in the following publications.

C. Schiller, Testing a conjecture on the origin of
space, gravity and mass, *Indian Journal of Physics* **96** (2022)
3047–3064. Read the published paper
online for free at rdcu.be/czpom. Dowload the
preprint here.

A dedicated discussion of the quantum gravity aspects of black holes appeared as a chapter in the book by A. Kenath ed., „A Guide to Black Holes“, Nova Science Publishers in January 2023: C. Schiller, Testing a microscopic model for black holes deduced from maximum force.

The first publication on gravitation from strands was C. Schiller, A conjecture on deducing general relativity and the standard model with its fundamental constants from rational tangles of strands, Physics of Particles and Nuclei 50 (2019) 259–299.

See also the dedicated page on quantum gravity.

**50 756 solar masses per second.** Strands provide a
microscopic model for space and horizons. This allows deriving the field
equations of general relativity and a model for quantum gravity. Numerous
tests of the strand conjecture in the domain of gravitation and quantum
gravity are deduced, starting from a single principle. All tests agree
with observations so far.

For example, strands confirm that gravitation – like nature itself
– has a power or luminosity limit c^{5}/4G,
a momentum flow or force limit c^{4}/4G,
a mass flow limit c^{3}/4G,
and a mass to length limit c^{2}/4G.
The limits are given by one
quarter Planck mass per Planck time, or 50 756 solar masses per second
(times c^{-1}, times c, or times c^{2}). No observation
ever exceeded these limits.

Many predictions about gravitons and quantum gravity are deduced, including a direct derivation of black hole entropy from strands. Above all, strands also explain the existence of gravitational masses of elementary particles, solve the hierarchy problem, and provide upper and lower limits for the mass values. All predictions agree with the data.

Strands seem to be the *simplest* quantum gravity proposal in the
literature. Strands *agree with and predict all observations:*
strands provide a microscopic model of space, black hole horizons and
gravitons, explain mass, particles and black hole radiation, imply general
relativity without modifications, prevent singularities and wormholes,
reproduce cosmology (see step 6), but predict the lack of elementary dark
matter particles. Strands are also *complete:* no question of quantum
gravity is unanswered.

Any complete description of nature has to be *strange*. To satisfy
this requirement for gravitation, the following animation, made by Jason
Hise, shows how black hole rotation is modelled in the strand conjecture.
(The flattening of the horizon at the poles is not shown.) With a bit
of imagination, you can determine the location of the ergosphere.

### Step 4. The strand conjecture about wave functions and field theory

**Pedagogical.**
For physicists, the best introduction to the strand tangle model is the
preprint showing how wave functions, quantum theory, fermions and bosons,
the three gauge interactions, the three elementary particle generations and
unique elementary particle masses follow from strands. A longer introduction is found on this page.

**Wave functions and interactions.** A single principle is
used to derive the Schrödinger equation, the Pauli equation and the Dirac
equation. Spin 1/2, fermion behaviour, mass and, above all, the observed
elementary particles and the observed gauge interactions are deduced -
without any modification.

**Complete.** The strand tangle model explains why all
measurements are electromagnetic, why only massive particles can have
electric charge, why the spin-statistics theorem holds, and why the origin
of gauge interactions settles the Yang-Mills millennium problem. (The latter topic is also explained here.)
In short, it is shown that all "laws" of physics are uniquely defined,
without any possible variation or alternative. There is only one possible
universe.

**From 9 lines to 1 line.** In 2022, the strand tangle
model, with its one line, explains about 8.2 lines of the 9 lines that describe all of physics and of
nature. The strand tangle model also agrees with all experiments. The
remaining constants of line 9 still have to be deduced. The task is not
finished. But: no other theory in the literature has achieved this much.
(In fact, two other approaches have similar results. The *octonion
model* by Singh, arxiv.org/pdf/2206.06911.pdf, and Connes'
*non-commutative geometry*, arxiv.org/pdf/1004.0464.pdf, both explain
more than 8 lines, but both appear to predict additional unobserved
particles. A more detailed evaluation
is found here.)

Using strands, nature is summarized in just 1 line: General relativity and the standard model are due to fluctuating strands of Planck radius, for which each crossing switch yields a quantum of action.

**Qubits.** The strand tangle model also shows how to
give concrete meaning to Zizzi’s expression “it from qubit”: qubits can be
modelled with strands. So can entanglement and decoherence. And gravity.

**Lepton tangles.**
Use your mouse to play with the 3d visualizations of the three simplest
tangles (derived in the various papers) for the electron neutrino, the muon
neutrino, and the tau neutrino:

and the simplest tangles for the **electron**, the
**muon** and the **tau**:

All these beautiful 3d visualizations were realized with Blender by Aleksandr, by Lucas and by Mitchell.

### Step 5a: The strand tangle model for the standard model of particle physics

**Beautiful.** When we look at the starry sky, we admire
the vast space, the coloured twinkling stars, and the deep blackness. The
strand conjecture proposes an explanation for their origin, their
properties and their motion. The foundations of what we find around us
– particles, space, horizons and colours of everything we see –
are explained.

C. Schiller, A conjecture on deducing general
relativity and the standard model with its fundamental constants from
rational tangles of strands, *Physics of Particles and Nuclei*
**50** (2019) 259–299. Download the published paper at
dx.doi.org/10.1134/S1063779619030055. Read the published paper online for free at
rdcu.be/cdCK7. Download the preprint here, with films.

**Testable.** The paper argues that modern physics
arises, directly and inevitably, from the Planck scale.
Below, the more pedagogical papers and preprints deduce
additional experimental predictions and tests.
A detailed list of experimental tests is found on the bet
page, by clicking here.

**Simple.** The strand conjecture starts with deducing
Dirac’s equation from Dirac’s trick for tangles. Then, tangle
classification yields the particle spectrum. Tangle deformations yield,
via the Reidemeister moves, the particle gauge interactions groups U(1),
SU(3) and broken SU(2). Working out the details gives usual particle physics,
with no additions, modifications, or omissions.
More details are found on this page.

A compact introduction for physicists and physics students is C. Schiller, On the relation between the three Reidemeister moves and the three gauge groups, preprint on Researchgate at https://www.researchgate.net/publication/369794894.

C. Schiller, Testing a conjecture on the origin of the
standard model, *European Physical Journal Plus* **136** (2021)
79. Download it at doi.org/10.1140/epjp/s13360-020-01046-8. Read the published paper online for free at
rdcu.be/cdwSI. Download the
preprint here.

**Elegant.** It is regularly claimed that the standard
model is complex, incomplete or even ugly. The strand conjecture argues
the exact opposite: all of particle physics is due to tangled strands
fluctuating at the Planck scale. A *single* fundamental process
appears to explain the principle of least action, the Dirac equation, the
observed interaction spectrum, the observed gauge symmetry groups, the
observed elementary particle spectrum, and the fundamental constants
(masses, mixing angles, and coupling constants) describing them. The
Lagrangian of the standard model arises, without modifications. Over 100
additional tests and predictions about particle physics beyond the standard
model are deduced. They agree with all experiments. So far, no other
approach in the research literature appears to make (almost) any of these
predictions. Indeed, it appears that the explanation of the standard model
using tangled strands is consistent, complete, correct, hard to vary, and
unique. Above all, it is beautifully simple.

A list of experimental predictions is also found here. Parity violation is illustrated in the videos on the topic.

### Step 5b: Quantum electrodynamics deduced from strands

C. Schiller, Testing a conjecture on quantum
electrodynamics,
*Journal of
Geometry and Physics* 178 (2022) 104551.
Download the preprint
here.

**Colours and beauty.** The strand conjecture shows how the
tangle model leads to quantum electrodynamics, including electricity,
magnetism and optics. Over 40 tests for the conjecture are given. So far,
they are all positive. In particular, the strand conjecture appears to
allow approaching two old challenges: how to calculate the fine structure
constant and how to calculate the lepton masses – both from first
principles. The preprint uses the tangle model of particles to deduce
estimates. The fine structure constant with its measured value
1/137.036(1) and the lepton masses,
in particular the electron mass, are the ingredients that determine all
colours, tastes, smells, sounds and most shapes around us. In other words:
it is argued that tangles of strands generate all beauty in
nature.

Presently, tangles lead to a crude estimate of the fine structure
constant that is correct within 30%. This is not good; but so far,
it is one of just two attempts worldwide to explain the value ab initio,
using a unified description of particle physics and general relativity.

**The spin of leptons.**
Leptons consist of three strands. The animation by Jason Hise gives
an impression about how they spin:

The central cube contains the tangle core of the specific lepton.

**The spinning electron tangle.**
Fabrice Neyret, inspired by Jason Hise, produced two animations
showing two options for the spinning electron. Use your mouse to change
point of view:

The radius of the strands is the Planck length.
The green bar is only added for better visualization; it shows
the orientation of the electron. The tangle tethers reproduce spin 1/2
and fermion behaviour under particle exchange.
The wave function arises from the blurring of the tangle crossings. The tangle
details determine electric charge (every chiral crossing produces an
electric charge e/3), parities (behaviour under mirror reflection and
rotation reversal), lepton number (results from the 3 strands), mass (not
visible directly, via the average rotation speed), electromagnetic
coupling and the fine structure constant (through the statistics of tangle
shapes), and the behaviour in particle reactions (due to the topology of
the rational tangle).
Positrons are mirror tangles rotating in the opposite sense.
More details are found in the published paper on
quantum electrodynamics linked a few paragraphs higher up, and also in the pdf found at step 4.
No other model of the electron achieves all these explanations.

Here is the photon, showing its rotating phase:

Here is a topologically *equivalent* version,
also showing the photon and its rotatig phase, animated by Mitchell
Wieringa and seen from two different directions:

### Step 5c: Quantum chromodynamics deduced from strands

**Quarks and nuclei.** The strand conjecture shows how the tangle
model leads to the strong interaction, the quark model, gluon flux tubes,
confinement and asymptotic freedom. The existence (new in 2022) of
glueballs is predicted. Many other tests for the tangle model are deduced,
including the lack of new generations, the lack of CP violation and the
lack of deviations from QCD. All consequences agree with the data. In
particular, the strand conjecture allows estimating the strong coupling
constant and the quark masses ab initio.

**The spinning motion of the simplest tangle of
the down quark.** Jason Hise also produced the animation for this
case:

Indeed, the tangle model is peculiar – to say the least.

### Step 6: Cosmology deduced from strands

C. Schiller, Testing a conjecture on cosmology and dark energy (preprint).

**The universe.** This and a subsequent preprint on cosmology
complete the topic of gravitation. In the strand conjecture, the universe
consists of a single closed strand that forms the cosmological horizon and
also the particles and the space inside it. Over time, this strand gets
more and more tangled. (As one reader said: the universe plays cat's
cradle.) This description reproduces usual cosmology and leads to numerous
tests and predictions: the universe expands;
nothing – no matter, no radiation and no space – exists beyond
the cosmological horizon; inflation did not occur; there are no cosmic
strings and no higher dimensions; there is no non-trivial topology; there
is no bouncing universe; there is just *one* universe; the luminosity
of the universe is always limited by c^{5}/4G; dark matter is not made of
unknown elementary particles; if dark matter exists at all,
it is made of known matter or black holes or both; dark energy, or vacuum
energy, does exist and is a natural consequence of strands; the density of
vacuum energy, the cosmological constant, is small;
baryogenesis appears to be due to non-perturbative effects.

The strand description of cosmology is promising. However, calculating
the vacuum energy density remains a challenge.
Therefore, clarifying the relation of strands to modified gravity and to the baryonic
Tully-Fisher relation remains a challenge as well.

### Outreach

**Particle size.**
As the animation at the very top of this page shows, there is no way to
define the *size* of an elementary particle. The wave function
describes its average extension. The electric charge describes its
interactions; but the
charge is a consequence of Planck-sized crossings. Finally, every electron
has tethers reaching the cosmological horizon. In short, an electron is at
the same time wave-function sized, of Planck size, and of the size of the
night sky.

The same is valid for all other elementary particles. Their size is always described by their wavelength, by the Planck length, and by the size of the universe. Everything in nature has three sizes. But not more. And not less.

*

**Blog.**
The blog on research about fundamental physics
and strand tangles tells more about general ideas, past mistakes,
objections, encountered difficulties, and progress.

*

**T-shirt.**
An important motivation for strand research has always
been the support for the ailing physics T-shirt industry.
For decades, it has been desperate for new designs. Now they are available.

*

**History.** The strand conjecture is a side result of the
free Motion Mountain physics book series, in particular of Dirac's spin 1/2
demonstration, of the principle of maximum force, of the strand
explanation of back hole entropy, and of the meditation time
offered by the Munich subway. Strands reduce the 9 lines describing textbook physics to a single
principle (that fits on a T-shirt), and make clear
predictions for experiments and calculations. If you want to bet about the
outcomes, to evaluate your chances, to comment, or if
you want to help with animations similar to these, write
to christoph@motionmountain.net.

*

**Technicalities.** The strand conjecture
reproduces the Lagrangians of the standard model and general relativity,
explains the number of generations and the particle spectrum,
deduces all Feynman diagrams and propagators,
explains the gauge groups U(1), SU(2) and SU(3),
explains the fundamental constants ab initio,
solves the hierarchy problem,
explains neutrino masses without a see-saw mechanism,
solves the strong CP problem,
predicts the validity of the standard model and of general relativity
up to the Planck scale without any intermediate energy scale,
implies that the weak interaction violates parity maximally,
explains the equality of proton and positron charge,
has no problems with anomalies,
predicts no issues with baryogenesis,
has no grand unification,
has no supersymmetry,
has no additional spatial dimensions,
has no inflation, no inflaton and no dilaton,
solves black hole and singularity issues,
implies gravitational waves,
has no dark matter particles,
has a naturally small cosmological constant,
solves various problems with gauge theories,
answers Hilbert's sixth problem,
and explains the principle of least action.

### Acknowledgements

Several of these articles were supported by grants from the Klaus Tschira Foundation: Eur Phys J Plus, Indian J Phys, J Geom Phys, IJGMMP, Z f Naturf, dark energy preprint, emergent quantum theory preprint, first Bronshtein cube preprint.

### Volume VI of Motion Mountain

**Older work.** A more extensive, more passionate, but also
older and less precise presentation is the original text on the strand
model. It was written as a *research* volume that continues the
adventure of the five *textbook* volumes.

### Strands in other languages

A `strand' is best translated in Dutch as *draad*, in French as
*fil*, in German as *Faden*, in Italian as *filo* and
in Spanish as *hilo*.
The mathematical concept of
`tangle' is best translated in Dutch as
*wirwar*, in French as *enchevêtrement*, in
German as *Gewirr*, in Italian as *groviglio* and in
Spanish as *enredo*.
A `tether' is best translated in Dutch as
*lijn*, in French as *lien*, in German as *Leine*, in Italian
as *nesso* and in Spanish, for example, as *vínculo*.

### Publication list

**On the Planck limits as foundations of the strand tangle model**

U. Hohm and C. Schiller, Testing the Minimum System Entropy and the Quantum of Entropy, Entropy 25 (2023) 1511.

A. Kenath, C. Schiller and C. Sivaram, From maximum force to the field equations of general relativity – and implications, International Journal of Modern Physics D 31 (2022) 2242019. This paper received an honourable mention in the 2022 competition of the Gravity Research Foundation.

C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity, Zeitschrift für Naturforschung A 78 (2022) 145–159.

C. Schiller, From maximum force via the hoop conjecture to inverse square gravity, Gravitation and Cosmology 28 (2022) 305–307.

C. Schiller, Tests for maximum force and maximum power, Physical Review D 104 (2021) 124079.

C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104 (2021) 068501.

Older papers on the topic can be found via Google Scholar and via ResearchGate.

**On deducing general relativity from strand tangles**

C. Schiller, Testing a microscopic model for black holes deduced from maximum force, chapter in the book „A Guide to Black Holes“, Nova Science Publishers (January 2023).

C. Schiller, Testing a conjecture on the origin of space, gravity and mass, Indian Journal of Physics 96 (2022) 3047–3064.

**On deducing the standard model from strand tangles**

C. Schiller, On the relation between the three Reidemeister moves and the three gauge groups, International Journal of Geometric Methods in Modern Physics (2023).

C. Schiller, From points to fluctuating strands: advancing theoretical physics, online publication at Essentia foundation (2023).

C. Schiller, Testing a conjecture on quantum chromodynamics, International Journal of Geometric Methods in Modern Physics, 20 (2023) 2350095.

C. Schiller, Testing a conjecture on quantum electrodynamics, Journal of Geometry and Physics 178 (2022) 104551.

C. Schiller, Testing a conjecture on the origin of the standard model, European Physical Journal Plus 136 (2021) 79.

C. Schiller, A conjecture on deducing general relativity and the standard model with its fundamental constants from rational tangles of strands, Physics of Particles and Nuclei 50 (2019) 259–299.

* *