1 Nature is maximally efficient
Teaser.Everything around us happens in a way that minimizes change.
Exploration.It took physicists many years to sort out the definition of "change". It was finalized around 1750.
If a system has a lot of kinetic energy, it moves a lot. The more time passes, the more it changes. Change increases both with kinetic energy and with time.
If a system stores energy, it reduces motion. Potential energy reduces change, and the more the longer it is stored. Change decreases both with potential energy and with time.
Therefore, physical "change" is kinetic energy minus potential energy, multiplied by the elapsed time. But physicists do not say "change" – they say "action". Think about an action movie. A large action value is for movies with a "lot of action". When many things, people, cars, planes move, exploding, burning, etc., then the action value – the integral over time of kinetic minus potential energy – is large. (For general relativity, the expression of the action is generalized.)
All equations of motion follow from the requirement that action, defined accordingly, is minimized. No counter-example is known.
Summary.Motion minimizes change. This is officially called the principle of least action. Everything that happens follows this rule, from the growth of trees to the motion of lightning. In simple English: nature is efficient; it dislikes squandering.
Possible falsification.Find an exception to least action.
More to ponder.Action W is the most fundamental quantity in physics. For example, the 10-volume Landau-Lifshitz textbook on physics starts with action on the very first page. Efficiency applies to every process in nature: action minimization is valid in classical physics, in quantum theory and in general relativity. It is valid for motion of stones, heat, light, particles, fields and empty space.
In short,dW=0 describes motion.
2 There is a maximum energy speed in nature
Teaser.Nature dislikes being in a hurry.
Exploration.It took physicists many years discover that there is a speed limit for energy. It was clarified in the early 20th century.
Throwing a stone while running yields a greater speed than throwing it standing still. However, in a vacuum, light from a moving lamp is not faster than light from a lamp at rest.
Even the lightest "stones", single electrons, cannot be accelerated faster than light, even using the largest amounts of energy. This also applies to protons, neutrinos, stones, rockets, X-rays or gravitational waves.
Despite high rewards, nobody has found a way to move energy faster than light in vacuum.
Summary.The speed of light in vacuum is the limit for the speed of any kind of energy. This is the principle on which special relativity is based. In simple language: nature refuses to hurry.
Possible falsification.Find a larger local energy speed.
More to ponder.The speed limit states that the locally measured energy speed value, relative to the observer, never exceeds c. Also, the observer performing the measurement must be physical and, e.g., cannot himself be superluminal. "Geometric" speeds can be higher than c, such as the speed of shadows, images or horizons.
The speed limit c is so fundamental that it is used to define the meter in terms of the second. The speed limit also means that time is different for different observers that move against each other; but this is another story. The principle of maximum local energy speed implies the full theory of special relativity.
In water, particles can move faster than light. What happens?
In short,v ≤ c implies special relativity.
3 There is a smallest observable change in nature
Teaser.Nature dislikes sloth. Something happens all the time.
Exploration.In the 1890s, Planck discovered photons, the particles of light. He also found that it is not possible to measure smaller changes than ℏ, a constant of nature that he called the elementary quantum of action. (Actually, he found that the smallest change was h=2πℏ, but nowadays the two quantities are often used interchangeably.)
The term "quantum" was made popular by Galileo, who explained that matter is made of "piccolissimi quanti", very small quanta that are not divisible. Planck took over the term. He was a quiet man, but was conscious that he made one of the greatest discoveries ever.
Attempt of a counter-example. To determine the action of a free particle requires to measure its energy at two different points in time. Even though, classically, action W is given by energy E times delta t, in nature and in quantum theory the action value W remains finite when delta t gets small, because of the uncertainty relation: the energy (difference) increases when delta t decreases. (Measuring energy requires time, that time must be shorter than delta t.) For small delta t, ℏ is at work: the uncertainty relation prevents that the measured action goes to zero when delta t goes to zero.
Further attempt of a counter-example. Detecting a spin 1/2 flip requires an action ℏ. There is no way to detect a spin flip with a smaller amount of action.
Another attempt. Every photon detection requires an action ℏ. In nature, there is just no way to detect a 1/2 or a 1/100 of a photon. Photons are elementary: they cannot be split.
The famous discussion match between Einstein and Bohr can be seen as a continuous attempt by Einstein to measure actions below ℏ. But nature does not allow this. Even Einstein, who in the beginning was adamant that this was possible, then changed his mind. There is a lower limit to action.
Summary.In 1899, quantum theory was born with Planck's discovery that action is quantized in multiples of ℏ, and that no smaller action value can be measured. The principle of quantized action was confirmed by all subsequent experiments.
Possible falsification.Measure a smaller action value than the lower limit given by the quantum of action.
More to ponder.If ℏ were not the smallest action value, atoms would not exist.
For physicists. Action is a scalar, a Lorentz invariant, and a quantum observable. The operator is found in many textbooks. It is self-adjoint and linear. There is also a canonically conjugated observable: angle.
Probabilities arise whenever one approaches the limit. In fact, the limit ℏ alone is sufficient to deduce wave functions and quantum mechanics.
In short,W ≥ ℏ implies quantum theory.
4 There is a maximum force in nature
Teaser.Nature dislikes excessive force. There is a limit to force.
In nature, c is the maximum local energy speed. Similarly, c^4/4G is the maximum local force, or maximum local momentum flow.
Exploration.The force limit c^4/4G implies that the locally measured force value, relative to the observer, never exceeds c^4/4G. The observer performing the measurement must be physical and, e.g., cannot be a black hole, or infinitesimally dense, or infinitesimally small. These statements can also be repeated for the maximum power or luminosity c^5/4G. This was clarified in the 1990s and 2000s.
Attempts to find counterexamples must be physical. An observer cannot add forces of distant mass configurations and claim that their sum exceeds the local limit c^4/4G. Such examples are easily found. It is thus wrong to claim that maximum force is valid for any physical surface.
The value c^4/4G is the largest possible gravitational force between two black holes.
The maximum force value implies inverse square gravity.
So far, also no counterexamples to the maximum luminosity c^5/4G have been published. Even the most recent observations of black hole mergers fail to exceed the luminosity limit. The highest instantaneous luminosity observed so far is about 3% of the maximum value.
Summary.No counter-examples or counter-arguments to maximum force or to maximum power have yet been observed or constructed.
The existence of a local maximum force is sufficient to deduce general relativity in all its entirety, including the field equations. This is the principle of maximum force.
Possible falsification.Find a larger local force or luminosity.
More to ponder.Can you find a way to distinguish experimentally whether the limit is c^4/G or c^4/4G?
Maximum force also applies to electromagnetism and nuclear interactions, and to any combination.
A maximum force also implies that space is curved.
A maximum force implies that perfectly straight motion does not exist.
In short,F ≤ c^4/4G implies general relativity.
5 Strands explain quantum theory
Teaser.Nature dislikes points - and the infinitely small.
Dirac's equation follows from invisible fluctuating strands.
Exploration.Dirac showed, around 1929, that spin 1/2 particles behave like tethered objects: they come back to the original state only after rotations by 4π, not after rotation by 2π. After 2π, the tethers remain tangled.
Such tethered particles can rotate continuously.
A rotation by 2π leads to a minus sign in front of the wave function. A rotation by 4π leads to a plus sign for the wave function.
Simply said, nature does not allow to observe tethers, but does allow to observe their crossing switches. And the crossing switches of invisible tethers define ℏ and c.
In 1980, Battey-Pratt and Racey showed that a Lorentz-transformation of this connection yields Dirac's equation. Kauffman suggested a relation between crossing switches and ℏ in 1987.
Tethered particles also behave as fermions: the system goes back to its original state only after a double particle exchange. Tethers confirm the spin-statistics theorem.
The geometric variables describing a strand crossing – position, density, orientation, phase – are the same variables that describe a wave function.
Dirac's trick also implies that the wave function is the average crossing density of invisible fluctuating tethers. We call them strands.
Summary.Dirac's trick implies Dirac's equation. Strands imply wave functions.
Possible falsification.Find a deviation from the Dirac equation, e.g., at high energy or in high gravitational fields.
More to ponder.Strands do not provide non-contextual hidden variables.
Strand entanglement explains quantum entanglement.
Particle rotation, or phase rotation, is the reason for the appearance of complex numbers in quantum theory.
In short,wave functions are blurred strand crossings.
6 Strands explain general relativity and quantum gravity
Teaser.Nature dislikes boredom - and flat space. Einstein's field equations follow from invisible fluctuating strands.
Exploration.Crossing switches of strands realize the limits c, c4/4G and the Boltzmann constant k.
Horizons are tight weaves of strands. Vacuum is made of untangled strands. Curvature is an inhomogeneous strand distribution.
Strands explain black hole energy, entropy and temperature. These relations allow to deduce the field equations. This was discovered around 2010 using a result by Jacobson from 1995.
The Hilbert Lagrangian of general relativity follows from strands.
There is no gravitational physics beyond general relativity at sub-galactic scales.
Space has three dimensions, because strand tangles do not exist in other dimensions.
On average, masses are surrounded by twisted tethers; the twits are due to Dirac's trick. The twisted tethers form clouds around every mass and behave as expected from virtual gravitons: twist clouds lead to attraction between masses. For small curvature and speeds and now horizon nearby, these graviton clouds imply universal 1/r2 gravitation.
Single gravitons are not detectable.
New quantum gravity effects are not observable.
Summary.Dirac's trick implies Einstein's field equations. Strands make up space and gravity.
Possible falsification.Find a deviation from Einstein's field equations at sub-galactic scales. Measure any (new) effect of quantum gravity. Find non-cummutative space. Find other dimensions.
More to ponder.Rotating black holes perform Dirac's trick with a huge number of tethers.
A mass is never smaller than its own black hole radius. Black holes have a finite moment of inertia.
There are no singularities and no physical points in space.
Topology leads to entropy and thus to gravitational dynamics.
In short,gravitons are moving twisted strand pairs.
7 Strands explain particles
Teaser.Particles are not points, but tethered to the rest of the world.
Particles are rotating rational tangles.
Exploration.Rational tangles imply three generations of elementary fermions, each with two particles.
Localized rational tangles of two strands are quarks.
Localized rational tangles of three strands are leptons; braids are Higgs bosons.
Each fermion is represented by an infinite family of rational tangles, a basic tangle and further family members that arise by addition of braids.
Tangle topology determines all quantum numbers: spin, parities, charges and flavours.
There is no particle beyond the standard model. This was clarified in the years around 2010.
Summary.Particles are spinning defects in space.
Possible falsification.Find any new elementary particle. Find another particle model that agrees with experiment. Find a new quantum number. Find new physics.
More to ponder.Only rational tangles reproduce how particles change quantum numbers when they interact. Other topological structures, such as knots, ribbons, graphs or even just other types of tangles, do not match observations.
Particles are not point-like, but at least of Planck size.
So far, no other model explains the particle spectrum.
In short,particles and space are complementary.
8 Strands explain gauge interactions
Teaser.Interactions of particles are due to strand deformations of tangles.
Exploration.In mathematics, deformations are described by gauge groups.
All tangle deformations can be mathematically classified into three types, called the three Reidemeister moves. This was discovered in 1926.
The three Reidemeister moves determine U(1), broken SU(2), and SU(3). This was found about 80 years later.
Tangle deformation implies a change of phase &ndah; this effect is required from any interaction.
The three Reidemeister moves also determine the strand structure of the vector bosons.
There is no interaction and no symmetry beyond the standard model.
Summary.The three known gauge groups derive from the three Reidemeister moves.
Possible falsification.Find another interaction, another gauge group, another fundamental symmetry or another gauge boson.
More to ponder.No other symmetry, gauge boson or interaction is predicted to exist. No supersymmetry, no unified gauge group, no Clifford algebra and no other internal space is predicted to describe nature.
So far, no other model explains the interaction spectrum.
In short,bending strands imply the observed gauge interactions.
9 Strands explain fundamental constants
Teaser.Tangle topology and shape determines interaction strengths.
Exploration.Deformation transfers explain Feynman diagrams. Transfer of Reidemeister moves between bosons and fermions explain the coupling constants.
Dirac's trick explains masses.
Exchanges of tether positions explain mixing angles.
The Lagrangian of the standard model of elementary particle physics follows from strands. This was discovered around 2010.
Summary.Tangle shapes determine the fundamental constants.
Possible falsification.Find a difference between calculated and measured values of the fundamental constants. Find a different explanation for the fundamental constants.
More to ponder.Average tangle shape is determined by tight tangle shape. But no non-trivial tight tangle shape is yet known analytically.
There is no multiverse.
So far, no other model explains interaction strengths.
In short,the hardest problem of 3d geometry is related to the fundamental constants.
10 Strands yield cosmology
Teaser.The universe plays cat's cradle.
Exploration.The universe consists of a closed strand that continuously increases in complexity. The increase defines time.
The universe is surrounded by a horizon.
Empty space expands, because with time, more strand segments enter it from the horizon.
Empty space contains a low, but positive energy density: dark energy.
Because strands do not allow other defects than the known elementary particles, there is no elementary dark matter particle.
Strands imply the lack of inflation, because there is no mechanism for such a process. Strands solve the flatness, homogeneity and isotropy problems; also, there is nothing behind the cosmological horizon.
Summary.The sky is dark at night. There is dark energy – but its time dependence is not clear yet. There is no elementary dark matter. The inverse square law might be modified at galactic and larger scales – this is not clear yet.
Possible falsification.Find elementary dark matter. Find anything beyond the cosmological horizon.
More to ponder.Could the universe be made of several strands?
Similar ideas on the structure of nature were proposed around 2000 by Kauffman.
In short,the universe is a closed strand.
11 Strands explain all motion
Teaser.No observed example of motion is unexplained.
Exploration.Strand tangles explain the three dimensions of space, the three gauge interactions and the three particle generations.
Strand tangles explain curvature of space and wave functions.
Strands explain all Lagrangians used in physics.
Local motion follows the principle of least crossing switch number. This explains the principle of least action.
Strands explain all mathematical concepts used in physics.
Strands explain that nature follows rules.
Summary.No observation about motion in fundamental physics seems to be unexplained by strands. Strands explain and describe physics and all natural sciences.
Possible falsification.Find an example of motion that differs from the strand tangle model. Find additional dimensions.
More to ponder.Topology leads to shapes, deformations and to dynamics. Or: dynamics results from algebra.
Is the strand explanation of nature satisfying? Does it fit on a T-shirt with sufficient elegance? Does it lead to say, like Wheeler requested, “Oh, how could it have been otherwise! How could we all have been so blind so long!”?
Strands do not need to have constant radius.
Alternatively, instead of strands, the space between strands can be taken as the fundamental evolving entity.
In short,motion locally minimizes crossing switches. On cosmological scale, the number of crossing switches increases.
12 The garden at the end of the street
Teaser.The tangle model implies all of nature and physics.
Exploration.Nature likes motion. Everything that happens in nature is a kind of motion.
To describe motion with precision, use the speed limit v≤c, the action limit W≥ℏ and the force limit F≤c⁴/4G: this yields quantum theory, general relativity and cosmology. Realize the limits with crossing switches of strands with Planck radius, form rational tangles, deform them with Reidemeister moves: this yields the particles, the interactions, the constants and the Lagrangian of the standard model.
Every equation of physics follows from strands.
Every type of motion, every measurement, and every observation follows from strands.
Strands and the Dirac trick at Planck scales together describe nature with full accuracy and full precision.
More precisely: tangled strands so far are the only approach in the literature that explains the first 7 lines summarizing physics, i.e., the only approach that explains the gauge groups and the particle spectrum. In addition, tangled strands so far are the only approach that allows calculating the fundamental constants (the 8th line). Strands predict that these calculations will agree with measurements. Tangled strands so far are the only approach that already produced estimates for a number of fundamental constants that are not in contrast with experiments.
Conclusion.Tangled strands describe accurately everything that happens. Tangles provide the tiniest theory of nature.
Possible falsification.Find a single difference between observations and the tangle model. Find physics beyond the standard model. Find a simpler model than the tangle model. Find a different model.
More to ponder.Strands imply: everything is connected to everything else.
Strands imply: every thing is made of everything.
One-line summary:Nature is a tangle of a long, invisible, fluctuating strand whose crossing switches define the Planck units and all observations.
Slides introducing strands
These slides provide an introduction
to the strand conjecture, a complete description of motion. The slides
contain five animations, make experimental predictions and explain the
origin of the fine structure constant, the electron mass and all
Bibliographic references about strands
E.P. Battey-Pratt and T.J. Racey, Geometric model for fundamental particles, International Journal of Theoretical Physics 19 (1980) 437-475.
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. Details here.
C. Schiller, Testing a conjecture on the origin of the standard model, European Physical Journal Plus 136 (2021) 79. Details here.
Jason Hise's animations are shown and strands are mentioned in this PBS television program, from 8:48 onwards (2021).
C. Schiller, Testing a conjecture on the origin of space, gravity and mass, preprint (2021). Download here.
Bibliographic references about the quantum of action
S. Boughn, Wherefore Quantum Mechanics?, arXiv:1910.08069.
L.J. Curtis, A 21st century perspective as a primer to introductory physics, European Journal of Physics 32 (2011) 1259–1274, section 5.2.
L.J. Curtis and D.G. Ellis, Use of the Einstein–Brillouin–Keller action quantization, American Journal of Physics 72 (2004) 1521.
J. Schwinger, B. Englert, Quantum Mechanics, 2003. This book on quantum theory explores the action operator in detail.
M.N. Sergeenko, Quantization of the classical action and eigenvalue problem, arXiv:quant-ph/0211099.
V. Hushwater, Quantum mechanics from the quantization of the action variable, Fortschr. Phys. 46 (1998) 6-8, 863-871.
V. Hushwater, A path from the quantization of the action variable to quantum mechanical formalism, Foundations of Physics 28 (1998) 167–184.
A. Zeilinger, On the interpretation and philosophical foundation of quantum mechanics, in "Vastakohtien todellisuus", Festschrift for K.V. Laurikainen, U. Ketvel et al. (Eds.), Helsinki University Press, 1996. In that paper, Zeilinger writes: "there is a universal smallest action which can be exchanged in a physical process".
J.-M. Levy-Leblond, Quantique, 1984. An excellent introduction to quantum mechanics.
Maslov's book from 1972 (in French translation) and his papers on the Maslov index added the final details to Einstein-Brillouin-Keller quantization.
J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Ann. Phys. (N.Y.) 4 (1958) 180–188.
W. Heisenberg, Das Plancksche Wirkungsquantum, 1945.
N. Bohr, Atomtheorie und Naturbeschreibung, Springer, 1931. The four articles in the book are based on the indivisibility of the quantum of action and continuously underline it.
L. Brillouin, Remarques sur la mécanique ondulatoire, J. Phys. Radium 7 (1926) 353–368.
A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. Dtsch. Phys. Ges. 19 (1917) 82–92.
M. Bronshtein, in his paper on the physics cube. See http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm.
O. Sackur, Die universelle Bedeutung des sog. elementaren Wirkungsquantums, Annalen der Physik, 345 (1913) 67-86. He writes on the first page: "Zu diesem Resultat gelangte ich mittels der Sommerfeldschen Hypothese, daß jede in der Natur ausgeübte Wirkung ein ganzzahliges Vielfache des elementaren Wirkungsquantums h ist."
A. Sommerfeld, Das Plancksche Wirkungsquantum und seine allgemeine Bedeutung für die Molekülphysik, Physikalische Zeitschrift 12 (1911) 1057-1069.
N. Bohr 1911, 1913.
M. Planck 1899, 1900.
Bibliographic references about maximum force
E.A. Rauscher, The Minkowski metric for a multidimensional geometry, Lett. Nuovo Cimento 7 (1973) 361-367, writing: "F can be considered an upper bound on force".
H.-J. Treder, The planckions as largest elementary particles and as smallest test bodies, Foundations of Physics 15 (1985) 161-166.
R. J. Heaston, Identification of a superforce in Einstein field equations, Journal of the Washington Academy of Sciences, 80 (1990) 25-36.
V. de Sabbata and C. Sivaram, On limiting field strengths in gravitation, Foundations of Physics Letters 6 (1993) 561-570.
L. Kostro and B. Lange, Is c^4/G the greatest possible force in nature?, Physics Essays 12 (1999) 182-189.
G.W. Gibbons, The maximum tension principle in general relativity, Foundations of Physics 32 (2002) 1891-1901.
C. Schiller, Maximum force and minimum distance: physics in limit statements, arXiv:physics/0309118.
C. Schiller, General relativity and cosmology derived from principle of maximum power or force, International Journal of Theoretical Physics 44 (2005) 1629-1647.
M.P. Dabrowski and H. Gohar, Abolishing the maximum tension principle, Physics Letters B 748 (2015) 428-431.
V. Cardoso, T. Ikeda, C.J. Moore and C.-M. Yoo, Remarks on the maximum luminosity, Physical Review D 97 (2018) 084013.
Many other papers treat the topic. Contrasting points of view are V. Faraoni, PRD 103 (2021) 124010 and A. Jowsey and M. Visser, arXiv:2102.01831. Because they added forces at different locations in space, they found apparent counter-examples. However, a detailed analysis shows that local maximum force and power are not exceeded, as explained in C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104 (2021) 068501 10.1103/PhysRevD.104.068501, free preprint at arxiv.org/abs/2109.07700.