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In the strand conjecture, the mass value of a fermion (in units of the Planck mass) is given by the average number of crossing switches that occur per Planck time. (Reason: mass is energy/c^2; energy, the ability to do work, is action per time; every crossing switch produces a quantum of action hbar.) A mass estimate for an elementary fermion tangle thus appears to require estimating two processes. First, one needs to estimate the probability p_bt of the spontaneous belt trick (due to strand fluctuations) and the number n_bt of crossing switches associated with it. Second, one needs to estimate the number n of crossing switches that occur on average *between* two belt tricks. When estimated for static masses, the approximation neglects the running of mass with energy. If only the simplest tangle is used, the approximation further neglects the Yukawa terms. To simplify calculations, the approximation of tight tangles is used. This triple approximation yields an estimate for elementary particle mass given by m/m_Pl = p_bt * n_bt * (1 + n) where m_Pl is the Planck mass. The probability p_bt of the belt trick for a lepton, with its 6 tethers, appears to be given by p_bt = ( e^(-l_bt/l_Pl) )^6 * O(1) where l_bt is the length in each tether that is required to go around the (tight) core during the belt trick, and l_Pl is the Planck length. (Note: improving the the estimate for p_bt is *hard*.) For the electron neutrino, which has the simplest tangle core among the leptons, the length ratio l_bt/l_Pl is around 12 (determined from actual ropes). The factor n_bt describes two main effects. First, it describes the crossing switches due to the tethers and the core segments rotating against each other during the belt trick. Second, it describes the crossing switches among the tethers that occur during the belt trick. For the electron neutrino, the factor n_bt is about 3 (also determined from actual ropes). For the electron neutrino, in a crude calculation, the number n of crossing changes between subsequent belt tricks can be neglected and be set to zero. Taken together, and inserting the value of the Planck mass, this yields an electron neutrino mass estimate of 2 meV. Given the crude approximation, the error estimate for this mass estimate should be at least a factor 1000 in both directions, giving the strand estimate for the electron neutrino's Dirac mass 0.002 meV < m_nu,e < 2 eV Yes, this is a terribly crude approximation. However, ponder this: (1) There is no way to estimate elementary particle masses at all, so far. (2) There is no way to calculate ropelengths and tangle shapes for any non-trivial tight tangle or knot in mathematics yet. All such results are only numerical. (3) There is no way to estimate the frequency p_bt in the complete research literature, so far: not in fluid vortex research, superconductivity, superfluidity, cosmic strings, polymer science, nor anywhere else. (4) Strands predict that the mass value is the same over space and time, across the universe and its history; that it is equal for the antineutrino; that it is a Dirac mass; that it runs with energy; that it is much smaller than the Planck mass, and thus that it solves the hierarchy problem without see-saw mechanism. In short, estimating the probability of the spontaneous belt trick - for a given tangle - is *the* challenge to solve.

Here are the main definitions of the strand conjecture: A strand is an almost one-dimensional "tube", of Planck radius, without ends and without any observable physical property, randomly fluctuating in three-dimensional background space. (Note: the Planck radius is also not observable. It is a help to imagination and makes things come out right - especially in quantum gravity and black holes. In QFT, the radius can usually be neglected and strands can be imagined as purely one-dimensional lines that fluctuate.) The basic physical observable is a crossing switch, the exchange of an under-pass by and over.pass that takes place at a specific position in space. A crossing switch defines the Planck units h-bar and also c and G. This allows to define all physical observables; these emerge form counting or time-averaging crossing switches. Energy is the number of crossing switches per time. (Because a crossing switch defines h-bar.) A few consequences: Only a certain groups of strands (e.g. a tangle) or a twisted strand (forming a crossing, e.g. a photon) have energy. A vacuum strand has none. Neither do many vacuum strands. Strands have no Hamiltonian, as they are not observable and have no observables. Therefore there is no evolution equation for a single strands and no variational principle for a single strand. The principle of least action that we use in QFT and GR is emergent, like energy itself is: it can be phrased as the principle that minimizes the number of crossing switches. Least action thus only applies to systems with energy, e.g., particles. The tangle model distinguishes clearly between elementary fermions (rational tangles, with spin 1/2 behaviour under rotations and proper fermion exchange behaviour, as illustrated in the videos at vimeo.com/62228139 and at vimeo.com/62143283) and elementary bosons (untangled strand groups with spin 1 and boson exchange behaviour). In the tangle model, all elementary fermions have Dirac mass, not Majorana mass (because they have localised energy and particles differ from antiparticles). Elementary bosons can have mass or be massless. About your questions on my physics background, just have a look at - and enjoy - my freely downloadable Motion Mountain physics text in 5 volumes.

M, all help and all suggestions are welcome. Thank you in advance. In fact, help in cosmology would be particularly welcome. I already tried to contact you, but one link in your signature is broken and the other has no contact data. Feel free to post or to email constructive advice.

In physics, proof is given by logical deduction from a single principle combined with agreement with observations. Go to the motion mountain website and look for the preprints Strands-QED.pdf and Strands-Gravitation.pdf , or for (the much longer) motionmountain-volume6.pdf . You will find the logical deduction of black holes from strands, of the neutrino mass sequence from tangle complexity, and of all the differences between the tangle model and the standard model. You can read that the range of the weak force depends on the W and Z mass (these are fundamental constants), and that estimates of these masses and their ratio are possible. The texts list many predictions and retrodictions; they are all deduced by inescapable logic from the strand conjecture. You and anybody else can check the logic of the deductions and criticize it. There can be errors! But it would help to be a little more specific than the quoted sentence. I do understand your anger. Deducing the standard model and general relativity from unobservable strands is hard to swallow. The only thing I can do is to point out the many results, the various new results, the astonishing consistency, the surprising completeness, and the fascinating simplicity of the conjecture. Maybe one further thought can underline this. Strands realize the old vision by Bronshtein: strands are simply a tool to deduce all of physics from h-bar, c and G.

Yes, I can: strands explain the number and gauge groups of the interactions, the number of elementary particles, and the fundamental constants. Strands explain why protons have the same charge as positrons. By explaining the fine structure constant and the mass of the electron, strands explain all colors in nature. Strands explain the lack of a Landau pole and a vanishing vacuum energy. Strands explain the field equations of general relativity. Strands explain black hole entropy and the no-hair theorem. Strands make predictions about the neutrino mass sequence. The standard model cannot do any of this.

We have different concept of what a "proof" is in physics. I have a simple one: a statement is correct if it agrees with experiment.

Of course you are entitled to you opinion. But I'm afraid it is not correct. There is *just one* assertion: fluctuating strands describe nature (the so-called "fundamental principle"). All other statements derive from this basic idea, with arguments and logic that anybody can question, discuss, check and refute. The detailed arguments lead to the Hilbert Lagrangian and to the standard model Lagrangian. I have summarized the main arguments in the previous messages. There are preprints with those arguments that can be read, checked and refuted, if there is something wrong. Strands predict the number of elementary particles (i.e., the number of quarks, leptons and elementary bosons), the number of interactions (3 gauge interactions plus gravity) and provides limits for the fundamental constants. No other proposal does this. About "zero proof". Every deduction from the fundamental principle of the strand conjecture agrees with all experiments. Strands predict that general relativity and the standard model describe nature completely at sub-galactic scales. In simple terms, strands predict that nothing will be discovered in this domain: no new particles, no dark matter partciles, no new forces, no new symmetries, no new energy scales, no higher dimensions, no new fundamental constants. Some new predictions are made. The ways to deduce the predictions is public and everybody can follow them and criticize them - as you are doing. Nothing is hidden. Everybody can perform experiments and test the predictions. This is exactly how research advances and proofs are made in physics.

I do not understand your point completely, so I will try. The pion and kaon are hadrons, made of two quarks; they are not elementary. The strand conjecture reproduces the quark model, in all its details. For example, strands successfully predict (retrodict) the mass sequence among hadrons (via the sequence of tangle complexity) and correctly predict (retrodict) which of the mesons violate CP. This is discussed on the motion mountain website. Thus, the hundreds of mesons and baryons are reproduced. (The tangles for a few dozen mesons and baryons are also found there.) Strands also seem to allow statements about Regge trajectories and tetraquarks. In retrospect, given that strands reproduce QFT, all this is not too surprising. Indeed, the gauge symmetry groups have nothing to do with the limits on the number of (really) elementary particles. Those limits come from the Reidemeister moves, and thus from the tangle model. And strands also limit the number of gauge groups - thus forbid grand unification, for example. These predictions come from strands, not from QFT.

About stiffness: I stressed that strands have no stiffness. Strands are not a material, they have no observable properties. Stiffness is not a property that they have. But maybe I misunderstand your point here? About composition: If something (A) is said to be made of something else (B), both A and B are observable. But in the strand conjecture strands are not observable. They are not composed of something else. They cannot be. In other terms, the strand conjecture claims that nature (space, fermions, bosons, horizons, black holes) is made of many strand segments. The total strand is nature. But maybe you asked about something else? About the statement that the strand conjecture "does not allow more particles, it does not allow more forces, and it does not allow different values of the fundamental constants." The proof is possible: 1. Forces are deformations of tangles; all deformations of tangles are composed of Reidemeister moves (this is a mathematical theorem); there are only three such moves (this is a mathematical theorem proven by Reidemeister in 1926); so there are only three gauge interactions, with U(1), broken SU(2), and SU(3). (Gravity is a separate story.) 2. Elementary particles are rational tangles. They can be elementary only if made of 1, 2 or 3 strands. Exploring the possible tangle options yields the known gauge bosons and the Higgs boson, the 6 quarks and the 6 leptons. More options are not possible. 3. The fundamental constants (masses, coupling constants and mixing angles) are due to (statistical) tangle core geometry. The (average) tangle geometry is unique, so the constants are unique. Strands do not allow different values of the fundamental constants. Still, the strand model can be wrong, of course. Many experiments listed in the bet page can prove the model wrong; for example, finding dark matter would invalidate the model, or finding grand unification, or finding that the fine structure constant varies across the universe. But also theory can prove the model wrong: one could find that the fundamental constants come out with the wrong values, for example. - I want to add that your questions are very useful to me. Thank you. Sorry, I did not see this. I know Lou Kauffman and I know his work well. He has inspired several ideas of the strand conjecture. Thank you for the link.

M: What is the strand composed of ? Nothing. Composition would imply observability. Nature is a single, tangled strand. M: Energy does not exist on its own and the strands give rise to the SM particles so they cannot be composed of those. See the problem here ? Your above description included materialistic terms ie stiffness etc. Strands have no observable properties: no stiffness, no energy, no mass, etc. It seems to me that there are no contradictions. Or maybe I misunderstand your remark. M: The criteria of your bet are also assertions. Of course. That is the nature of any bet. What I just want to stress is that all of them follow from one basic assertion, namely that strands describe nature. M: QFT doesn't even declare the SM model is complete no physics theory does. Of course QFT does not declare this. One of the interests for the strand conjecture is that it does so, in contrast to QFT, because it does not allow more particles, it does not allow more forces, and it does not allow different values of the fundamental constants. These arguments are all not possible in QFT.

D : You also said electric charge moves slower than light, how fast does it move exactly? Things move as fast as you accelerate them. The point is that in nature, there are thing that move at light speed (such as photons) and others that do not, such as massive particles. All electric charges have mass. This is a natural result in the strand conjecture. D : What is a nearly one-dimensional tube? That is a tube that is so thin (Planck radius, 10^-35 m) that in most cases its diameter can be neglected. D : You assert, assert, and assert. No. There is only *one* assertion: that the fundamental principle of the strand conjecture allows to derive the Lagrangian of the standard model. Above I gave the mathematical arguments step by step that lead to the Dirac equation (done in 1980 by Battey-Pratt & Racey), to Maxwell's equations (done by Heras in 2007), to the gauge groups via Reidemeister moves, to the particle spectrum via tangle classification, and to the fundamental constants, via statistical tangle geometry. You can discuss every single argument in this argument chain; there might be one or even several errors in each step. But writing "assert, assert, assert" is not the same as finding an error. M: You should at least be able to apply wave equations to describe how the tube would bend and twist. Alas, this is not possible. It is easy to see that this cannot work. First, tubes are not observable. Second, tubes have no tension, no mass, etc and cannot be described by wave equations. Third, if one could, then they would be a kind of hidden variables. And then one could not get quantum theory, which forbids non-contextual hidden variables.

You are right. The original statement was sloppy. A strand is a nearly one-dimensional tube of Planck radius without ends - and without mass, colour, stiffness, energy, charge or any other observable physical property - randomly fluctuating in shape in three-dimensional background space. You are right that a probability distribution must be specified. But this topic leads too far. In short, it must and can be specified in such a way that known physics is reproduced.

M: Provide the equations showing how the Reidemeister moves correspond to any of the Feymann diagrams you posted provide at least something that proves it can do what you claim. The logic is this: Tangle core shape determine the quantum phase of a fermion such as the electron. Shape deformations behave like interactions. Shape deformations come in three kinds: Reidemeister moves I, II and III. The first Reidemeister move, the twist deformation, defines U(1). The proof is simple: twists can be generalized to arbitrary angles and can be added (in the same way as rotations can be added). And two standard twists (by 180°) are equivalent to none. Twists thus reproduce the U(1) algebra. Thus the generalized twists (to arbitrary angles) generate the U(1) group. (Indeed, twist rotations *represent* the U(1) group: their composition/multiplication is a U(1) *representation*.) Twists naturally act only on chiral tangles. So it becomes natural to define tangle chirality as electric charge. Topologically, electric charge (=chirality) is conserved. And also topologically, charge can only appear on tangles with 2 or 3 strands. Thus electric charges only appears on particles with mass. Thus electric charges move slower than light. Localised charges (at rest) naturally lead to 1/r^2 in three dimensions. Together (and by mathematical theorems cited in the paper) these three properties about electric charge imply the Maxwell's equations and the Lagrangian (-1/4)FF for the free electromagnetic field plus minimal coupling to electric charge. Minimal coupling implies the Feynman vertex of QED (and only this one). Together with the Dirac equation, this implies perturbative QED. (The preprint on QED gives some more details, including the arguments for deriving the fermion/Dirac Lagrangian.) This is the argument train that appears to imply that Reidemeister move I leads to perturbative QED. M: If the only equations that are involved are those used by QFT already then there literally is no purpose to use your tangle model. This is completely true! The only equations that arise from the logic just given are exactly those of perturbative QFT, or perturbative QED in the case of electrodynamics. Only the QED Lagrangian arises, without any change whatsoever. There is no deviation at all. There is indeed no purpose to use the tangle model for any calculation in perturbative QED. But there is a small addition: you can *determine* the masses of elementary particles and the fine structure constant from (statistical) tangle geometry. It is a small addition, because it does not change perturbative QED. But it is an intriguing small addition, because perturbative QED does not determine these numbers. The tangle model states that these numbers are not free parameters, but unique, fixed and calculable constants. One reason to be interested in the fundamental constants such as the fine structure constant and the electron mass is the following. All colours in nature are due to these two constants. As long as we do not understand the origin of these two constants, we do not understand the origin of the colours around us.

M. Can you not supply a mathematical basis for a single strand ? This is a mathematical description of a strand, even if you do not like it: A strand is a one-dimensional line of Planck radius, without ends, without stiffness, without any observable physical property, randomly fluctuating in three-dimensional background space. M: You have no idea how many ppl state this or that alternative model is just like the standard model. Yes, I have an idea: zero. I bet that you cannot name a single person with an alternative model that is able to deduce a single fundamental constant (mass, coupling, mixing) that appears in the standard model. M: I can mathematically describe the range of each force or the decay rates and mean lifetime of a particle. Good. Many can do this. For the range of QED and QCD I summarized it above. Mean lifetimes are determined by branching ratios. These in turn are determined by Feynman vertices, masses and coupling constants. M: I can mathematically describe particle generations. Many can do this: you just need their number, namely 3, their mass values, their mixings and their coupling constant values. The challenge of any model is to explain where the number 3 comes from, where the coupling constants come from, where the mixings come from, and where the masses come from. These numbers determine decay rates, branching ratios etc. The tangle model says that the number 3 comes from tangle topology. The tangle model says that couplings derive from the probability of Reidemeister moves, speaking loosely. The tangle model states that mixings derive from the specific tangle structure. The tangle model also says that mass derive from tangle complexity, and that the tangle structure determines the mass value.

M: I'm still waiting for a mathematical description for a strand. A strand is a one-dimensional line of Planck radius, without ends, without stiffness, *without any observable physical property*, randomly fluctuating in three-dimensional background space. The only observable arises when at a skew strand crossing, an underpass changes to an overpass. M: What determines how many strands are required etc. A quark is made of 2 strands -- because this allows to deduce the quark model and yields spin 1/2. A lepton is made of 3 strands -- because this allows no composites but spin 1/2. A gauge boson is made of 1, 2, or 3 untangle strands, because this yields spin 1, and the gauge groups. M: Is that not clear enough, you can use whatever method you prefer canonical as per QFT, conformal etc but mathematically prove your model. M: I know what QFT is capable of and how the SM Lagrangians apply I want a demonstration of your model to compare to. Seeing how the Lagrangian of electromagnetism arises is quite straightforward and explained in the preprint on QED. Electric fields are twist densities; magnetic fields are twist flow densities; charge is 1/3 of crossing number of a tangle. Coulomb's law arises. Maxwell's equations arise. The free Dirac Lagrangian also arises. And step by step, all the other components of the standard model Lagrangian also follow. In total, strands reproduce the Feynman vertices and the standard model Lagrangian. M: I want a clear and concise mathematical demonstration of the advantages you claim your model produces that the standard methodology cannot produce mathematically. 1. It explains why the charge of the proton is the same as the charge of the electron. 2. It explains the particle spectrum and properties (3 generations, quantum numbers), as well as the interaction spectrum and their properties (U(1), broken SU(2), and SU(3)). 3. It explains the fundamental constants: mass of the particles, the three coupling constants and the mixing angles. All three points are explained are in the mentioned preprints. The math and the logic is not hard and not so long. If you take the words away, the logic is very short and concise. M: You have no idea how many ppl state this or that alternative model is just like the standard model. To attempt to avoid the required mathematics. I have to guess the meaning of these two sentences. I am not trying to avoid mathematics; I am just trying to avoid *superfluous* mathematics. As far as I know, there is no other model in the research literature or even on the internet that deduces the standard model. More precisely, there is *no* model that deduces the points 1, 2 and 3 just mentioned. If there is such a model, let me know. And if there would be such a model, then the tangle model would not be needed any more. In fact, in that case, the tangle model would probably be *false*. On my betting page, I mention this option explicitly as a way to win a bet *against* the tangle model.

Here is a collections of answers to your remarks. M: Then you should have no problem demonstrating how to derive a vertex for a fermion here without having to go through your links. The two pdf linked above show how to deduce a vertex involving fermions or bosons. More precisely, a fermion is a tight tangle (tethers/ends are not observable, just the core is, the region where all the crossings are). The path "traced" by the tangle core corresponds to the propagator. The vertex itself corresponds to a deformation of the involved tangle core or cores; the deformation leads to one or more additional tangles leaving. Due to topological restrictions (only deformations are allowed, no cutting or regluing of strands), only the observed vertices are possible. M: There is a formula behind every vertex can you reproduce those formulas using your tangles? Yes, the formulae are exactly the same. As you mentioned, the formulae are uniquely fixed by the coupling constants, the coupling structure, and by the propagators. And these properties are reproduced. To be even clearer, there is no reason at all to use the tangle model, as it is completely equivalent to QFT: there are no deviations from the standard model with Dirac neutrinos. The only reason to explore the tangle model is that is appears to explain the particle spectrum and the three gauge groups of the interactions. So far, no other model can do that. M: Can you determine the range of a force? Yes, with the usual arguments that involve the vertices. The Abelian U(1) force (the group is *derived* from strands) obeys minimal coupling and thus reproduces Coulomb's law, Maxwell's equations, and infinite range. The bosons of the non-Abelian SU(3) (also this group is *derived* from strands) is short-range because of the lack of commutativity. M: Can you determine which decays are allowed and which are not ? Yes. Given that the correct vertices -- and the known conservation laws for quantum numbers -- are reproduced by the tangle model, you get all the allowed decays, and none else. M: Why does a strand twist? Strands fluctuate all the time; for example, they are being pushed by other vacuums strands.

You are right, my list is not the complete list of Feynman diagrams - because that list is infinite. The list is the complete list of Feynman vertices. And you can look it up in any book on the standard model. The list is complete and not "made up". With the vertices you can construct all Feynman diagrams, including the Penguin diagrams. And then you can do the calculations that you mention. Indeed, as you write, the gauge and Yukawa couplings are needed, as well as the mixing angles. These constants, and the conservation laws at each vertex, are discussed in the preprints.

Mordred, sorry, but I do not understand neither your English nor your physics. Just to answer the last point: no, I did not make up the Feynman diagrams myself. The Feynman diagrams in the papers (and in the two attached pdf in a previous post) are the complete list of Feynman diagrams of the standard model of elementary particle physics. Just look up any book or review article on the standard model, and you will find them there - not more, not less.

Yes. There are several steps. 1: Reproducing the Dirac equation. 2. This implies reproducing the Dirac propagator. 3. Reproducing the free field Lagrangian for gauge fields. 4. Showing that the observed gauge symmetry groups (unbroken or broken) are valid. 5. Showing that the coupling with gauge fields obeys the usual relations and gauge freedom. 6. Showing that only the observed particles exist. 7. Showing that only the observed vertices of the standard model are possible.

Half a year later, I cannot edit the post any more. Please free to take the web page out and put the link http://dx.doi.org/10.1134/S1063779619030055 to the paper in instead. There is a rational tangle assigned to each fermion and each boson. When playing with them, one notes that only certain vertices are possible. They turn out to be exactly the observed ones. I attach two pdfs showing the pictures of the vertices that follow from the tangles. All this happens in just three spatial dimensions, like for any tangle. Quantum numbers such as electric charge, parity, baryon number, lepton number, etc. are topological properties of the tangles. The possible families of rational tangles limit the particles to three generations and to the observed fermions and leptons. i-lagr1.pdf i-lagr2.pdf

The strand conjecture deduces all the known Feynman diagrams, so doing QFT remains correct. The number of generations also arises; so does a geometric picture for the CKM and PMNS mixing angles. See http://dx.doi.org/10.1134/S1063779619030055 One difference to string theory is that everything takes place in 3 spatial dimensions.

The strand conjecture, unifying and deriving the Lagrangians of both particle physics and general relativity, has been published in http://dx.doi.org/10.1134/S1063779619030055. Like in any unification attempt, predictions must be made and tests proposed. They are listed on http://www.motionmountain.net/bet . The page also offers a bet: the bet is lost if anything new is ever discovered in physics: new particles, new forces, new symmetries, new dimensions, new energy scales, new parameters, or anything else that invalidates the high energy desert. The bet is also lost if any other unified theory is ever found. Furthermore, the bet is lost if the strand conjecture is found to be inconsistent. Many other ways to invalidate the prediction are given. The bet is written so as to make it as easy as possible to win for anybody who disagrees.

The tangle model has been improved in the past years. A research preprint is now available at http://www.motionmountain.net/research.html It makes the derivation of the standard model of particle physics evident. The known particle spectrum, the known gauge interactions and the Feynman diagrams of the standard model are deduced. Enjoy.

The new edition of the free physics text in pdf format is now online at http://www.motionmountain.net The new edition includes many new topics, pictures, puzzles and tables; it keeps the promise to be fascinating and challenging on every page. The text now explains the geometric phase and its uses in optics, the limited, three-dimensional colour space of humans, how artists use it, and the twelve-dimensional colour space of mantis shrimps, the fascinating physics of the human voice, the singing of high-voltage lines, the knots possible in Maxwell's equations, and the reasons not take Descartes' philosophy seriously, following his own advice, because of the finiteness of the speed of light. The text also tells how to build a mirror that moves almost as fast as light itself, explains why the signals between the eye and the brain should be explored further, presents new findings on how birds feel magnetic fields, introduces the the nine mineral classes and the nine chemical elements that make up almost all rocks, shows the shadow produced by a single atom, tells how to measure the air temperature in the summer using a clock and add various topics on forensic physics, especially on the ways to make hidden fingerprints visible. Enjoy the reading. Christoph Schiller

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