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FQXi FORUM
September 2, 2010
CATEGORY:
What's Ultimately Possible in Physics? Essay Contest
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TOPIC: What if ... fundamental mathematics constrains the physics? by Franklin Potter [refresh]
TOPIC: What if ... fundamental mathematics constrains the physics? by Franklin Potter [refresh]
Essay Abstract
A hint of where to peek into what promises to be the ultimate source of the physics rules of the Universe and what present day physics speculation can be eliminated. If a 4th quark family appears at the LHC, then this ultimate source has been revealed!
Author Bio
Frank Potter is a Research Physicist at Sciencegems.com who was formerly at the University of California, Irvine for 25 years. His latest popular science books are Mad About Physics and Mad About Modern Physics with co-author Christopher Jargodzky, in which one can find traditional and new challenges requiring at least one logical step beyond the physics textbook exercise. His research includes extensions of the Standard Model of particle physics to discrete spacetime and modifications of the general theory of relativity.
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A hint of where to peek into what promises to be the ultimate source of the physics rules of the Universe and what present day physics speculation can be eliminated. If a 4th quark family appears at the LHC, then this ultimate source has been revealed!
Author Bio
Frank Potter is a Research Physicist at Sciencegems.com who was formerly at the University of California, Irvine for 25 years. His latest popular science books are Mad About Physics and Mad About Modern Physics with co-author Christopher Jargodzky, in which one can find traditional and new challenges requiring at least one logical step beyond the physics textbook exercise. His research includes extensions of the Standard Model of particle physics to discrete spacetime and modifications of the general theory of relativity.
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Fundamental symmetries elaborated by pure mathematics into physical theory is a disaster given empirical observation (e.g., Yang and Lee, 1957). The SI standard of mass is a physical artifact, Newton's G cannot be calculated. The Standard Model arrives massless, the Higgs boson is faery dust. Supersymmetry's partners refuse to appear (solar axion telescope), protons do not decay (Super-Kamiokande). Supergravity, lattice and loop quantum gravity, and above all string and M-theory predict nothing.
Physics is drunk with symmetry - Noether's theorems, Newton and Green's function. Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. Quantum field theories (QFT) with hermitian hamiltonians are invariant under the Poincaré group containing spatial reflections. Parity is a spatial reflection and parity is not a QFT symmetry! QFT are invariant under the identity component of the Poincaré group - the subgroup consisting of elements that can be continuous path joined to the Poincaré group identity; only an orthochronous Poincaré group representation. This subgroup excludes parity and time reversal. Supersymmetric (SUSY, gauge symmetry plus spacetime symmetry) grand unified theories relating fermions and bosons to each other contain allowances for symmetry breaking (inserted soft breaking terms into the Lagrangian where they maintain the cancellation of quadratic divergences).
Noether's theorems demand continuous symmetries or at least approximation by a Taylor series. Noether fails for parity. Quantum gravitation theories supplement Einstein-Hilbert action with an odd-parity Chern-Simons term. Physics cannot abide parity, adding symmetry breakings to make theory consistent with observation. An axiomatic system is no stronger than its weakest axiom. Empirical reality is parity divergent for all but the strongest interactions. CERN will be a massive disappointment. Physical theory is fundamentally wrong for postulating intrinsic parity symmetry. That is physics' self-imposed limit, that is why it fails.
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Physics is drunk with symmetry - Noether's theorems, Newton and Green's function. Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. Quantum field theories (QFT) with hermitian hamiltonians are invariant under the Poincaré group containing spatial reflections. Parity is a spatial reflection and parity is not a QFT symmetry! QFT are invariant under the identity component of the Poincaré group - the subgroup consisting of elements that can be continuous path joined to the Poincaré group identity; only an orthochronous Poincaré group representation. This subgroup excludes parity and time reversal. Supersymmetric (SUSY, gauge symmetry plus spacetime symmetry) grand unified theories relating fermions and bosons to each other contain allowances for symmetry breaking (inserted soft breaking terms into the Lagrangian where they maintain the cancellation of quadratic divergences).
Noether's theorems demand continuous symmetries or at least approximation by a Taylor series. Noether fails for parity. Quantum gravitation theories supplement Einstein-Hilbert action with an odd-parity Chern-Simons term. Physics cannot abide parity, adding symmetry breakings to make theory consistent with observation. An axiomatic system is no stronger than its weakest axiom. Empirical reality is parity divergent for all but the strongest interactions. CERN will be a massive disappointment. Physical theory is fundamentally wrong for postulating intrinsic parity symmetry. That is physics' self-imposed limit, that is why it fails.
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Dear Uncle Al,
Thank you for your comments, most of which I understand. Mixed in with some of your obviously true statements are many that cannot be challenged without further developments. However, I would like to point out a few related options.
You might be willing to consider a different approach, i.e., using discrete symmetries and their finite groups instead of continuous symmetries, as well as a discrete spacetime. Some consequences already worked out are
(1) there would be no need for the Higgs boson to provide masses because mass ratios of the leptons and of the quarks in the Standard Model are determined by the relationships to the j-invariant of elliptic modular functions, i.e., the masses are invariants under the linear fractional transformations of the finite binary rotation groups in 3-D for the leptons and in 4-D for the quarks;
(2) the Standard Model gauge group is simply a very good continuous approximation to the finite gauge group at the Planck scale, thereby explaining electroweak symmetry breaking, etc., in the simplest possible manner;
(3) by using finite rotational groups for the lepton families and the quark families - and I point out that they are subgroups of the Standard Model gauge group - one can make the unique mathematical connection from our 4-D spacetime via special quaternions called icosians to 8-D space and 10-D spacetime, thereby connecting to a discretized version of M-theory. This mathematical process ensures that the 4-D physical world is all we need but it has connections to the mathematical world of the Monster group and all its richness.
(4) If the b' quark appears at the LHC, then CERN will be a huge success! And Nature has discrete symmetries everywhere, even at the Planck scale.
I could fill in many more details of the advantages of the finite groups for the basis of the physics rules in the Universe, but I direct you to the references.
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Thank you for your comments, most of which I understand. Mixed in with some of your obviously true statements are many that cannot be challenged without further developments. However, I would like to point out a few related options.
You might be willing to consider a different approach, i.e., using discrete symmetries and their finite groups instead of continuous symmetries, as well as a discrete spacetime. Some consequences already worked out are
(1) there would be no need for the Higgs boson to provide masses because mass ratios of the leptons and of the quarks in the Standard Model are determined by the relationships to the j-invariant of elliptic modular functions, i.e., the masses are invariants under the linear fractional transformations of the finite binary rotation groups in 3-D for the leptons and in 4-D for the quarks;
(2) the Standard Model gauge group is simply a very good continuous approximation to the finite gauge group at the Planck scale, thereby explaining electroweak symmetry breaking, etc., in the simplest possible manner;
(3) by using finite rotational groups for the lepton families and the quark families - and I point out that they are subgroups of the Standard Model gauge group - one can make the unique mathematical connection from our 4-D spacetime via special quaternions called icosians to 8-D space and 10-D spacetime, thereby connecting to a discretized version of M-theory. This mathematical process ensures that the 4-D physical world is all we need but it has connections to the mathematical world of the Monster group and all its richness.
(4) If the b' quark appears at the LHC, then CERN will be a huge success! And Nature has discrete symmetries everywhere, even at the Planck scale.
I could fill in many more details of the advantages of the finite groups for the basis of the physics rules in the Universe, but I direct you to the references.
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Your essay at the end touched somewhat with what I have been working on. An aspect of this with respect of a quantum phase transition and the cosmological constant I wrote up at:
http://www.fqxi.org/community/forum/topic/494
This discusses the E_8 or the exceptional Jordan matrix group. I don't discuss this much here in this essay, but the triality of octonions in the 3x3 J^3(O) Jordan matrix points to the Mathieu group M_{24} or Leech lattice as potentially more fundamental. A break down of this is the (H_4xE_8)^2, where the two H_4 are the 120 or 600 cells (dual to each other) with the icosian group of quaternions. The 120-cell also tessellates hyperbolic manifolds whuch as the AdS spacetime, and this has relationships to AdS/CFT correspondence issues.
I don't discuss these issue terribly much in the essay here, but these issues are in certain way I think related.
I will say I have some doubts about the existence of 4-th quark and lepton doublets. The problem is that this would have increased the number of species in the post inflationary quark-gluon plasm of the unvierse. This would have lowered temperatures in the early universe and have changed the deuteron abundance we observe.
Cheers LC
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http://www.fqxi.org/community/forum/topic/494
This discusses the E_8 or the exceptional Jordan matrix group. I don't discuss this much here in this essay, but the triality of octonions in the 3x3 J^3(O) Jordan matrix points to the Mathieu group M_{24} or Leech lattice as potentially more fundamental. A break down of this is the (H_4xE_8)^2, where the two H_4 are the 120 or 600 cells (dual to each other) with the icosian group of quaternions. The 120-cell also tessellates hyperbolic manifolds whuch as the AdS spacetime, and this has relationships to AdS/CFT correspondence issues.
I don't discuss these issue terribly much in the essay here, but these issues are in certain way I think related.
I will say I have some doubts about the existence of 4-th quark and lepton doublets. The problem is that this would have increased the number of species in the post inflationary quark-gluon plasm of the unvierse. This would have lowered temperatures in the early universe and have changed the deuteron abundance we observe.
Cheers LC
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Dear Lawrence,
Thank you for your comments and insight. I can make the following responses:
(1) The existence of the 4th quark family of quarks representing 4-D binary rotation symmetries is the key to understanding the mathematical connections, solves several outstanding problems, AND it allows the YES/NO prediction of something to be found at the LHC: the b' quark at about 80 GeV decaying to b quark plus photon, and the t' quark. There is a slight hint of this b' quark in the Fermilab data at the acceptable energy range but nothing conclusive, so the enormous current at the LHC and judicious selection procedures should resolve the issue. There may be some initial confusion with people claiming a possible Higgs decay but deciphering the spin of the decaying particle will clear this up. By the way, I expect no Higgs because it is not needed.
(2) If my approach is correct, there will be no 4th lepton family for two reasons: (i) no more 3-D binary rotation groups exist for a 4th lepton family to exist, and (ii) astrophysics limits the number of fundamental particles to 15, so a 4th lepton family would exceed this limit by one.
(3) Yes, the Leech lattice and triality of the octonions (here the icosians) play a very important role, especially in discrete spacetimes, and I have been considering them for quite some time. The beauty here is how all of this mathematics is interconnected.
(4) One more very important point. If one sticks to continuous symmetries in 10-D spacetime and tries to 'divide down' to 4-D spacetime and a 6-D internal symmetry space, one gets the 10^500 or so possibilities string theorists are struggling with. However, if one uses DISCRETE 10-D spacetime, the division is easy and unique to a 4-D discrete spacetime and a 4-D discrete internal symmetry space - just what is needed!
(5) Both the spacetime and the internal symmetry space will turn out to be discrete at the Planck scale, and we are observing these in their continuum limits. The successes of the Standard Model are because it is such a good approximation to the real symmetries, those represented by finite subgroups of the Standard Model gauge group. That is, the actual lepton and quark family groups are finite subgroups of SU(2)L x U(1)Y = SU(2) x I = SU'(2), where I is the 2-element inversion group, and there are 3 in 3-D and 4 in 4-D, so 3 lepton families and 4 quark families.
The LHC will let the truth be told!
Cheers FP
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Thank you for your comments and insight. I can make the following responses:
(1) The existence of the 4th quark family of quarks representing 4-D binary rotation symmetries is the key to understanding the mathematical connections, solves several outstanding problems, AND it allows the YES/NO prediction of something to be found at the LHC: the b' quark at about 80 GeV decaying to b quark plus photon, and the t' quark. There is a slight hint of this b' quark in the Fermilab data at the acceptable energy range but nothing conclusive, so the enormous current at the LHC and judicious selection procedures should resolve the issue. There may be some initial confusion with people claiming a possible Higgs decay but deciphering the spin of the decaying particle will clear this up. By the way, I expect no Higgs because it is not needed.
(2) If my approach is correct, there will be no 4th lepton family for two reasons: (i) no more 3-D binary rotation groups exist for a 4th lepton family to exist, and (ii) astrophysics limits the number of fundamental particles to 15, so a 4th lepton family would exceed this limit by one.
(3) Yes, the Leech lattice and triality of the octonions (here the icosians) play a very important role, especially in discrete spacetimes, and I have been considering them for quite some time. The beauty here is how all of this mathematics is interconnected.
(4) One more very important point. If one sticks to continuous symmetries in 10-D spacetime and tries to 'divide down' to 4-D spacetime and a 6-D internal symmetry space, one gets the 10^500 or so possibilities string theorists are struggling with. However, if one uses DISCRETE 10-D spacetime, the division is easy and unique to a 4-D discrete spacetime and a 4-D discrete internal symmetry space - just what is needed!
(5) Both the spacetime and the internal symmetry space will turn out to be discrete at the Planck scale, and we are observing these in their continuum limits. The successes of the Standard Model are because it is such a good approximation to the real symmetries, those represented by finite subgroups of the Standard Model gauge group. That is, the actual lepton and quark family groups are finite subgroups of SU(2)L x U(1)Y = SU(2) x I = SU'(2), where I is the 2-element inversion group, and there are 3 in 3-D and 4 in 4-D, so 3 lepton families and 4 quark families.
The LHC will let the truth be told!
Cheers FP
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I submit two additions to my two previous response posts:
(1) The Frank Potter posts above with black background are mine, the author. I simply forgot to login first.
(2) I read that many other submissions talk about unification of the fundamental interactions with great complexities of mathematics. With my approach the game is so much simpler mathematically. If the b' quark appears at the LHC at around 80 GeV, then my 1st and 2nd references in my submission show how a UNIQUE UNIFICATION is accomplished in discrete 10-D spacetime dictated by Weyl E8 x Weyl E8 = 'discrete' SO(9,1). The beauty here is that the physical spacetime is 4-D and so is the internal symmetry space for the Standard Model, all as expected. One simply needs to consider discreteness as dictated by finite symmetry groups instead of Lie groups. One doesn't need to complicate matters further.
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(1) The Frank Potter posts above with black background are mine, the author. I simply forgot to login first.
(2) I read that many other submissions talk about unification of the fundamental interactions with great complexities of mathematics. With my approach the game is so much simpler mathematically. If the b' quark appears at the LHC at around 80 GeV, then my 1st and 2nd references in my submission show how a UNIQUE UNIFICATION is accomplished in discrete 10-D spacetime dictated by Weyl E8 x Weyl E8 = 'discrete' SO(9,1). The beauty here is that the physical spacetime is 4-D and so is the internal symmetry space for the Standard Model, all as expected. One simply needs to consider discreteness as dictated by finite symmetry groups instead of Lie groups. One doesn't need to complicate matters further.
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Dear Franklin,
I am having trouble understanding how a discrete symmetry can become a continuous one. I have read your references, but it is still unclear. I am picturing a perfect crystal. At macroscopic regions it can still displays the same discrete symmetries. Does this mean that space time at larger scales has discrete symmetry defects? If yes, is this related to mass? (I am thinking here about the mechanism for electric resistance in a metal due to crystalline defects.) Why are the masses different for different generations then? Also if yes, why is the dimensionality still 4? If there are “crystalline defects” would the long range dimensionality not be able to change? Can you please clarify?
Thank you.
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I am having trouble understanding how a discrete symmetry can become a continuous one. I have read your references, but it is still unclear. I am picturing a perfect crystal. At macroscopic regions it can still displays the same discrete symmetries. Does this mean that space time at larger scales has discrete symmetry defects? If yes, is this related to mass? (I am thinking here about the mechanism for electric resistance in a metal due to crystalline defects.) Why are the masses different for different generations then? Also if yes, why is the dimensionality still 4? If there are “crystalline defects” would the long range dimensionality not be able to change? Can you please clarify?
Thank you.
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Dear Florin,
Thank you for your questions. Perhaps my comments below with help somewhat. As you can see, one must accept discrete symmetry at all scales, although we may have difficulty resolving the discreteness above some scale.
(1) We would probably agree that if the b' quark at about 80 GeV shows up at the LHC then there is an excellent possibility that spacetime is discrete and...
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Thank you for your questions. Perhaps my comments below with help somewhat. As you can see, one must accept discrete symmetry at all scales, although we may have difficulty resolving the discreteness above some scale.
(1) We would probably agree that if the b' quark at about 80 GeV shows up at the LHC then there is an excellent possibility that spacetime is discrete and...
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Dear Franklin,
Thank you for the quick and detailed answer. I understand your model has good predictive power, but I still do not understand the physical mechanism of the continuous approximation. I am able to rotate any object in a continuous fashion, and not in discrete angles. Also the continuous Lorenz symmetry seems to hold in all experiments ever performed. How is this possible when the underlying symmetry is discrete? You mentioned in one of your papers that the continuous symmetry acts like a cover for the discrete ones. Is this term identical with let’s say SU(2) acts as a double cover for SO(3)?
My thinking is that if the underlying space-time is discrete, then you either ca have an exact tessalation of the continuous space by the discrete one (but this may be at odds with general relativity because the curvature can destroy the tessalation), or you have space-time local defects. In the second case would you not expect light diffusion in vacuum via a Rayleigh scattering for example?
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Thank you for the quick and detailed answer. I understand your model has good predictive power, but I still do not understand the physical mechanism of the continuous approximation. I am able to rotate any object in a continuous fashion, and not in discrete angles. Also the continuous Lorenz symmetry seems to hold in all experiments ever performed. How is this possible when the underlying symmetry is discrete? You mentioned in one of your papers that the continuous symmetry acts like a cover for the discrete ones. Is this term identical with let’s say SU(2) acts as a double cover for SO(3)?
My thinking is that if the underlying space-time is discrete, then you either ca have an exact tessalation of the continuous space by the discrete one (but this may be at odds with general relativity because the curvature can destroy the tessalation), or you have space-time local defects. In the second case would you not expect light diffusion in vacuum via a Rayleigh scattering for example?
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Dear Florin,
Thank you again for your comments and questions.
(1) At what size scale are you rotating "any object in a continuous fashion"? It's a long way from the Planck scale of 10^-35 meters to even the miniscule scale of the LHC probing at about 10^-23 meters. And it's further still to objects of centimeter size.
(2) The "cover" I mention has nothing to do with the double cover of SO(3) by SU(2). I'm simply saying that at our macro-scale wrt the Planck scale we cannot resolve the discreteness easily - like looking at my sheet of paper with 1 cm rulings from 10^12 meters away, i.e., from Jupiter, say. And furthermore, if I have a regular octahedron rotating in my hand and you are looking with your naked eye, at some distance you will not be able to distinguish its shape, and probably assume that it is spherical for simplicity, leading to the use of a continuous group for the spherical symmetry.
(3) If the underlying spacetime is discrete, then Penrose's heavenly sphere is tesselated by the finite subgroups of SO(3,1), for example. If you look carefully, these are the same ones as for subgroups of SU(2), the ones tesselating the Riemann sphere, and therein begins the connections via icosians to Weyl E8 x Weyl E8 = 'discrete' SO(9,1), etc.
(4) I haven't worried yet about curvature destroying these tesselations because the fundamental particles (leptons, quarks, etc.) themselves must draw together 'nodes' of some type (mathematical?) to form the subgroup symmetric entities, i.e., curve the space. Otherwise one is left with a space without particles.
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Thank you again for your comments and questions.
(1) At what size scale are you rotating "any object in a continuous fashion"? It's a long way from the Planck scale of 10^-35 meters to even the miniscule scale of the LHC probing at about 10^-23 meters. And it's further still to objects of centimeter size.
(2) The "cover" I mention has nothing to do with the double cover of SO(3) by SU(2). I'm simply saying that at our macro-scale wrt the Planck scale we cannot resolve the discreteness easily - like looking at my sheet of paper with 1 cm rulings from 10^12 meters away, i.e., from Jupiter, say. And furthermore, if I have a regular octahedron rotating in my hand and you are looking with your naked eye, at some distance you will not be able to distinguish its shape, and probably assume that it is spherical for simplicity, leading to the use of a continuous group for the spherical symmetry.
(3) If the underlying spacetime is discrete, then Penrose's heavenly sphere is tesselated by the finite subgroups of SO(3,1), for example. If you look carefully, these are the same ones as for subgroups of SU(2), the ones tesselating the Riemann sphere, and therein begins the connections via icosians to Weyl E8 x Weyl E8 = 'discrete' SO(9,1), etc.
(4) I haven't worried yet about curvature destroying these tesselations because the fundamental particles (leptons, quarks, etc.) themselves must draw together 'nodes' of some type (mathematical?) to form the subgroup symmetric entities, i.e., curve the space. Otherwise one is left with a space without particles.
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Dear Franklin,
Thank you again for the clarifications and for the quick answer. Also thank you for your patience in answering my questions, because I still do not understand.
Suppose for the sake of argument that we are dealing with a discrete Newtonian space-time and that the smallest space cell is a cube with the side equal with the Plank length. From Jupiter, a small cube on Earth will look like a sphere. I agree with this. But if the space tessellation is exact, space will have a very strong anisotropy because the cube angles are 90 degree no matter what size the overall cube has. In general, any tessellation will still have relative large angles involved regardless of the size of the basic unit. Because of this I would expect in any exact discrete space time model to see big anisotropies (which are not observed). One (maybe the only) way out is to have a disordered tessellation (partial, like a pentagonal one, or total like let’s say a spin glass). Then a local disorder will be imperceptible from Jupiter, and you still get your theoretical predictions. However, the point you have to address in this scenario is why light does not scatter around like it does in the air.
We are able to observe distant stars and we also do not experience any space anisotropies. How do you reconcile your proposal with those 2 experimental facts?
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Thank you again for the clarifications and for the quick answer. Also thank you for your patience in answering my questions, because I still do not understand.
Suppose for the sake of argument that we are dealing with a discrete Newtonian space-time and that the smallest space cell is a cube with the side equal with the Plank length. From Jupiter, a small cube on Earth will look like a sphere. I agree with this. But if the space tessellation is exact, space will have a very strong anisotropy because the cube angles are 90 degree no matter what size the overall cube has. In general, any tessellation will still have relative large angles involved regardless of the size of the basic unit. Because of this I would expect in any exact discrete space time model to see big anisotropies (which are not observed). One (maybe the only) way out is to have a disordered tessellation (partial, like a pentagonal one, or total like let’s say a spin glass). Then a local disorder will be imperceptible from Jupiter, and you still get your theoretical predictions. However, the point you have to address in this scenario is why light does not scatter around like it does in the air.
We are able to observe distant stars and we also do not experience any space anisotropies. How do you reconcile your proposal with those 2 experimental facts?
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Dear Florin,
Thank you for more questions and comments. I'm not sure where your measurable space anisotropies would arise as measurable quantities, and I point out that they must be measurable. The underlying space/spacetime is discrete with anisotropies but these are not yet measurable anisotropies.
(1) Regarding a photon traveling a straight path from a distance star. Why no scatter and why no deviation from a straight line? If spacetime is discrete at the Planck scale of distance, say, and its path is determined by linear fractional transformations (i.e., Mobius transformations), every transformation of the form tau -> 1 + tau (where tau is the ratio of lattice sides) moves the photon through the lattice undeviated from a straight line path. No measurable deviations here.
(2) Regarding rotations, since they are linear fractional transformations also, then they would occur as expected, progressing around the axis. If the scale is very small - such as the Planck length scale - I doubt whether any measurement with present day apparatus will be able to reveal the discreteness.
(3) Consider your cube example. You state that "because the cube angles are 90 degrees no matter what size the overall cube has", a statement I do not understand. If the space is a cubic lattice with cube lengths of about 10^-35 meters and the physical cube to be rotated has side lengths of 10^-2 meters, one would need to be able to resolve angle changes of about 10^-33 radians to observe the effects of any discreteness. The fundamental particles simply move to the next node points via tau -> 1 + tau. I know of no such measurement apparatus capable of doing this!
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Thank you for more questions and comments. I'm not sure where your measurable space anisotropies would arise as measurable quantities, and I point out that they must be measurable. The underlying space/spacetime is discrete with anisotropies but these are not yet measurable anisotropies.
(1) Regarding a photon traveling a straight path from a distance star. Why no scatter and why no deviation from a straight line? If spacetime is discrete at the Planck scale of distance, say, and its path is determined by linear fractional transformations (i.e., Mobius transformations), every transformation of the form tau -> 1 + tau (where tau is the ratio of lattice sides) moves the photon through the lattice undeviated from a straight line path. No measurable deviations here.
(2) Regarding rotations, since they are linear fractional transformations also, then they would occur as expected, progressing around the axis. If the scale is very small - such as the Planck length scale - I doubt whether any measurement with present day apparatus will be able to reveal the discreteness.
(3) Consider your cube example. You state that "because the cube angles are 90 degrees no matter what size the overall cube has", a statement I do not understand. If the space is a cubic lattice with cube lengths of about 10^-35 meters and the physical cube to be rotated has side lengths of 10^-2 meters, one would need to be able to resolve angle changes of about 10^-33 radians to observe the effects of any discreteness. The fundamental particles simply move to the next node points via tau -> 1 + tau. I know of no such measurement apparatus capable of doing this!
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Dear Franklin,
You are right, the anisotropies do happen, but only at energies high enough to resolve the small distances (which we cannot access today). This begs the question then: would this discreetness leave any imprint on the cosmic background radiation? Probably not because by the time the universe became transparent to radiation it was already too cold. Then the only hurdle remaining is the GR curvature, and I am thinking outloud, what would be the relationship with Unruh radiation?
Well, thank you again for your answers and good luck in this contest. My questions only reflected my strong interest in your paper and I think that your essay is one of the best.
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You are right, the anisotropies do happen, but only at energies high enough to resolve the small distances (which we cannot access today). This begs the question then: would this discreetness leave any imprint on the cosmic background radiation? Probably not because by the time the universe became transparent to radiation it was already too cold. Then the only hurdle remaining is the GR curvature, and I am thinking outloud, what would be the relationship with Unruh radiation?
Well, thank you again for your answers and good luck in this contest. My questions only reflected my strong interest in your paper and I think that your essay is one of the best.
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I agree there can't be more than 15 fundamental particles. I suppose we could consider a larger number of such particles as being of such high mass they don't ontribute to a partition function in the state of the universe at lower energy we can reasonably model today.
If you read my essay I am trying to address whether the issue of black hole complementarity can tell us something about...
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If you read my essay I am trying to address whether the issue of black hole complementarity can tell us something about...
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PS: I forgot to conclude with the matter of this as some penultimate theory. The Mathieu group is a subset of the automorphism over the Fischer-Greis "monster" group. This might be the mathematics which codifies the ultimate theory of physics we can know. There are some interesting suggestions for this. The full automorphism appears to be a 26-dimensional Lorentzian system, which is remarkably similar to the 26-dimensional bosonic string. This is stuff that Borcherds, Conway and Sloane have worked out in the last few decades.
Cheers LC
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Cheers LC
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Dear Potter,
i am a bit puzzled about my physics update. What i learnt some years back that the first family of quarks is constitutted by u,d,s quarks with masses in the range of 3-5 Gev. The second family belongs to c,b,t quarks with mass of 1.5, 4.5 and 175 Gev(latter is an estimate value). You claim that b quark mass will be around 80 Gev and t quark estimated at 2600 Gev.The first family constitute the matter that forms the visible matter of baryons while the second family remains illusive in this respect. Does it constitute dark matter and remain in their free state there! Or there exists probability of existence of another family of very very heavy quarks that we still need to postulate.
If one looks at the emergence of the force fields, these came in sequence of gravity,nuclear strong, electromagnetic and nuclear weak, as demanded by universe evolution. Quarks /gluons started forming nuclei and finally atoms/molecules before the first star came into being.
What was the nature of primordial matter that got created at Big Bang! It may well be something that no accelerator that we may ever build on earth can regenerate for us. Thus, it is hard to believe that fundamental Maths can help solve all the mysteries of Physics. Physics deal with physical reality and to me mathematics is a mere tool like the experimental syatems and has no way that it can govern the explanation of what happens/happened/will happen in Physics.
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i am a bit puzzled about my physics update. What i learnt some years back that the first family of quarks is constitutted by u,d,s quarks with masses in the range of 3-5 Gev. The second family belongs to c,b,t quarks with mass of 1.5, 4.5 and 175 Gev(latter is an estimate value). You claim that b quark mass will be around 80 Gev and t quark estimated at 2600 Gev.The first family constitute the matter that forms the visible matter of baryons while the second family remains illusive in this respect. Does it constitute dark matter and remain in their free state there! Or there exists probability of existence of another family of very very heavy quarks that we still need to postulate.
If one looks at the emergence of the force fields, these came in sequence of gravity,nuclear strong, electromagnetic and nuclear weak, as demanded by universe evolution. Quarks /gluons started forming nuclei and finally atoms/molecules before the first star came into being.
What was the nature of primordial matter that got created at Big Bang! It may well be something that no accelerator that we may ever build on earth can regenerate for us. Thus, it is hard to believe that fundamental Maths can help solve all the mysteries of Physics. Physics deal with physical reality and to me mathematics is a mere tool like the experimental syatems and has no way that it can govern the explanation of what happens/happened/will happen in Physics.
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Dear Lawrence,
I agree with your thinking about the connections mathematically, especially those connections to the Leech lattice, triality, and the Monster. The Golay-24 code and the Mathieu-24 group may also be important. Since the discrete 4-D physical spacetime telescopes up mathematically to discrete 8-D space and discrete 10-D spacetime as proposed, then the triality connection to Feynman diagrams,etc., brings up mathematically the discrete 24-D space and discrete 26-D spacetime. But first we need some evidence for the discreteness - i.e., the b' quark (and the t' quark) that represents the [3,3,3] discrete symmetry group.
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I agree with your thinking about the connections mathematically, especially those connections to the Leech lattice, triality, and the Monster. The Golay-24 code and the Mathieu-24 group may also be important. Since the discrete 4-D physical spacetime telescopes up mathematically to discrete 8-D space and discrete 10-D spacetime as proposed, then the triality connection to Feynman diagrams,etc., brings up mathematically the discrete 24-D space and discrete 26-D spacetime. But first we need some evidence for the discreteness - i.e., the b' quark (and the t' quark) that represents the [3,3,3] discrete symmetry group.
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Dear Narendra,
Thank you for your questions. Here's a brief review:
(1) We now have 3 families of leptons [e, e neutrino]; [muon, muon neutrino]; [tau, tau neutrino] and 3 families of quarks [up, down]; [charm, strange]; [top, bottom].
(2) The Standard Model makes no prediction for any families beyond the first lepton one and the first quark one. AND, knowing that 3 families of each exist, the Standard Model does not make any clear connections between the families but is obviously extremely close to the ultimate description of Nature. My geometrical approach makes very clear connections among the families, fits mathematically within the aegis of the Standard Model, but 4 quark families are predicted.
(3) I need the 4th quark family to appear at the LHC as the b' quark (read as b-prime quark). A hint may have shown up at Fermilab but not good enough for any discovery claim - too much background.
(4) The Big Bang with inflation plus other parts is the present standard model of the evolution of the universe, but as more and more data is accumulated, there has been some fuzziness developing which may allowed some alternative approaches to be considered. Nucleosynthesis is much of the game to be better understood by any alternatives to the standard model.
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Thank you for your questions. Here's a brief review:
(1) We now have 3 families of leptons [e, e neutrino]; [muon, muon neutrino]; [tau, tau neutrino] and 3 families of quarks [up, down]; [charm, strange]; [top, bottom].
(2) The Standard Model makes no prediction for any families beyond the first lepton one and the first quark one. AND, knowing that 3 families of each exist, the Standard Model does not make any clear connections between the families but is obviously extremely close to the ultimate description of Nature. My geometrical approach makes very clear connections among the families, fits mathematically within the aegis of the Standard Model, but 4 quark families are predicted.
(3) I need the 4th quark family to appear at the LHC as the b' quark (read as b-prime quark). A hint may have shown up at Fermilab but not good enough for any discovery claim - too much background.
(4) The Big Bang with inflation plus other parts is the present standard model of the evolution of the universe, but as more and more data is accumulated, there has been some fuzziness developing which may allowed some alternative approaches to be considered. Nucleosynthesis is much of the game to be better understood by any alternatives to the standard model.
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As i understand you gave the masses of the 4th quark family,b prime and t prime that are still to be seen experimentally. You claim to predict these as per your approach. May i have the salient features of your geometrical approach that predicts the fourth quark family. Besides the mass, what are the charges associated with the fourth family, in contrast with the third quark family. Can you visualise the time period that such quarks may have existed subsequent to the birth of the universe with Big Bang. That may provide you a comological way to look for them, instead of the LHC. The latter remains a dobtful starter.
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‘… However, Godels incompleteness theorem may be an impediment. It proves that there is no such thing as a complete logic system, that every logic system contains TRUE statements which cannot be proven true…’
‘Why is the last line of a proof surprising, if its truth is already hiding tautologically in the lines above?’ Richard Powers, The Gold Bug Variations.
If to explain some...
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‘Why is the last line of a proof surprising, if its truth is already hiding tautologically in the lines above?’ Richard Powers, The Gold Bug Variations.
If to explain some...
view entire post
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Greetings Frank,
I enjoyed reading your essay, and even more once I'd read some of your previous papers, which are cited therein. I would have liked it better if you explained your idea more fully in the essay itself, rather than making me look at your sources to make sense of things. It seems like an interesting idea though. Witten had a paper on BTZ Black Holes (arXiv: 0706.3359) that used the Monster group as a generating object.
I have long been a fan of the "It from Bit" and "Digital Physics" line of reasoning, and a chronicler of what Wigner called the unreasonable effectiveness of Math, in connection with my theory of the Cosmos based on the Mandelbrot Set. It seems perfectly reasonable for me to believe that the interesting objects of Math do act as generators of order in the Physical sciences, although I champion a more pragmatic view in my contest essay.
If I were to pursue a similar road to that suggested by the opening comments of your essay, I would probably note that Gravity (and the motion of material objects) is connected with the Reals, EM waves seem to be well modeled by the Complex numbers, which would suggest we look to Quaternions to understand spin and the Weak force and to Octonions for the Strong force interactions. There seems to be some connection in this reasoning with your essay, as E8 comes out of the Octonions mapped onto themselves. Am I right?
And please a mention on Icosions, as I hadn't heard that term until now. I thought there were only four normed division algebras, and this seems to imply a fifth set in that family.
All the Best,
Jonathan J. Dickau
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I enjoyed reading your essay, and even more once I'd read some of your previous papers, which are cited therein. I would have liked it better if you explained your idea more fully in the essay itself, rather than making me look at your sources to make sense of things. It seems like an interesting idea though. Witten had a paper on BTZ Black Holes (arXiv: 0706.3359) that used the Monster group as a generating object.
I have long been a fan of the "It from Bit" and "Digital Physics" line of reasoning, and a chronicler of what Wigner called the unreasonable effectiveness of Math, in connection with my theory of the Cosmos based on the Mandelbrot Set. It seems perfectly reasonable for me to believe that the interesting objects of Math do act as generators of order in the Physical sciences, although I champion a more pragmatic view in my contest essay.
If I were to pursue a similar road to that suggested by the opening comments of your essay, I would probably note that Gravity (and the motion of material objects) is connected with the Reals, EM waves seem to be well modeled by the Complex numbers, which would suggest we look to Quaternions to understand spin and the Weak force and to Octonions for the Strong force interactions. There seems to be some connection in this reasoning with your essay, as E8 comes out of the Octonions mapped onto themselves. Am I right?
And please a mention on Icosions, as I hadn't heard that term until now. I thought there were only four normed division algebras, and this seems to imply a fifth set in that family.
All the Best,
Jonathan J. Dickau
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Thanks for your reply. The monster group is way too much for physics. In my essay
http://www.fqxi.org/community/forum/topic/494
I do some work with the Jordan exceptional algebra. The triality on the three octonions leads to an embedding into the Mathieu or Leech structure. This is I think the penultimate (because the monster might be the ultimate) quantum error correction code for quantum cosmology.
QCD is a breakdown from the G_2 group, where SU(3) is the maximal subgroup of G_2 plus 3 and bar-3. These might be interpreted as the families, where you consider the prospect there are four. G_2 though is the automorphis of the Jordan group, as F_4 and G_2 are centralizer of E_8. The extension to the E_6 valued Jordan matrix leads to twistor space structures and that the norm of states obeys
(φ|ψ) = (2ħ)^{-1}{Ω(φ,ψ) + iΩ(φ,gψ)],
where g is a group operation and Ω the symplectic operation. Parentheses are used for bra-ket operations because this editor does not like carrot symbols. The group operation is g_2, which constructs the E_6 Jordan algebra from the Hermitian and anti-Hermitian Jordan matrices.
Cheers LC
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http://www.fqxi.org/community/forum/topic/494
I do some work with the Jordan exceptional algebra. The triality on the three octonions leads to an embedding into the Mathieu or Leech structure. This is I think the penultimate (because the monster might be the ultimate) quantum error correction code for quantum cosmology.
QCD is a breakdown from the G_2 group, where SU(3) is the maximal subgroup of G_2 plus 3 and bar-3. These might be interpreted as the families, where you consider the prospect there are four. G_2 though is the automorphis of the Jordan group, as F_4 and G_2 are centralizer of E_8. The extension to the E_6 valued Jordan matrix leads to twistor space structures and that the norm of states obeys
(φ|ψ) = (2ħ)^{-1}{Ω(φ,ψ) + iΩ(φ,gψ)],
where g is a group operation and Ω the symplectic operation. Parentheses are used for bra-ket operations because this editor does not like carrot symbols. The group operation is g_2, which constructs the E_6 Jordan algebra from the Hermitian and anti-Hermitian Jordan matrices.
Cheers LC
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Hello again Frank,
As luck would have it; one of the display articles on the FQXi homepage is germane to your essay's topic and to my comments above. It talks about a pair of researchers using Octonions to model quarks and telescope 10 dimensions to 8-d and then to 4-d space-time.
Taking on String Theory's 10-D Universe with 8-D Math
I thought this might be of interest.
All the Best,
Jonathan
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As luck would have it; one of the display articles on the FQXi homepage is germane to your essay's topic and to my comments above. It talks about a pair of researchers using Octonions to model quarks and telescope 10 dimensions to 8-d and then to 4-d space-time.
Taking on String Theory's 10-D Universe with 8-D Math
I thought this might be of interest.
All the Best,
Jonathan
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The icosians are 120 quaterions on H_4. There exist the 120 and 600 cells which are dual to each other. The exceptional group decomposes into two copise of H_4 plus permutations in the Weyl group. The Leech lattice embeds three copies of the E_8, and so this decomposition suggest that the heterotic string is more fundamentally a lattice system similar to solid state physics with a gauge E_8xE_8 field theory on it. It is similar to Bloch waves in solid state mechanics. It is also interesting that the 120-cell will tessellate the AdS_4 spacetime. The AdS_5 spacetime dual to S^5 is then AdS with an additional dimension which has a G_2 holonomy symmetry in the AdS_4xR^1 structure. This G_2 holonomy is the automorphism of the Jordan exceptional algebra.
This is a discrete system, yet the lattice is determined to within a gauge choice. This is the basic departure from solid state physics. String physics emerges in a Skymrion type of model, similar to bundles of relativistic electrons in solid state systems with large ion charges or with graphene.
My article at
http://www.fqxi.org/community/forum/topic/494
discusses the Jordan algebra and its role with black hole complementarity and the cosmological constant.
Cheers LC
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This is a discrete system, yet the lattice is determined to within a gauge choice. This is the basic departure from solid state physics. String physics emerges in a Skymrion type of model, similar to bundles of relativistic electrons in solid state systems with large ion charges or with graphene.
My article at
http://www.fqxi.org/community/forum/topic/494
discusses the Jordan algebra and its role with black hole complementarity and the cosmological constant.
Cheers LC
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Hello,
Thanks for the concise description of the Icosians, Lawrence. This is exceptionally cool stuff (pun intended). I shall give your essay a read soon, though I daresay a complete understanding may elude me. However; with each foray into unfamiliar mathematical territory, the better become my chances to come out with something meaningful. This is especially so with folks like you and Frank who delight in the grandeur and beauty of it.
I'd like to see your comments on the above too, Franklin Potter.
All the Best,
Jonathan
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Thanks for the concise description of the Icosians, Lawrence. This is exceptionally cool stuff (pun intended). I shall give your essay a read soon, though I daresay a complete understanding may elude me. However; with each foray into unfamiliar mathematical territory, the better become my chances to come out with something meaningful. This is especially so with folks like you and Frank who delight in the grandeur and beauty of it.
I'd like to see your comments on the above too, Franklin Potter.
All the Best,
Jonathan
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The icosians are a four dimensional version of the dodecahedron and icosahedron. These are called the dodecahedrachoron and icosahedrachoron. They are dual to each other, and tessellate hyperbolic spaces, as does the dodecahedron in three dimensions, as seen in the attachment.
LC
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LC
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For some reason the attachment did not take
Cheers LC
attachments: 1_Hyperbolic_orthogonal_dodecahedral_honeycomb.JPG
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Cheers LC
attachments: 1_Hyperbolic_orthogonal_dodecahedral_honeycomb.JPG
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Dear Narendra,
Thank you for more questions in your October 1 post. I have been away for about a week with no Internet access, so I will answer them briefly now.
(1) Why a 4th quark family? The quark states represent the 4 binary rotational symmetry groups that span the 4-D internal symmetry space, each group having two orthogonal states, i.e., the family pair. There are no more such groups available. The groups are [3,3,3], [4,3,3], [3,4,3], and [5,3,3], the last one for the 4th quark family (to correct my error in the Oct 1 response to Lawrence Crowell where I accidentally listed [3,3,3] for the t'-quark and b'-quark family.]
(2) The charges are the same for all quark families, one state with +1/3 and one state with -2/3 unit charges. Of course, there is in 4-D a conjugate internal symmetry space which has the antiparticle states with opposite charge signs and the same mass signs, the latter because these antiparticle states are gauge equivalent and not simply equivalent - that's what the U(1)Y does.
(3) Regarding a look at cosmological effects of this 4th quark family, it is potentially the source of the baryon asymmetry of the universe, as pointed out by W.-S. Hou, because the Jarlskog constant would have the correct value, about a factor of 10^13 larger than with 3 quark families.
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Thank you for more questions in your October 1 post. I have been away for about a week with no Internet access, so I will answer them briefly now.
(1) Why a 4th quark family? The quark states represent the 4 binary rotational symmetry groups that span the 4-D internal symmetry space, each group having two orthogonal states, i.e., the family pair. There are no more such groups available. The groups are [3,3,3], [4,3,3], [3,4,3], and [5,3,3], the last one for the 4th quark family (to correct my error in the Oct 1 response to Lawrence Crowell where I accidentally listed [3,3,3] for the t'-quark and b'-quark family.]
(2) The charges are the same for all quark families, one state with +1/3 and one state with -2/3 unit charges. Of course, there is in 4-D a conjugate internal symmetry space which has the antiparticle states with opposite charge signs and the same mass signs, the latter because these antiparticle states are gauge equivalent and not simply equivalent - that's what the U(1)Y does.
(3) Regarding a look at cosmological effects of this 4th quark family, it is potentially the source of the baryon asymmetry of the universe, as pointed out by W.-S. Hou, because the Jarlskog constant would have the correct value, about a factor of 10^13 larger than with 3 quark families.
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Dear Anton,
I have been away for about a week, so I will now respond to one of your many excellent comments in your October 2 post.
(1) Regarding "the universe has to create itself out of nothing..." Mathematically, there is a good possibility for nothing to become something. The fundamental particles known as leptons and quarks are spinors, i.e., spin 1/2 particles whose states must be rotated twice for the wave function to return to its initial value, not just one rotation for vectors.
(2) From a zero-length vector arises two orthogonal spinors, as originally derived by E. Cartan in the 1920's. Here, then, is the mathematical source of something out of nothing. There is a great book by S.L. Altmann called Rotations, Quaternions, and the Double Groups (1986) that provides the details with great insight.
(3) From the zero-length vector in spacetime can arise a quark-antiquark pair. What happens to energy conservation? There is a balance between the mass-energy, charge energy, etc. with the negative potential energy so that the sum remains zero.
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I have been away for about a week, so I will now respond to one of your many excellent comments in your October 2 post.
(1) Regarding "the universe has to create itself out of nothing..." Mathematically, there is a good possibility for nothing to become something. The fundamental particles known as leptons and quarks are spinors, i.e., spin 1/2 particles whose states must be rotated twice for the wave function to return to its initial value, not just one rotation for vectors.
(2) From a zero-length vector arises two orthogonal spinors, as originally derived by E. Cartan in the 1920's. Here, then, is the mathematical source of something out of nothing. There is a great book by S.L. Altmann called Rotations, Quaternions, and the Double Groups (1986) that provides the details with great insight.
(3) From the zero-length vector in spacetime can arise a quark-antiquark pair. What happens to energy conservation? There is a balance between the mass-energy, charge energy, etc. with the negative potential energy so that the sum remains zero.
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Dear Jonathan,
I am back from about a week absence, so I can responds to some of your questions and comments in your October 4 post.
(1) I think that your normed division algebra sequence suggestion may be important, but I might think in a slightly different manner: (i) Reals for the EM vector potential with A because its phase factor involves i eA/c; (ii) Complex for the weak interaction, thereby having a 2x2 matrix in the phase factor; (iii) Quaternions for the color interaction for quarks involving 4x4 matrix rotations spanning the complete 4-D internal symmetry space; and (iv) Octonions in the phase factor for the gravitational interaction.
(2) Note that I am just speculating in (1), for I have not seriously engaged my thoughts with regard to the consequences.
(3) The icosians are the key, at least to connecting our discrete 4-D physical space to discrete 8-D, telescoping upward, yet remaining both a quaternion (because there are only 4 non-zero coefficients) and also being an octonion. No, icosians are not a fifth normed division algebra.
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I am back from about a week absence, so I can responds to some of your questions and comments in your October 4 post.
(1) I think that your normed division algebra sequence suggestion may be important, but I might think in a slightly different manner: (i) Reals for the EM vector potential with A because its phase factor involves i eA/c; (ii) Complex for the weak interaction, thereby having a 2x2 matrix in the phase factor; (iii) Quaternions for the color interaction for quarks involving 4x4 matrix rotations spanning the complete 4-D internal symmetry space; and (iv) Octonions in the phase factor for the gravitational interaction.
(2) Note that I am just speculating in (1), for I have not seriously engaged my thoughts with regard to the consequences.
(3) The icosians are the key, at least to connecting our discrete 4-D physical space to discrete 8-D, telescoping upward, yet remaining both a quaternion (because there are only 4 non-zero coefficients) and also being an octonion. No, icosians are not a fifth normed division algebra.
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Dear Lawrence,
I have been away, so I can now respond briefly to your interesting comments in your October 5 posting.
(1) Triality of the 3 octonions (telescoped up from discrete 4-D via icosians) for a fundamental interaction of spinor-vector-spinor definitely leads to the Leech lattice and its symmetry, as you state. Also the Golay-24 coding related to the Leech lattice is important for errorless progression, probably through the discrete lattice.
(2) You remain fixated on Lie groups and may be correct, but if the b' quark appears at around 80 GeV, then you will definitely need to consider their finite subgroups and the discrete symmetries which I consider to be the underlying basis for the Standard Model.
(3) The Monster has more fingers into many areas of mathematics than I can appreciate. But the Monster has THE intimate connection to elliptic modular functions which definitely play a key role in leptons and quarks, the coefficients of the expansion of the j-invariant in a Fourier series are related to dimensions of its irreducible representations.
(4) And as I have shown, each family has an invariant syzygy with proportionality to the j-function, producing ratios of family masses. There is no need for a Higgs.
(5) The first indication that I might be on the right path was the ability to predict the top quark mass of about 160 GeV in 1992, a few years before its discovery at Fermilab in 1995 at 170 GeV. You may recall that everyone else was predicting values under 95 GeV before 1994.
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I have been away, so I can now respond briefly to your interesting comments in your October 5 posting.
(1) Triality of the 3 octonions (telescoped up from discrete 4-D via icosians) for a fundamental interaction of spinor-vector-spinor definitely leads to the Leech lattice and its symmetry, as you state. Also the Golay-24 coding related to the Leech lattice is important for errorless progression, probably through the discrete lattice.
(2) You remain fixated on Lie groups and may be correct, but if the b' quark appears at around 80 GeV, then you will definitely need to consider their finite subgroups and the discrete symmetries which I consider to be the underlying basis for the Standard Model.
(3) The Monster has more fingers into many areas of mathematics than I can appreciate. But the Monster has THE intimate connection to elliptic modular functions which definitely play a key role in leptons and quarks, the coefficients of the expansion of the j-invariant in a Fourier series are related to dimensions of its irreducible representations.
(4) And as I have shown, each family has an invariant syzygy with proportionality to the j-function, producing ratios of family masses. There is no need for a Higgs.
(5) The first indication that I might be on the right path was the ability to predict the top quark mass of about 160 GeV in 1992, a few years before its discovery at Fermilab in 1995 at 170 GeV. You may recall that everyone else was predicting values under 95 GeV before 1994.
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Hello Frank,
Welcome back. Thanks for the answers to my questions. Some very interesting stuff. No need for a Higgs, with your approach? Way cool.
It seems I will have to do some further study into the Icosians, and may understand your work better after that. This is fun Math to explore! But since it's supposed to be predictive, it will be interesting to see if a fourth quark family appears.
All the Best,
Jonathan
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Welcome back. Thanks for the answers to my questions. Some very interesting stuff. No need for a Higgs, with your approach? Way cool.
It seems I will have to do some further study into the Icosians, and may understand your work better after that. This is fun Math to explore! But since it's supposed to be predictive, it will be interesting to see if a fourth quark family appears.
All the Best,
Jonathan
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I agree about the importance of modular forms. The Mathieu M_{24} or Leech lattice system has modular structure. The monster F-G group appears to be the ultimate goal. I have thought this was physics and cosmology for the mid 21st century or maybe 100 years from now. Of course given some thing I see coming in the world I question whether we will be around long enough. Of course the M_{24} or Leech is the automorphism sporadic group for the F-G, which makes it terribly important in its own right. In part with the Jordan exceptional algebra I have been focused somewhat on G_2 at the automorphism there.
I think it all comse down to scale. Here is how I see it. String theory operates on a scale of about 10-20 Planck lengths, with the Hagedorn temperatures at sqrt{5pi} times the Planck length. My thinking is that strings are emergent structures, similar to Skymrion field theory, where there are emergent knots of fields. In fact if you read my article
http://www.fqxi.org/community/forum/topic/494
you will see the role of Chern-Simons Lagrangian in the exceptional algebra, which Skymrion theory is based on. This theory extends to a scale maybe down to 5-10 Planck units of scale. This theory is set up to fold it all into the Leech structure which I think holds down to a scale of nearly 2 Planck lengths. Now this is the automorphism of the monster group, so a quantum cosmology based on this might take us to 1 + epsilon times the Planck scale.
The monster group contains lots of stuff! An automorphism of it is the 26-dimensional Lorentz group --- the 26 dimensional bosonic string?! It is the ultimate form of group theory, and maybe what ever physical principle or structure can ever possibly exist is coded in it. I will say for now I question whether we can do much with it. You might check out Tony Smith's paper below and his website. he has all sorts of ideas about this, but I question whether this is physics at this time.
If you have ideas about how to actually do physics with the F-G group let me know.
As for discrete and continuum, I see things as an interplay between the two.
Cheers LC
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I think it all comse down to scale. Here is how I see it. String theory operates on a scale of about 10-20 Planck lengths, with the Hagedorn temperatures at sqrt{5pi} times the Planck length. My thinking is that strings are emergent structures, similar to Skymrion field theory, where there are emergent knots of fields. In fact if you read my article
http://www.fqxi.org/community/forum/topic/494
you will see the role of Chern-Simons Lagrangian in the exceptional algebra, which Skymrion theory is based on. This theory extends to a scale maybe down to 5-10 Planck units of scale. This theory is set up to fold it all into the Leech structure which I think holds down to a scale of nearly 2 Planck lengths. Now this is the automorphism of the monster group, so a quantum cosmology based on this might take us to 1 + epsilon times the Planck scale.
The monster group contains lots of stuff! An automorphism of it is the 26-dimensional Lorentz group --- the 26 dimensional bosonic string?! It is the ultimate form of group theory, and maybe what ever physical principle or structure can ever possibly exist is coded in it. I will say for now I question whether we can do much with it. You might check out Tony Smith's paper below and his website. he has all sorts of ideas about this, but I question whether this is physics at this time.
If you have ideas about how to actually do physics with the F-G group let me know.
As for discrete and continuum, I see things as an interplay between the two.
Cheers LC
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i am happy to see your response to m posting. i can understand that you may have something in nothing that existed before the creation of the universe but that is awaited. Though i do not comprehend the 4 th family of quarks indicated by you, i see that these may well be the primordial aprticles generated at the Big bang which subsequently disappeared through ultra fast decay that initiated the ultrafast evolution of the created point during Big bang. he third family and others are then resoonsible for the generation of baryons in the visible matter and the dark matter simply consists of frozen quark matter that no longer has the strong field strength to glue into baryons. Such a picture may help explain initial extra quick evolution of the universe as welll as production of huge amount of dark amtter and associated dark energy. in my esssay i have attempted to understand it through the changing initial strengths of the gravitation and strong nuclear fields the firstb to arrive on the scene in a sequential manner. The Physics of that early universe can not be based on the Phydics we have worked out for the later universe in the past few hundred years! The ehavier quarks decayed ultra fast into lower mass families under very strong field strengths present in those extremely high temperatures cum mass densities, no accelerator in earth can possibly simulate in the years to come!
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Dear Narendra,
Thank you for your further comments. I can respond thusly:
(1) Once I discovered that the 3 lepton families when assigned to the 3 finite binary rotational subgroups of SU(2)L x U(1)Y known as [3,3,2], [4,3,2] and [5,3,2], i.e., the 3-D rotational symmetries, provided the mass ratios 1:108:1728 via connections with the j-invariant function, I had no choice on assigning the quark families to the 4-D rotational symmetries of the 4 finite binary rotational subgroups. They are the only ones remaining that fit the same scheme, therefore 4 quark families. Notice that the leptons are 3-D entities and the quarks are 4-D entities in this scheme, not point particles.
(2) Then I discovered that the SU(3)-color symmetries can also fit in 4-D by working out the 4-D rotations which occur in two planes simultaneously, thereby reproducing the color interaction and the three color charges. So the Standard Model fits into a 4-D internal symmetry space and does not require a larger space.
(3) If I now ask whether other fundamental particles are possible, I would say NO: no supersymmetry particles, no unparticles, no dark matter, no Higgs, etc. They would not fit this same scheme.
(4) Consequently, I would doubt that your suggestion "dark matter simply consists of frozen quark matter that no longer has the strong field strength..." will survive. The color interaction is built into the origin of the quark particle states, i.e., their 4-D quality itself. Putting 3 4-D quark states together mathematically makes one 3-D baryon, etc.
(5) I would also doubt that any of the fundamental interactions, including gravitation, have changed their strengths over time, but I cannot provide an argument yet.
(6) For any alternative approach to cosmology, one needs to explain baryogenesis. One way is the one provided by the standard model of cosmology - big bang, inflation, etc., which is a complicated model with lots of patchwork over the years. What needs to be explained by alternatives is essentially the creation of about ONE hydrogen atom per cubic meter per 10 million years to account for the matter/energy density of the universe as we know it.
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Thank you for your further comments. I can respond thusly:
(1) Once I discovered that the 3 lepton families when assigned to the 3 finite binary rotational subgroups of SU(2)L x U(1)Y known as [3,3,2], [4,3,2] and [5,3,2], i.e., the 3-D rotational symmetries, provided the mass ratios 1:108:1728 via connections with the j-invariant function, I had no choice on assigning the quark families to the 4-D rotational symmetries of the 4 finite binary rotational subgroups. They are the only ones remaining that fit the same scheme, therefore 4 quark families. Notice that the leptons are 3-D entities and the quarks are 4-D entities in this scheme, not point particles.
(2) Then I discovered that the SU(3)-color symmetries can also fit in 4-D by working out the 4-D rotations which occur in two planes simultaneously, thereby reproducing the color interaction and the three color charges. So the Standard Model fits into a 4-D internal symmetry space and does not require a larger space.
(3) If I now ask whether other fundamental particles are possible, I would say NO: no supersymmetry particles, no unparticles, no dark matter, no Higgs, etc. They would not fit this same scheme.
(4) Consequently, I would doubt that your suggestion "dark matter simply consists of frozen quark matter that no longer has the strong field strength..." will survive. The color interaction is built into the origin of the quark particle states, i.e., their 4-D quality itself. Putting 3 4-D quark states together mathematically makes one 3-D baryon, etc.
(5) I would also doubt that any of the fundamental interactions, including gravitation, have changed their strengths over time, but I cannot provide an argument yet.
(6) For any alternative approach to cosmology, one needs to explain baryogenesis. One way is the one provided by the standard model of cosmology - big bang, inflation, etc., which is a complicated model with lots of patchwork over the years. What needs to be explained by alternatives is essentially the creation of about ONE hydrogen atom per cubic meter per 10 million years to account for the matter/energy density of the universe as we know it.
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I have an idea about how to work the monster into things, or at least indicate how it is in the background of things. The J^3(O) eceptional matrix can be extended to the C* valued J^3(CxO). This may be further extended to the quaterionic J^3(HxO) with the inclusion of anti-Hermitian cubic matrix terms and a G_2 fibration between the C* ant anti-C* elements. This leads to 26-dimensional lightcone realizations, which are Lorentzian group realizations of the Leech lattice. This is a 26-dimensional automorphism group over the Fischer-Greiss group (monster).
More later,
Lawrence B. Crowell
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More later,
Lawrence B. Crowell
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Dear Franklin Potter,
Most of the outcomes from collider experimentations are probabilistic rather than deterministic, may be due to Gödel's incompleteness and it's contradictory to Einstein's statement, 'God does not play dice'. This may be of mathematical constrains in observability or physical inconsistency in reality and I think this impediment may be of physical inconsistency in reality. I agree with you that nature is numerical, but the numeric order what nature exhibit is rather different from the mathematical systems we adapt to observe nature by theoretical formulations.
If we think of a wave propagation through spin matrix in that quaternions number system and norms of geometric origin are applicable, then we may have to identify 18 hadrons in pairs with opposite spins, to observe a segment of surface of a section in the numeric hierarchy the nature have, in that the finite simple groups may be applicable.
With best wishes,
Jayakar
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Most of the outcomes from collider experimentations are probabilistic rather than deterministic, may be due to Gödel's incompleteness and it's contradictory to Einstein's statement, 'God does not play dice'. This may be of mathematical constrains in observability or physical inconsistency in reality and I think this impediment may be of physical inconsistency in reality. I agree with you that nature is numerical, but the numeric order what nature exhibit is rather different from the mathematical systems we adapt to observe nature by theoretical formulations.
If we think of a wave propagation through spin matrix in that quaternions number system and norms of geometric origin are applicable, then we may have to identify 18 hadrons in pairs with opposite spins, to observe a segment of surface of a section in the numeric hierarchy the nature have, in that the finite simple groups may be applicable.
With best wishes,
Jayakar
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Dear Jayakar,
Thank you for your comments. I'll try to respond to some of them.
(1) Quantum mechanics has two parts (i) a deterministic part which Penrose calls Schrödinger evolution U and (ii) a non-deterministic part called state reduction R. I.e., one first has a linear combination of possible state outcomes according to the rules of QM, and then in the apparatus appears one of them with its probability of occurence.
(2) In my approach, all particles that can be isolated such as an electron, proton, etc. require a 3-dimensional space for their state definition. But we live in a 4-D spacetime which can be shown to behave like a 4-D space when both are discrete - i.e., can both be seen as rotations of a tesselated Riemann sphere. What I'm getting to is that our fundamental particles may really have that 4th spatial dimension for their use even though somehow it is shared with out time dimension. What we interpret as probabilistic collapse to a single outcome may be more determined than we think. I'm only guessing here. Then perhaps 'God does not play dice.'
(3) One other advantage of the specific finite groups for the leptons and quarks as the basis for all physics, other than being subgroups of the gauge group of the Standard Model, is that their linear fractional transformations in the discrete spacetime can be shown to lead to the conservation laws in the limit of a continuous spacetime. The connection may not be so direct with other groups.
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Thank you for your comments. I'll try to respond to some of them.
(1) Quantum mechanics has two parts (i) a deterministic part which Penrose calls Schrödinger evolution U and (ii) a non-deterministic part called state reduction R. I.e., one first has a linear combination of possible state outcomes according to the rules of QM, and then in the apparatus appears one of them with its probability of occurence.
(2) In my approach, all particles that can be isolated such as an electron, proton, etc. require a 3-dimensional space for their state definition. But we live in a 4-D spacetime which can be shown to behave like a 4-D space when both are discrete - i.e., can both be seen as rotations of a tesselated Riemann sphere. What I'm getting to is that our fundamental particles may really have that 4th spatial dimension for their use even though somehow it is shared with out time dimension. What we interpret as probabilistic collapse to a single outcome may be more determined than we think. I'm only guessing here. Then perhaps 'God does not play dice.'
(3) One other advantage of the specific finite groups for the leptons and quarks as the basis for all physics, other than being subgroups of the gauge group of the Standard Model, is that their linear fractional transformations in the discrete spacetime can be shown to lead to the conservation laws in the limit of a continuous spacetime. The connection may not be so direct with other groups.
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Franklin,
I happened to look in on your site and found you responded to a post of mine 3 weeks ago.
You are covering some of the same ground I work in. My work on the exceptional Jordan matrix is meant to take things to the Leech lattice and the 26-dimensional Lorentz group, which is the automorphism of the Monster group. I am not sure exactly why the gauge structure for QCD is...
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I happened to look in on your site and found you responded to a post of mine 3 weeks ago.
You are covering some of the same ground I work in. My work on the exceptional Jordan matrix is meant to take things to the Leech lattice and the 26-dimensional Lorentz group, which is the automorphism of the Monster group. I am not sure exactly why the gauge structure for QCD is...
view entire post
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Dear Lawrence,
Thank you for your further comments about the larger spaces and some of the mathematical relations among the finite simple groups, Jordan exceptional algebra, etc. that act in them. I also note your comment about gravitation. Let me respond simply.
(1) Although the higher dimensional spaces are important mathematically, and they can be related to lower dimensional spaces by such entities as icosians, etc., I find that if I can define the lepton, quarks, and interaction bosons as states that span the 3-D and 4-D real spaces (or the unitary plane C2), then I may have a better grasp of their physical properties and behavior.
(2) In fact, if the b' quark shows up at the LHC, then leptons as 3-D geometrical entities and quarks as 4-D geometrical entities have ALL the right mathematical and physical properties. One only needs to go to higher dimensional spaces such a 10-D because the icosians telescope one up from 4-D to 8-D to 10-D space-time. There are important mathematical relations up in 10-D spacetime that seem to dictate why 4-D works so well for the physical properties of our universe, both as the dimension of the internal symmetry space in which the Standard Model acts and as the dimensions of space-time. I know some of them, but I suspect that a few more are yet to be uncovered.
(2) I also suspect that gravitation does not need to be quantized directly. There is a lot more to be done with the general relativistic Hamilton-Jacobi equation and a simple transformation than most people realize.
(3) I will read some of your posts.
Cheers
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Thank you for your further comments about the larger spaces and some of the mathematical relations among the finite simple groups, Jordan exceptional algebra, etc. that act in them. I also note your comment about gravitation. Let me respond simply.
(1) Although the higher dimensional spaces are important mathematically, and they can be related to lower dimensional spaces by such entities as icosians, etc., I find that if I can define the lepton, quarks, and interaction bosons as states that span the 3-D and 4-D real spaces (or the unitary plane C2), then I may have a better grasp of their physical properties and behavior.
(2) In fact, if the b' quark shows up at the LHC, then leptons as 3-D geometrical entities and quarks as 4-D geometrical entities have ALL the right mathematical and physical properties. One only needs to go to higher dimensional spaces such a 10-D because the icosians telescope one up from 4-D to 8-D to 10-D space-time. There are important mathematical relations up in 10-D spacetime that seem to dictate why 4-D works so well for the physical properties of our universe, both as the dimension of the internal symmetry space in which the Standard Model acts and as the dimensions of space-time. I know some of them, but I suspect that a few more are yet to be uncovered.
(2) I also suspect that gravitation does not need to be quantized directly. There is a lot more to be done with the general relativistic Hamilton-Jacobi equation and a simple transformation than most people realize.
(3) I will read some of your posts.
Cheers
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There are some interesting developments along these lines, though in different guises. I attach a paper by Baez and Barrett on the quantum tetrahedron. This paper is from 1999, but it has some interesting results. Also recently Dirac monopoles have been observed in solids with tetrahedral symmetry, where I include a Perspectives article in Science on this. I can send the full articles if you are so interested.
Cheers LC
attachments: 2_375.pdf, 2_9903060v1.pdf
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Cheers LC
attachments: 2_375.pdf, 2_9903060v1.pdf
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Hello Frank, et al;
If you like what Baez and Barrett have cooked up in the paper Lawrence recommends above, you may find value in the follow-up paper by B&B on Relativistic Spin Networks arXiv:gr-qc/0101107, and in Baez' paper with Christensen and Egan on Asymptotics of 10j Symbols arXiv:gr-qc/0208010, which I recommended to Ray Monroe. I'm happy to see this thread continuing to evolve, as there is some cool stuff being discussed here.
All the Best,
Jonathan
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If you like what Baez and Barrett have cooked up in the paper Lawrence recommends above, you may find value in the follow-up paper by B&B on Relativistic Spin Networks arXiv:gr-qc/0101107, and in Baez' paper with Christensen and Egan on Asymptotics of 10j Symbols arXiv:gr-qc/0208010, which I recommended to Ray Monroe. I'm happy to see this thread continuing to evolve, as there is some cool stuff being discussed here.
All the Best,
Jonathan
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In looking at these results in the papers by Baez et al, I am pondering whether they can be applied to minimal sphere packing configurations. The 24-cell has a B_4, D_4 and F_4 representation. The D_4 is a triality of 8 tetrahedra, the B_4, which defines an SO(9) group with the quotient on F_4 is an 8-tetrahedral and 16 octahedral sysstem (16-cell).
Cheers LC
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Cheers LC
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Dear Lawrence & Jonathan,
Thank you both for your comments and reading suggestions.
(1) Several of the suggested papers I had skimmed a long while ago but did not pay much attention to the details. So I was aware of them for a short time and then forgot about them. Thank you for jogging my memory and suggesting that I look at them.
(2) Sphere packings in real space dimensions 4, 8, and 24 probably have important links to fundamental particles, although I would bet upon the rotational symmetries of the finite binary rotational groups in 3-D and 4-D as being the key mathematical concepts for defining lepton and quark families with the sphere packings as ancillary.
(3) The local operations of gauge group of the Standard Model, considered in a discrete internal symmetry space, can all be handled with the binary icosahedral group taken twice as I x I, which leads to the E8 lattice, which is related to sphere packing in 8-D.
(4) There is a history of suggestions linking particle physics to condensed matter physics. Although there are also great differences to consider also, one similarity may be in determining the speed of light value from first principles. As you know, phonons, magnons, etc., i.e., all the pseudo-particles in the crystal lattice, have upper limits to their propagation speeds.
(5) If indeed the speed of light in a discrete space is analogous to pseudo-particle propagation in an atomic lattice, then one can work backwards to use the light speed value to determine the properties of the discrete space lattice itself with respect to photon propagation. There are some interesting results!
Cheers
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Thank you both for your comments and reading suggestions.
(1) Several of the suggested papers I had skimmed a long while ago but did not pay much attention to the details. So I was aware of them for a short time and then forgot about them. Thank you for jogging my memory and suggesting that I look at them.
(2) Sphere packings in real space dimensions 4, 8, and 24 probably have important links to fundamental particles, although I would bet upon the rotational symmetries of the finite binary rotational groups in 3-D and 4-D as being the key mathematical concepts for defining lepton and quark families with the sphere packings as ancillary.
(3) The local operations of gauge group of the Standard Model, considered in a discrete internal symmetry space, can all be handled with the binary icosahedral group taken twice as I x I, which leads to the E8 lattice, which is related to sphere packing in 8-D.
(4) There is a history of suggestions linking particle physics to condensed matter physics. Although there are also great differences to consider also, one similarity may be in determining the speed of light value from first principles. As you know, phonons, magnons, etc., i.e., all the pseudo-particles in the crystal lattice, have upper limits to their propagation speeds.
(5) If indeed the speed of light in a discrete space is analogous to pseudo-particle propagation in an atomic lattice, then one can work backwards to use the light speed value to determine the properties of the discrete space lattice itself with respect to photon propagation. There are some interesting results!
Cheers
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Sphere packings are equivalent to quantum error correction codes. These codes preserve quantum information. This is what makes them particularly of interest. The F_4 group describes the 24-cell which is involved with the spatial part of the J^3(O) in 26 dimensions, and serves as a tessellation of four dimensions. The E_8 has under the Weyl group description diag[H_4, H_4], which gives the 120 and 600 cells. These will tessellate hyperbolic spaces, such as the AdS spacetime.
Cheers LC
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Cheers LC
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