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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  • Post #43

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 LCAttachment #1: 2_375.pdfAttachment #2: 2_9903060v1.pdf

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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  • Post #45

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

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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  • Post #47

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

2 years later

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  • Post #48

Have you checked PDG about current limits?

b'->b+gamma => mass > 96 GeV

Or are they using the wrong simulators for the b' decay?

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