Sunday, October 17, 2010
Tricks and Treats
I was honored with a good answer and a reply to my comment to Matti Pitkanen: (quite a trick of number theory, and yes, quite a treat.)
I have got too accustomed to p-adic length scale hypothesis. p-Adic mass calculations lead to p-adic length scale hypothesis. Particles are characterized by p-adic length scale L_p=sqrt(p)*R, p prime and R the radius of CP_2 of order 10^4 Planck lengths. p-Adic length scale is of order Compton length and characterizes the size of the space-time sheet at which particle topologically condenses.
Physically preferred primes correspond to p= about 2^k, k integer. Mersenne primes p= M_k = 2^k-1,
k also prime but not any prime, are especially preferred ones. For instance, electrion corresponds to M_127, the largest Mersenne which does not correspond to completely super-astrophysical p-adic length scale. Also Gaussian Mersennes (1+i)^k-1 are expected to be very preferred.
Electron, muon, and tau correspond to M_127, Gaussian M_113, and M_107. Hadronic space-time sheets to M_107 and weak gauge bosons to M_89. This inspires the hypothesis that all Mersennes are physically important and that one has a hierarchy of fractally scaled up copies of weak and hadronic physics. In living matter this hierarchy would show up itself in the richest manner since as many as 4 Gaussian Mersennes are in the length scale range 5 nm-2.5 micrometers.
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If find it a most interesting, and a convincing direction for theory. An interesting alternative to how mass enters our equations. I especially like how it is close to our ideas of number theory (and perhaps the algebra of complex numbers although I feel our physics too long dependent on this concept- a variation perhaps like Whitehead's ideas on the axioms of Euclid adjusted- overlapping congruences.
My last post links to what I think as a novel principle: *m that in information conveyed by light, an analogy to sound in cathedrals, the concavity may change the shape of spheres for within that space "light seems to go faster than in flatland." This then another questioning of what happens in shifting scales of a finite but boundless expanding universe to some fixed state of energy. Certainly, we can imagine that in some medium light would be impeded. In any case, the jazz behind the portal at the edge of the universe (face of God?) may as well be a flux of some sorts- but I hardly think it the foundational case.
There are posts on sciencechatforum asking about the idea of a Torus universe. For me one should ask first about the nature of tori. Certainly it should be obvious that a hypercube contains two such tori. That these can be flat and Euclidean. That the usual computation of volume in the hypersphere is that of the Torus formula. And so on. Very good questions that only begin to sort out what we mean by any of the geometry's of hyperspace. (can the p-adic idea relate to the linear diagonal of a series of quasic hypercubes?)
Now, in the idea of a flat (Riemann sphere) plane we clearly see in the quasic grid that higher dimensions may be represented, including full natural spaces of any dimension and genus). So, can we not imagine effects as if space were indeed two dimensional quasically, and Euclidean, even in the complex planes and whatever compacting of matrices that examines the surfaces and boundaries of such things between dimensions. What would be true of an analog, of at least the low dimensions, to the Spinozan like spheres may also be true to the Infinite tessellations or the twists when we break things down- classically or not- in other words beyond our vision there is a religious element possible in the interpretations and metaphysics if not the physics- most clearly expressed as having substance by the peoples of the book.
I vaguely intuit again that these Merscene or other primes, arithmetical thus intelligible, the nature of primes and so on, would do well to consider double factorials as they correspond to orthogonal quasic structures.
Linguistically, such a quasic grid has me considering the parallels and perpendiculars say of a symbol if not a notion- somewhere in the quasic field of large Clifford like dimension ie n in 2 to the n, where our linguistic universe begins by two words compared and their quasic coordinates.
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Sometimes, to add here instead of a recent post, the simple way we set up the algebra of a problem (one that say leads to a convergence to phi but as a result of the way we set up the equation and not necessarily the observation of that value) will actually have physical meaning where we still have less than adequate notations, notions, and pictures in the foundations of mathematics and logic. In this sense some spurious and creative results may prove true in our human centered intuitions even if hard to prove. But I offer the above illustration as a general question and not a profound breakthrough in detail. Still, a lot more is needed beyond our idols of the various authors and generalizations of spaces of any of the various concepts of dimension. At this point one could question, even without alternatives, if topology and number theory are enough to explain things.