The fine-structure constant might be the most popular target of physics numerologists. α numerology has a long history, such as Eddington's efforts and Feynman's remark. It's a recurring topic in this long thread which might be the high point of Internet-era physics numerology.
Today on vixra there is an article which speculates about how to obtain one of the numerological formulas for α, 4π3+π2+π. It's unusual for two reasons. First, the author (Amir Mulic) speaks the technical language of M-theory; he proposes to "interpret... this expression in terms of the volumes of lp-sized three-cycles on G2 holonomy manifolds". (lp would be the Planck length.)
Second, he mentions that the coupling has to "run", i.e. change its value with energy scale. This aspect of quantum field theory is why particle physics professionals tend to ignore even Koide's relation, to say nothing of the more baroque formulae invented by amateur numerologists. The modern paradigm is that simple relations among particle masses and coupling constants exist at ultra-high energies, but that at low energies these relations will be obscured by complicated corrections, e.g. extra terms containing a logarithm of the energy, described by "beta functions" which can be derived from fundamental theory.
I haven't really gone over Mulic's article (I note that he had a similar one on arxiv years ago), and I am apriori skeptical that this particular idea will work out. But what's noteworthy here is just that someone is making this sort of effort - trying to explain the numerological formulas using the full conceptual apparatus of modern mathematical physics.
Before I comment further, it might help to show how things look without such a bridge. On one side, we have the efforts of someone like Angel Garcés Doz, already mentioned several times on this blog. Garcés Doz works hard, and like Mulic, draws inspiration from 7-dimensional geometry. Still, I find his formulas more interesting than his physics.
On the other side, consider this item of F-theory phenomenology (via Lubos). Here we have a genuine example of how a string-theory background geometry might determine a particular value of α: in this case, it's "the number of fuzzy points" in "a non-commutative four-cycle" wrapped by a 7-brane. But the value of α thereby obtained is the high-energy value, the value at the grand unification scale - perhaps 1/24 or 1/25, says Lubos. It only approaches 1/137 at low energies because of those messy correction terms.
Incidentally, this "fuzzy F-theory phenomenology" played a role at the dawn of my own attempts to make sense of what Marni Sheppeard was doing. One day she exhibited a parametrization of the CKM matrix, in terms of circulant matrices, and I was interested in whether this could fit into an existing framework like F-theory. It was very interesting to see that number 24 appearing as one of her parameters, but at the time none of us knew enough to judge whether Brannen and Sheppeard's circulants, and Heckman and H. Verlinde's fuzzy points, could fit into the same theoretical synthesis.
One of the interesting ideas about deriving fundamental constants from first principles is to start with some sort of expected boundary condition (e.g. Planck or GUT energy) and then to use the beta function to back out the conventional value. Have you seen any scholarship for alpha along those lines?
ReplyDeleteI can't think of any. In grand unification, you use the beta function in the other direction, but the actual value of the unified force coupling has no explanation known to me, in field theory.
DeleteIn string theory, coupling constants can correspond to the volume of branes. For example, a stack of N branes gives you a U(N) gauge field - one brane per "color" (e.g. N=3 is QCD), and with gauge bosons being strings between the branes in the stack, e.g. red-antiblue gluon would be a string from "red brane" to "blue brane". And the strength of the force depends on the size of the branes - i.e. they wrap around some part of the compact extra dimensions, and the coupling is proportional to the volume of this part. This may be seen in "intersecting braneworld" models.
The F-theory paper above provides an exotic example of this, in which the volume is discrete, because the branes are wrapped on a noncommutative "space" consisting of a finite number of "points".
There seem to be several formulas, equivalent to the electromagnetic fine structure constant (zero momentum). This is due to the richness of the theory of unification; yet to discover a whole (although there are partial results), and mathematical flexibility, which is expected from a unified theory, in which all physical concepts, are actually different manifestations of the same mathematical entity, whose main basis it is pure geometry, and therefore, the theory of numbers, etc.
ReplyDeleteThe result of Amir Mulic is very interesting, since it is a sum of factors sphere surfaces in dimension 7, 4 and 1: ie 11 dimensions (7 + 4 = 11)
And, 7 x 4 x 1 = dimSO(8) = 28
(dimSU(4)/4) x S(7d) = [(16π^3)15] x dimSU(4)/4= 4π^3
2S(4d)= π^2
2S(1d)= π
dimSU(4)= Sum ( All of the projections of all spins ) = Ss( 2s+1)=15
(dimSU(4)/4) x S(6d) +2S(4d) + 2S(1d) = 4π3+π2+π
The correction factor for the final value of the fine structure constant to zero momentum, is due to the masses of tau, muon and electron.
In this brief commentary I can not expose all the entire result, with its theoretical development, but implies the existence of a hyperbolic geometry for empty states. This hyperbolic geometry implies infinite speeds, or tachyon zero energy: This means that the infinite speeds are indistinguishable or equivalent to zero speed, for the obvious fact that an observer may never measure whether a point is static (circular compactifications) or moves at infinite speed. This implies the existence, in the absence of energy (energy zero tachyon = pure space, annulment of the time), the existence of quantum worm holes, which are in themselves pure vacuum state without energy.
These wormholes, geometry of a hyperboloid of one sheet; open; are connecting the states of entangled particles.
This intertwining of wormholes rotating at infinite speed, due to the lack of energy or energy tachyon zero, is what allows the instantaneous transmission state change wave function of one of the entangled particles, to make a measurement on one of them. But there's no violation of the conservation of energy (zero energy the wormhole, pure space with zero time)
These quantum worm holes, which are manifested in the entanglements, nor violate causality, since no transfer of energy and they are not observable, or measurable. And time is stop, zero. Time is emergent. There are not "fricction" ( zero energy = pure space with not time )
("Masses" imaginary).
This vacuum hyperbolic geometry is fully consistent with of Sitter space, solution of Einstein's equations of GR.
These hyperbolic zero vacuum states of these quantum worm holes, are intimately related to the zeros of the Riemann zeta function. All these results I show in my next article, I am preparing.
These states of wormholes, which produce entanglement; depend on the maximum amount of entangled states for dimension seven, three and two. And are the (not factorable, not factorizable probability = entanglement) prime numbers: 2 ^ 7-1 = 127, 2^3 -1=7, 2 ^ 2 -1 = 3
137= 127 +7 +3 ; SU(3)==> 2^3 -1 ; SU(2)==> 2^2 -1
137 prime number ; dimE8( non zero roots, lattice ) -137= 240- 137 = 103
103 prime number. 103=[ 2 x In(mpk/me) ]
Finally I put the formula for the fine structure constant at zero momentum,
, And its value to the scale of electroweak unification.
The formula for alpha ( momentun zero ):
angel777's uploaded images - Imgur
L(alpha) = sqr( 137/4Pi )
alpha^-1(Mz)(ms) = 127.962 = 137.035999073 -In(2 x 240 )/ [sqr(2) x sin_Weff(Mz)(ms)]
sin_Weff(Mz)(ms) = In( Phi)=0.481211.. ; Phi = 1.618033988...
Best regards
====Relativity Extension===
ReplyDeletechange factor of special relativity is obtained directly, as the derivative or infinitesimal change of the arc sine of a circle, the circle of the contour of a hyperboloid.
d(arcsin(x) ) = 1/sqr(1-x^2) ; x= v/c
The higher the speed of light is the distance of the geodesic or infinitesimal outer surface of the hyperboloid of one sheet (open), quantum wormhole (open). That is, the derivative of the arc cosh:
d(arccosh )= 1/sqr(x^2-1) ; x = V/c
==============================================
As you have pointed out that the fine-structure constant should be a running constant with (1/α) = 4π3+π2+π, at low energy while it approaches to 128 at energy higher than 90 Gev. . Thus, the 4π3+π2+π formula meets only one point of that spectrum. Furthermore, that formula can only be interpreted in terms of geometry or topology, hinting an eleven (11) dimension universe. But, the following formula has the following points.
ReplyDelete1. It has a ‘physics’ parameter, the Weinberg angle.
2. As this Weinberg mixing angle is a function depending on the energy, this formula encompasses the entire (1/α) spectrum (from 4π3+π2+π to 128).
3. It is also a formula for an eleven dimension universe.
Beta = 1/alpha = 64 ( 1 + first order mixing + sum of the higher order mixing)
= 64 (1 + 1/Cos A(2) + .00065737 + …)
= 137.0359 …
A(2) is the Weinberg angle, A(2) = 28.743 degree
The sum of the higher order mixing = 2(1/48)[(1/64) + (1/2)(1/64)^2 + ...+(1/n)(1/64)^n +...]
= .00065737 + …
Amir Mulic’s formula (4π^3+π^2+π) and his interpretation “…but only in a setup which is clearly different from the standard model of particle physics (SM) … interprets the formula in terms of volumes of three-cycles on the compactification manifold” are indeed very interesting. But, I would like to take it one step further.
ReplyDeleteLet 2 π = the circumference of a unit disk (with radius = 1) = CUD
π = half CUD = HCUD
Then, his formula can be written as,
(1/α) = (1/2) {CUD * [(CUD + 1/CUD)^2 + (HCUD – 1/HCUD) – ((1/CUD) – 1)^2]}, and this is indeed a topological description of the Alpha but is going way beyond “…in terms of volumes of three-cycles on the compactification manifold”.
In my formula, the Alpha is the function of the Weinberg mixing angle which again was derived via a topological model (see Axiomatic physics, the revolutionary physics epistemology, at http://prebabel.blogspot.com/2012/05/axiomatic-physics-revolutionary-physics.html ). Furthermore, in the article “Law of Creation (http://www.prequark.org/Create.htm )”, it shows three points.
1. The topology of this universe is described as a “Garden hose model” (see http://www.prequark.org/tcone.JPG ), and it has 11 dimensions.
2. This 11-dimension is manifested as a ball-donut (torus) transformation (by punching two holes on a ball). Thus, the Euler number for ball is 2 while it is 0 for torus. Yet, ball has four-dimensions [needs 4-codes, such as, (A, G, T, C) or (Red, Yellow, Blue, White)] and torus has seven-dimensions, such as (A, G, T, C, male, female, kids)]. Thus life is the combination of ball and donut. The four codes (A, G, T, C) provide the “individuality” of life while the seven codes for the “immorality” of life, and (4 + 7 = 11).
3. In the G-string model (G-STRING AND DARK ENERGY,
http://tienzengong.wordpress.com/2013/08/24/g-string-and-dark-energy/ ), the generations are viewed as a new color charge (G1, G2, G3). Thus, the Standard Model particle zoo is described with 7-codes (Red, Yellow, Blue, While, G1, G2, G3), forming a topological donut. Together with the cosmology sphere (ball, with 4-dimensions), this universe is described with this 11 (4 + 7 = 11) dimensions.
Thus, (4π^3+π^2+π) is a topological (at zero momentum) description of my formula.
This is an issue going way beyond Alpha-numerology. From this, we can actually discuss the falsifiability of the multiverse.
ReplyDeleteFor consecutive cycling universes (with different boundary or initial conditions for each universe), this multiverse could will be a ‘fact’. For some simultaneously co-existing universes (with different physics laws or nature constants), this simultaneous-multiverse (S-multiverse) can of course be falsified although by definition that those other universes are unobservable from ‘this’ universe.
First, for the S-multiverse as a ‘fact’, it has produced a special universe which we are living in. For the S-multiverse-theory, it has failed to describe that ‘fact’, as being unable thus far to find out how ‘this’ universe came about from the S-multiverse fact. That is, the S-multiverse-theory is useless and nonsense.
Second, if we can positively show that the life of this universe does arise from the physics laws of ‘this’ universe, then the S-multiverse ‘fact’ is no longer relevant to ‘this’ universe. And, the S-multiverse-theory can comfortably be thrown into the trash can; see “The great divide (http://prebabel.blogspot.com/2013/12/the-great-divide_1900.html )”.
Third, if we can show that the nature constant (such as Alpha) of ‘this’ universe is not ‘bubble’ dependent, there is no reason for the nature constants of the other universes to be bubble-dependent.
The Alpha-formula (with the Weinberg angle) which I showed in the comment above is derived without any bubble factor in it, and it consists of only two steps.
One, ‘uncountable-infinity to finite’ concretization process, see "Multiverse bubbles are now all burst by the math of Nature”. That is, the uncountable-infinity is concretized as a circle (or a pie).
Two, this pie (the highest infinity) is ‘divided’ or shared by two group members (forming the structure of this universe), that is, 64 kids (of this ultimate infinity); one group (16 kids) takes the energy (dark energy), the other group (48 kids) takes the landmass as landlords. See “Pimple Model; BARKED UP THE WRONG TREES (M-THEORY AND SUSY);” and “DARK ENERGY, MYSTERY NO MORE!” Thus, the numbers (64, 48 and 24) play the key roles in my Alpha-formula. The whole point is about the ‘dividing’ or sharing the pie, and there is nothing about the bubble.
Although Mulic has M-string interpretation for his formula, it is still basically a numerological formula. Yet, when it is rewritten as the following equation, the physics significance is now all clear.
Let 2 π = the circumference of a unit disk (with radius = 1) = Pie
π = half Pie = HPie
Then, his formula can be rewritten as,
(1/α) = (1/2) {Pie * [(Pie + 1/Pie)^2 + (HPie – 1/HPie) – ((1/Pie) – 1)^2]} … equation A
The equation A above is a topological (static) description of the pie-sharing.
(Pie + 1/Pie), type 1 mixing (division); (Pie + 1/Pie)^2, the first order mixing (sharing)
(HPie – 1/HPie), type 2 mixing (division), the second order mixing (sharing)
[(1/Pie) – 1]^2, the ‘remainder’ (indivisible) of the division (sharing)
So, equation A = (1/2) Pie * (the first order sharing + the second order sharing - the ‘remainder’ of sharing)
That is, Mulic’s new Alpha formula is also bubble-independent. The fact that the Alpha is now written with two formulas (one static and one dynamic) while both of them are bubble-independent. Now, we can be very certain that S-multiverse is not a fact.
While by definition that S-multiverse is unobservable from this universe, it can still be falsified with the three points above. This could be the key contribution for Mulic’s new formula.
J.S. Markovich points out the cute fact that 1/alpha is rather close to 137 plus 1/27. Expressions of the form 2n/27 show up in Brannen-type Koide formulas, see elsewhere on this blog.
ReplyDeleteFor any given number, there are unlimited number of way to reach it with numerological formulas. For a nature constant, its formula must encompass a ‘physics’. As Alpha is the ‘lock’ of three nature constant, its ‘physics’ must be all encompassing, giving rise to ‘all’ physics. The following is a short list.
ReplyDelete1. gives rise to nature constants {Alpha, e (electric charge), c (light speed), ħ (Planck constant), etc.},
2. gives rise to the particle zoo of the Standard Model,
3. gives rise to quantum principle,
4. gives rise to unified force equation,
5. gives rise to dark mass and dark energy, Planck data (dark energy = 69.2; dark matter = 25.8; and visible matter = 4.82)
6. gives rise to life,
7. gives rise to arithmetic,
8. gives rise to baryongenesis,
9. gives rise to cosmological constant (Λ),
10. gives rise to … everything in ‘this’ universe.
That is, there is a ‘criterion’ for the Alpha formula. The failure of any one of the ‘gives rise to …’ by the Alpha physics, its validity will be in question. For this ‘ultimate’ physics, it should connect to all issues above. Thus, we should start from any issue above and walk to all other issues via the ‘bridges’ of this ultimate physics (the Alpha physics, in this case).
Just for the fun of it, I will start with the cosmological constant (Λ) by introducing a “Ghost-rascal conjecture”:
“For a coin flipping game (head vs tail), T is the number times flip as one ‘game’, N is the number times that that ‘game’ is played. If T >= 10 and N >= 10^500, then no amount of sabotage from a Ghost can change the outcome of this game.”
This Ghost-rascal game is very much ‘physical’ which can be played by a first grader. And, the probability of each game-pattern at any given number N (i) can be defined with a probability function Λ, such as,
Function Λ (i) = P (top, the largest probability of a game-pattern) – P (bottom, the smallest probability of a game-pattern) at any given N (i).
Yet, by the definition of the conjecture, the probability of each game-pattern should all be the same at a number N (z), that is Λ (z)= 0 (exactly) at N (z). For a number N (i) < N (z), Λ (i) > 0. When this N (i) is very large, this Λ (i) should be almost to be zero.
Thus, the bigger the T is, the smaller the N is needed. When T = 1, the conjecture could fail unless the ‘N’ goes to infinite. When T >= 3, the power of immutable becomes strong. Thus, I will show a game with T = 3, as below.
Game 1: (tail, head, head)
Game 2: (head, tail, head)
Game 3: - (tail, tail, head)
Can these three games make contact to this physical universe? When we make this game a bit spicier, we can truly surprise ourselves. Let’s add three more spices (hot, color, and twister) for this game.
One, hot-juice: the head carries 1/3 of electric charge, the tail with zero (0).
Two, color: the first flip (or spin) is red, yellow the second and the blue the third. Then, every game carries a color-tag which is the color of the ‘single’, such as the Game 1 is red; Game 2 is yellow; Game 3 is blue.
Three, twister: flipped by left hand (sabotaged by the Ghost) is marked with a negative sign. Flipped by the right hand carries a positive sign for the game.
With these additional spices, this game can actually describe all the Standard Model particles (excluding the bosons) symbolically. That is, this game has made one ‘gives rise to …’ of the above list.
Then, when we make the three game (1, 2, 3) as a compound game, it actually is the symbolical description of ‘proton’. Yet, for a color-blind person who is unable to see (the color), to feel (the hot juice) or to know (the twisting), he sees that the three games above are as below.
Game 1: (# * *)
Game 2: (* # *)
Game 3: (# # *)
This new compound game is, in fact, a glider of Conway’s Life game which can be the base for building a Turing computer. That is, there is a chance for this game to ‘give rise to life’.
More, later…
There are two points on this numerology issue.
DeleteA. For any given number, there are unlimited ways of reaching it with numerological formulas. Yet, a numerological formula is, in general, unable to reach a different number.
B. For the ‘true’ physics of ‘this’ universe, it can reach all ‘numbers’ of ‘this’ universe.
My Alpha formula has three numbers (64, 48, 24), and they are the consequences of the Alpha-physics (the Ghost-rascal conjecture). For this Ghost-rascal game,
i. My hand (the right hand, marked with a ‘positive’ sign) is totally random. The Ghost-hand (the left hand, marked with a ‘negative’ sign) is ‘planned and ordered’ sabotage. While the maximum outcomes of the Ghost-hand cannot go beyond the scope of my hand, it is not a symmetry of my-hand (total randomness) but is the source of the orderliness. This Ghost-hand can give rise to a baryongenesis type of universe (see http://prebabel.blogspot.com/2013/12/baryongenesis-master-key-of-all.html ).
ii. When T = 3 with three spices (hot, color, twister), the total outcome-patterns of the Ghost-rascal game are 48, which can give rise to the 48 ‘matter particles’ (not including the bosons) of the Standard model, and Neff = 48/16 = 3 (exactly).
iii. By adding one additional spice {the Perfect Symmetry (real/imaginary time symmetry, see http://www.prequark.org/Mphy.htm )}, the total outcome-patterns of this Ghost-rascal game become 64 (see http://prebabel.blogspot.com/2012/04/48-exact-number-for-number-of.html ). From here, the Cabibbo, Weinberg angles and Alpha are derived (see http://prebabel.blogspot.com/2011/10/theoretical-calculation-of-cabibbo-and.html ).
iv. The imaginary time gives rise to ‘dark energy’ while the ’48 matter outcome-patterns’ give rise to ‘dark mass’. These 48 outcome-patterns are having ‘equal rights’ in terms of the ‘land-mass’ in this outcome-universe. Even while there is only one-half of the one-generation {(1/2) x (48/16) = 8} is playing on the ‘stage’ (visible), all other patterns (not on stage, invisible) are still carrying the same land-mass right. Thus, the visible/invisible ratio should be {(48 - 8 + 1) x (100 – w)%/7} = 5.33 when w is chosen as ‘9’ according to the AMS 2 data. The ‘7’ = 8 – 1, as the one of the outcome-pattern (neutrino-like) of this visible group is not visible. This v/iv ratio is identical to the Planck data (= 5.3526). Furthermore, any revision of the Planck data can still be accommodated with this Ghost-rascal equation when the true value of ‘w’ is measured (see http://tienzengong.wordpress.com/2013/09/19/barked-up-the-wrong-trees-m-theory-and-susy/ ).
Now, this Alpha-physics (Ghost-rascal conjecture) has given rise to ‘all’ numbers of ‘this’ universe.
1. The cosmological constant (Λ),
2. Cabibbo angle,
3. Weinberg angles,
4. Alpha,
5. Neff = 3,
6. Planck data (for dark energy, see http://tienzengong.wordpress.com/2013/09/25/dark-energy-mystery-no-more/ ).
7. gives rise to the particle zoo of the Standard Model (see http://tienzengong.wordpress.com/2013/08/24/g-string-and-dark-energy/ ),
8. gives rise to Baryongenesis (see above),
9. gives rise to quantum principle (see http://tienzen.livejournal.com/973.html ),
10. gives rise to unified force equation (see http://prebabel.blogspot.com/2013/11/why-does-dark-energy-make-universe.html ),
11. gives rise to life (see http://www.prequark.org/Biolife.htm ),
12. gives rise to arithmetic (see http://www.prequark.org/Mlaw.htm ).
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