Wednesday, July 13, 2011

Inspiration strikes!

What is the situation? We have a functioning field theory, the standard model, in which there are over a dozen numbers that are just input parameters. We have a supposed derivation of a few of those parameters, in terms of quantities associated with the icosahedron. How are we to give these derivations causal significance in a more fundamental theory?

The basic idea is as follows. We have an icosahedron associated with each point in space. (The exact nature of the association doesn't matter at this stage.) Then, we have a field whose expectation value has a specific functional dependence on the properties of the icosahedron. Or rather, we have several such fields, each with its special functional relation. Then, we combine those fields so as to give rise to the standard model parameters. And we're done!

For example, consider the problematic relationship: dihedral angle equals sum of three physical parameters (let's overlook the peculiarity of mr nothing's "GUT angle" for now), plus another quantity. If you were "measuring" an icosahedron through, say, a TQFT, there's simply no reason why the TQFT would directly detect the existence of that decomposition.

So instead, we suppose there is a "dihedral angle field", with a VEV equal to the icosahedron's dihedral angle, and a "Weinberg angle field", which gets its VEV from, say, that DFQ angle (we may suppose that it's a slightly non-Euclidean icosahedron). And then we have a "Cabibbo angle field", whose VEV equals "d.a.f. VEV - W.a.f. VEV - 'GUT angle' - pi/3" - with this relationship being enforced, not by any property of the icosahedron, but simply by algebraic fiat. Voila, we now have the Cabibbo angle as an output from our "icosahedral theory of nothing", ready in turn to be an input to the set of relationships which reproduces the standard model.

Analysis of nothing, part 2

So, let's recapitulate. The dihedral angle of the icosahedron is supposed to be the sum of three angles of physical significance, plus 60 degrees. The exponential of the surface area of the icosahedron, divided by the mass of all leptons expressed in units of electron mass, allegedly equals ... another whimsical expression relying on the Weinberg angle. And, counting all the fermions and gauge bosons, we get 24, and this has something to do with SU(5) and the icosahedron.

Also, there are a bunch of alleged identities involving the Weinberg angle, which I have not yet bothered to check numerically, nor have I checked whether a simultaneous solution to these equations is even possible. Also, I suspect that the closeness of the "DFQ angle" mentioned in the previous post, and the experimentally measured value of the Weinberg angle, may subliminally be at work here. (Let me note in passing that if you embedded an icosahedron in a slightly non-Euclidean geometry, it should be possible to make the DFQ angle exactly equal to the measured Weinberg angle - at one's preferred energy scale, that is, since the value of the Weinberg angle flows.)

The attentive reader may recall that the premise of this blog was, that abstracts from the snarxiv could - if used in moderation - serve as genuine inspiration for physics. The current experiment is to see whether an authentic bit of numerological physics, found in the wild (the physics blogosphere), can similarly provide inspiration. So rather than proceed with the numerological analysis, for a moment I want to switch tracks and ask, in what sort of physical theory could the relationships listed at the start of this post actually exist and actually play a role in physical causation and explanation?

Since Weinberg angle, Cabibbo angle, Higgs VEV, etc., all acquire physical significance on account of the roles they play in a particular quantum field theory (the standard model), we are presumably looking for a beyond-standard-model theory which reduces to the standard model in some limit, and in which icosahedra matter. Perhaps there are compact dimensions shaped like icosahedra; perhaps there are icosahedral branes. Perhaps there are interaction vertices dual to icosahedra, or perhaps we calculate certain amplitudes by integrating over icosahedra.

See, this is the fun part: take the work of mr nothing, and try to hybridize it with the conventional apparatus of physical theory. But I must say that the "sum of angles" worries me. We are supposed to be using the icosahedron to explain the standard model, not vice versa. And while the dihedral angle is certainly a natural property of the icosahedron, the peculiar decomposition into a sum of four angles does not appear to be natural. Even supposing that an icosahedral structure appears at some level of our theory, why would the theory be sensitive to the existence of that decomposition of the dihedral angle? Unfortunately, the similarity of the DFQ angle and the Weinberg angle doesn't seem to help, because the DFQ angle isn't a natural part of the dihedral angle, so far as I can see. I shall need to meditate on the geometry of the icosahedron for a little while...

Analysis of nothing, part 1

What are we dealing with here? An attempt to derive various basic parameters of physical theory from quantities associated with the icosahedron. Which quantities? The dihedral angle (angle between two faces), and what I'll call the "inscribed radius" and the "circumscribed radius" (really, these are distances from the 3d center of the icosahedron to the middle of any face and to any vertex, respectively).

I'm a bit puzzled by "angle w". Although it's close to the "DFQ" angle in the link above, I can't see that it corresponds to any natural property of the icosahedron. Yet we have these two alleged identities

5 x ( 1 + sin^2(W) +sin(W) ) = cos ( 2pi/10 ) / [ cos( 2pi/5 ) ]^2

(3 + sqr(5) ) x [ sqr(3)/12] = cos(w)/[ (1 + sin(w) )^2 -1]

If these are both genuinely, exactly true for some w, then that's interesting... OK, duh, I bet "angle w" is supposed to be the Weinberg angle. In fact, mr nothing says so explicitly: "mW/mZ = cos(W)"

Because I'm lazy, I will not try, right away, to see if a w does exist that exactly satisfies the two equations above, or to otherwise guess what the geometric inspiration for those formulae might be. Instead, let's continue identifying the physical interpretation of the three icosahedral quantities proposed. For example, we are told that

The dihedral angle of the icosahedron = Weinberg angle + Cabibbo angle + "GUT angle" + pi/3

Despite its name, the "GUT angle" appears to be mr nothing's discovery, rather than, say, the value of the Weinberg angle in some GUT - it is defined as the dihedral angle divided by the square root of "one plus (half the Higgs VEV in units of electron mass)".

The other crucial element of the physical interpretation of the icosahedron turns out to involve its surface area A:

exp( A ) / (sum mass all leptons/ electron mass ) = [ ( sin(w) + cos(w) ) x sin(w) ]^-1

The "inscribed radius" and "circumscribed radius" don't seem to be playing much of a role...

Conceptually crucial, I believe, will also be the statement that

6 leptons + 6 quarks + 8 gluons + 1 fotón + 3 B ( w+, w- , z ) = 24 = 4! ===> SU(5)

followed by the introduction of the icosahedron. This is where mr nothing is trying to turn his identities into physics.

The numerology of mr nothing

Of course, physics can't just be elegant mathematical constructions without quantitative output. And so today we shall pause to consider the observations of a commenter at Lubos's blog, called "mr nothing". Presently I will make some remarks about how much sense can be extracted from them (or else I will tire of the exercise and delete this post). But for now, let's just hear from mr nothing himself.

Comment #1

The Mass Higss bosón: the mass is 119,61 Gev
There are five Higgs Bosons: 2 charged ( +1, -1 ) and 3 not charged
Fermi constant/ sqr(sqr(2)) = 246 Gev [246 Gev x cos ( 2pi/5)]/(1+cos(2pi/5)]= mH= 119,61 Gev
( 246 Gev)^2 = ( Sum mass all leptons )^2 + (Sum mass all quarks)^2 + (mW)^2 + (mZ)^2 + (mH )^2

Comment #2

5 x ( 1 + sin^2(W) +sin(W) ) = cos ( 2pi/10 ) / [ cos( 2pi/5 ) ]^2 ; angel w = 28,15648º ; mW/mZ = cos(W)
6 leptons + 6 quarks + 8 gluons + 1 fotón + 3 B ( w+, w- , z ) = 24 = 4! ===> SU(5)
Icosahedral symmetry : very important

If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is : sin(2pi/5); thus: 1-sin^2 (2pi/5) = cos (2pi/5)

Comment #3

and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is
(3 + sqr(5) ) x [ sqr(3)/12] = cos(w)/[ (1 + sin(w) )^2 -1]
The surface area A
of a regular icosahedron of edge length a are: 5 x sqr(3) ; exp( A ) / (sum mass all leptons/ electron mass ) = [ ( sin(w) + cos(w) ) x sin(w) ]^-1
diedral angle = 138,189685º = angle w + cabibo angle + angle GUT + angle ( 360/6 )
diedral angle / sqr[ In ( mass vacuum higgs/ ( 2 x me ) ) +1] = angle GUT
Mass Vacuum Higgs / 2 x electron mass = 481841,46525 / 2