M Theory Lesson 119
Carl Brannen has related the Koide mass formula to a field energy expression, from which he has derived the tribimaximal mixing matrix. Just another unpublishable crackpot coincidence. This matrix (more or less) also appears in the new and much discussed paper by Garrett Lisi on an E8 theory of everything. Of course, TOEs are only theories of what little we knew before, but you must look at the pretty hexagonal pictures in the paper. 
posted by Kea | 10:29 AM
19 Comments:
Kea,
thanks for writing about Garrett's work and for linking to Bee's blog about it.
I put up a comment on Bee's blog, and I would like to also put it here (you can delete it for redundancy if you want to, but I thought it might be interesting to your readers also). Here it is:
Bee, with respect to Garrett's paper about "... E8 ... over a four dimensional base manifold ...",
and
your remark "... Kaluza-Klein theory ... is an approach to unify GR with gauge theories. This works very nicely for the vector fields, but the difficulty is to get the fermions in ...",
and
"... the exceptional Lie-groups ... offers the possibility to arrange both, the fermions as well as the gauge fields, in the Lie algebra and root diagram of a single group ...":
Instead of just putting E8 and 4-dim spacetime together by hand,
what about
taking the 248 dimensions of E8
and
combining them with a 4+4 = 8 dimensional Kaluza-Klein Spacetime similar to that described by N. A. Batakis in Class. Quantum Grav. 3 (1986) L99-L105, which was successful with respect to the Standard Model gauge bosons and was not generally accepted because of difficulties of adding fermions.
Perhaps the boson-fermion structure of E8 could solve those difficulties of the Batakis model.
By combining the 248 E8 dimensions with the 8 Batakis Kaluza-Klein spacetime dimensions,
you could form the 256 dimensional Cl(1,7) Clifford algebra,
whose structure could be used as a guide for naturally constructing a Lagrangian that could reproduce the successes of the Standard Model and Gravity based (as Garrett does) on MacDowell-Mansouri.
Tony Smith
PS - Maybe I should have said that structurally the 248 dimensions of E8 break down into
128 Spin(16) half-spinor fermionic dimensions
and
120 dimensions of the Spin(16) adjoint bosonic dimensions,
and
the other 8 dimensions (Batakis Kaluza-Klein spacetime) of Cl(1,7) could correspond to the 8-dimensional root vector space of the Spin(16) Lie algebra, which is the same root vector space as that of E8.
In short,
256 Cl(1,7) generators = 248 E8 generators + 8 root vector space dimensions,
and the 8 root vector space dimensions correspond to 4+4 dimensional K-K spacetime.
I see at least the following problematic aspects in E_8 model (actually noncompact version of E_8).
1. Putting bosons and fermions in the same Lie algebra is problematic since Lie algebra structure is lost if coefficients become anti-commutative. If this is not made, one has difficulties with spin and statistics.
The only way out that I can see is to assume that all generators are bosonic and assume additional supersymmetry predicting superpartners of standard model bosons and fermions.
2. The assigment of fermion generations to octonionic triality is questioned also by Lisi and again is not in spirit with the interpretation of 8-D representations. As a rule all unifications neglect the fact that different generations correspond to hugely different mass scales. Somewhat surprisingly - and if I understood correctly - Lubos seems to be finally beginning to consider seriously the possibility that particle mass scale might be a new degree of freedom and that some kind of natural selection could have selected elementary particle mass scales (see the Wegener post). Forcing everything to multiplets of a bigger group is to me just this methological thinking were facts must bow to the power of method.
3. The unitary representations of noncompact version of E_8 are infinite-dimensional. The finite dimensional representations are non-unitary. The problem can be overcome by using fields possessing E_8 spin and living in some space-time in which E_8 acts: some coset space of E_8 perhaps. In the case of Lorentz group similar problem is encountered and one ends up to the extension of Lorentz group to Poincare group. Something similar should occur now since one wants Poincare invariance. But does there exist any space with group structure analogous to that defined by translations with E_8 action?
Thanks for the link to Carl Brannen's use of electromagnetic field equations to explain the Koide mass formula.
The Koide mass formula says take the three generations of masses such as leptons, square root the mass of each and then add these square roots together, square the result, then multiply that sum by 2/3, you obtain the sum of the three lepton generation masses!
As Carl suggests, because for an electromagnetic field the energy density (Joules per cubic metre) of the field is equal to half the product of the permittivity of the vacuum and the square of the electric field strength (volts/metre), you get
Energy density (energy E, unit volume X)
= E/X
= (mc^2)/X = (1/2)*{permittivity}*{electric field strength in volts/metre}^2
Hence,
m ~ e^2
i.e. mass is directly proportional to the square of the field strength.
If masses depend on field strengths (i.e., if some vacuum field bosons which interact with gravitons to produce mass, are associated with electric charges), then because the electron, muon and tauon all have similar electric charges (with slight differences due to vacuum polarization loops, the kind of thing that means that differemt leptons have slightly different intrinsic magnetic moments), you would expect to have to square root the masses:
m ~ e^2
so
m^{1/2} ~ {+/-}e
I think this is true because there is evidence I've seen that is consistent with it, and that evidence comes from several calculations and many "coincidences".
What is occurring here physically is very simple indeed. The core of the electron (or muon or tauon) has electric charge only and no mass; hence the core of the electron/muon/tauon is a common entity.
The differences between these three leptons stem from the surrounding vacuum field.
It is the surrounding vacuum field, some kind of Higgs field or a Dirac sea, which interacts with gravitons to produce the "space time curvature" of the vacuum.
It interacts also with the cores of leptons and quarks to give them their masses.
m^{1/2} ~ {+/-}e
can be seen to be a consequence of the idea that masses arise from electromagnetic field interactions with a vacuum field that interacts with the gravitons,
hence the two-step interaction for masses/gravity: charged core interacts with surrounding field in vacuum (Higgs-like or Dirac-sea like) which in turn interacts with gravitons.
The fact that the Koide mass formula says m1 + m2 + m3 = (2/3)*[(m1^1/2) + (m2^1/2) + (m3^1/2)]^2 looks to me like a WEIGHTED AVERAGING of the ratio of relative masses to relative field strengths (expressed in mass units).
I.e., it should be written as:
(m1 + m2 + m3)/[(m1^1/2) + (m2^1/2) + (m3^1/2)]^2
= 2/3.
In plain English: the sum of the masses divided by the square of the sum of the relative field strengths (when expressed in units of mass via E=mc^2) gives you a dimensionless number, 2/3.
The reciprocal of number also comes up in other "coincidences", e.g., muon mass is roughly {electron mass}/[(2/3)*{alpha)] = 205.5*{electron mass, 0.511 MeV} ~105 MeV.
I've tried to come up with a mechanism, although it is still sketchy in detail, for hadron masses as well as leptons, e.g. see the "about" link on the sidebar of my nige.wordpress.com blog, but it needs to be rewritten and there are lots of loose ends.
The Koide formula is impressive and I didn't think about the square roots coming in via the electromagnetic field equation, so I'm grateful for Carl to pointing this out. Hopefully it is the missing link ....
Matti said, about Garrett's E8 physics model:
"... Putting bosons and fermions in the same Lie algebra [ E8 ] is problematic ...".
The "problematic" problem is overcome because the exceptional Lie algebras F4, E6, E7, and E8 have some unusual (for Lie algebras) characteristics:
they are based on Octonions
and
they are combinations of adjoint representations and spinor (or half-spinor) representations of Spin(n) Lie algebras.
It is the second characteristic that lets you use both fermions and bosons.
As Garrett said (over on Bee's blog):
"... The cool thing about the exceptional [ Lie ] groups [ including E8 ] is that the femions come out, algebraically, with the correct spins -- so they do satisfy the spin statistics relation as Grassmann fields ...
a beautiful thing about the exceptional Lie algebras ... The spinors we need are built into the Lie algebra structure of the exceptional groups ...".
An easy way to see this is to note that 248-dimensional E8 is the sum of
120-dimensional Spin(16) the elements of which, being in the adjoint representation of Spin(16), transfrom like vectors (gauge bosons etc)
and
a 128-dimensional half-spinor of Spin(16) the elements of which, being spinors, transform like spinors (fermions).
Tony Smith
Matti asks:
"... does there exist any space with group structure analogous to that defined by translations with E_8 action? ...".
The easiest way to see such a thing is to look at the graded structure of the Lie algebra.
For example,
the Lie algebra Spin(2,8) has a 3-grading
G = g(-1) + G(0) + g(+1)
Spin(2,8) = R8 + Spin(1,7) + R8
so that Spin(1,7) acts like a gauge group on the 1-level space R8.
For E8, Thomas Larsson said on sci.physics.research
"... "... e_8 also seems to admit a 7-grading
g = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3
of the form
e_8 = 8 + 28* + 56 + (sl(8) + 1) + 56* + 28 + 8*
... g_-3 is identified with spacetime translations ... spacetime has 8 dimensions ...".
That 8-dimensional spacetime is the right dimenison for the Kaluza-Klein spacetime that Matti and I like to use: M4 + CP2.
Tony Smith
Thank you Tony, Matti and Nigel for interesting comments. As you know, I don't view Garrett's TOE as a final picture, by any means, but every piece of the puzzle makes things clearer. Also, I am not surprised by Lubos' apparent change of heart, because he is a smart guy.
Kea, you should have mentioned that your latest paper on the discrete Fourier transform applications to generations of particle masses is tied up with all this too.
I haven't found your copy on WordPress so I'm giving the link to the old copy I put on WordPress. Please edit out the link and point it to where you keep your latest copy, or reply and give a decent link.
And by the way, you should add a link to it at your Blog's home, maybe under "Other Links".
Carl, the paper is here.
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