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May 30, 2016

CATEGORY: Blog [back]
TOPIC: The Art of Combining Degrees-of-Freedom [refresh]

Blogger Daniel Doro Ferrante wrote on Nov. 11, 2009 @ 17:28 GMT
What do physicists do? I have been thinking a bit about what I do and what the true job of a physicist is and I think I can state it as finding the fundamental variables that most appropriately describe a physical problem. Usually, once these so-called “degrees-of-freedom” are found, the problem is solved.

Sounds easy enough, however, Nature can be tricky. The degrees of freedom that we call “fundamental” in one setting may not be the same in a slightly different setting. What may be important to describe the system under some conditions may not be so under different conditions; which poses the question: How do you transform one set of degrees of freedom into another? Telling these situations apart and learning how to “combine” one set into another is quite complicated. Essentially, we need to have a deeper understanding of our theories, well beyond the one used in daily applications and technology. That is, we need to sharpen our ability to go beyond approximations and stable situations, moving towards conclusions which acknowledge the system as a whole more completely. It is not enough, anymore, to only look at bits and pieces and try and infer the behavior of the whole system.

Which brings me to the topic of phase transitions. Phase transitions are well understood classically (the whole topic is called “thermodynamics”) but not so much so quantum mechanically. But something has been brewing for quite some time, and has now found its way into more glorious pastures: Quantum Phases--something that FQXi researcher Subir Sachdev has been looking into (see the article “The Black Hole and the Babel Fish” for more details) and which has caught the attention of the media recently because it uses string theory techniques to explain condensed matter phenomena. I have been working on a related area at Syracuse University. Of course, this is a project still in its speculatory infancy. But, there are some positive and encouraging results and I will outline these here.

In condensed matter physics, quantum phases are defined to be the different quantum states at zero temperature. A quantum phase transition (QPT) happens when a system undergoes a change from one quantum phase to another, at zero temperature. It describes an abrupt change in the ground-state of the system caused by its quantum fluctuations. A quantum critical point (QCP) is a point in the phase diagram of a system (at zero temperature) that separates two quantum phases. (The image above is taken from Piers Coleman and Andrew J. Schofield, Nature 433, 226-229, shows the phase diagram (a) and the growth of droplets of quantum critical matter near the quantum critical point (b).)

In addition to this, QCPs distort the fabric of the phase diagram creating a phase of “quantum critical matter” fanning out to finite temperatures from the QCP. In an analogy with a Black Hole, no information about the microscopic nature of the system affects the quantum critical matter, that is, the passage from a non-critical to a critical quantum phase requires crossing an “event horizon”. As expected, at the QCP, the system exhibits spacetime scale invariance, justifying the idea that it can be modeled via a Conformal Field Theory (CFT)--because of this, sometimes the QCP is referred only as a “conformal point” of the system.

At this point, we are almost ready to make a connection with high energy physics, namely with quantum field theory (QFT), but to set the stage, let us start with an example: an ultralocal spin 0 bosonic field, also known as a scalar field in 0-dimensions, which I’ll give the following potential:

$V(\phi) = \frac{\mu}{2}\, \phi^{2} + \frac{\lambda}{4}\, \phi^{4} \;.$

All of the desired objects in this little example are well defined: Path Integral (aka, Partition Function), Schwinger-Dyson equations, Fourier Transforms, etc.

The Partition Function is given by,
$\mathcal{Z}[\mu, \lambda ;\, J] = \int_{\Gamma} e^{i\, (\frac{\mu}{2}\, \phi^{2} + \frac{\lambda}{4}\, \phi^{4}) - i\, J\, \phi}\, \mathrm{d}\phi \; ;$

where Γ is the integration region (or integration contour, as I’ll explain shortly), while the Schwinger-Dyson equation is:

$(\lambda\, \partial^3_J + \mu\, \partial_J)\, \mathcal{Z} = J$

where we have used
$\phi \mapsto -i\, \partial_J .$

The only meaningful parameter in this problem is a combination of both coupling constants, namely
$g = \mu/\lambda$
and
$\lambda \neq 0.$

The Schwinger-Dyson equation and the Partition Function are just two expressions of the same problem. So, given that we have a 3rd order differential equation, we must have 3 different solutions. Which brings me back to the integration contour Γ above: we must have 3 distinct contours in order to find all possible solutions to our problem. Note that we can Fourier Transform the Schwinger-Dyson equation above and obtain a polynomial equation, with complex-valued solutions labelled by g.

$p^3 + g\, p = 0 \; .$

So, the bottom line is this little quantum toy model of ours has three different solutions, i.e., its moduli space has three points.

The three solutions of the differential equation above (Schwinger-Dyson) are called Parabolic Cylinder Functions,
$U(g,J),\, V(g,J),\, W(g,J),$
all dependent on g.

Let us make some straightforward observations. We needed three different boundary conditions (i.e., D-branes) to find all possible solutions to the Schwinger-Dyson equations, which is analogous to saying that we needed three different contours
$\Gamma_1,\, \Gamma_2,\, \Gamma_3$
in order to appropriately define the Partition Function. More explicitly, we have three different Partition Functions, all originating from the same toy model, with three different “quantum systems” as solutions, where each system is determined by the allowed values of g. Furthermore, each one of these allowed ranges of g corresponds to a point in the moduli space of the polynomial equation we found above.

We can now try and adopt a dynamical systems viewpoint. Essentially, in a graph of
$\partial\phi = \pi,$
we would get three attractors whose basins and flows are not trivial. (The image, right, is a visual aid showing attractors, taken from Wikipedia, but it’s not directly related to the construction mentioned here.) If we playing with the values of g inside each of the three distinct ranges, these flows and basins vary smoothly; however, when the values get closer and closer to the boundaries of the different ranges, we start to see the appearance of “folds,” “cusps,” and “catastrophes”. This is equivalent to studying the singularities of the polynomial we found above; which is also analogous to finding the singular points formed in the level sets of the Potential function. At the same time, the Schwinger-Dyson equations for these different values of the parameter g, undergo Stokes phenomena along lines of accumulation of Lee-Yang zeros, which creates regions of analyticity for the Partition Function.

At this point, we are ready for the punchline: Each inequivalent vacua of this toy model of ours, delimited by the equivalence class of the allowed values of g, corresponds to a different quantum phase. After all, each quantum phase is nothing but the ground state of the model. Indeed, each of the parabolic cylinder functions above has a very distinct behavior, defining vacuum states that have quite different properties. In fact, the asymptotic expansion of each of these functions yields very different behavior: one is amenable to perturbation theory (i.e., coupling constant expansion, known as the symmetric phase); the other one behaves like a soliton; and the last one is known as the broken-symmetric phase.

To be continued in my next post. Stay tuned.

this post has been edited by the author since its original submission

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Lawrence B. Crowell wrote on Nov. 12, 2009 @ 00:44 GMT
This looks interesting. I found your paper

http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.2205v1.pd
f

which is a pretty massive thesis. This appears to involve quantum criticality and quantum phase states of matter. I am particularly interested in this with respect to how the so called breakdown of quasi-particle states (or a renormalized mass of the electron) determines the cosmological constant. This is a Skyrmion field theory, which has an underlying fermionic structure. The M^4xS^7 supergravity has basis elements on S^7 for a G_2 holonomy define as

e^a = ψ^+(x)γ^aψ(x)

for γ^a Dirac matrix elements of Cl_{7,1}, and gauge connections are

A_μ = ψ^+(x)∂_μψ(x).

Cheers LC

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Lawrence B. Crowell wrote on Nov. 14, 2009 @ 02:54 GMT
People, come on --- this blog thread is one of the more interesting ones we have had in a while. This stuff is very deep and profound. It connects up with some theory of mine about quantum Hall effects invovled with M2-branes.

Cheers LC

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Georgina Parry wrote on Nov. 14, 2009 @ 05:35 GMT
Daniel Doro Ferrante wrote "What do physicists do? I have been thinking a bit about what I do and what the true job of a physicist is and I think I can state it as finding the fundamental variables that most appropriately describe a physical problem. Usually, once these so-called "degrees-of-freedom" are found, the problem is solved."

Good, then my work is done!

Lawrence,

this is too mathematical for me. Please enjoy and then perhaps you can explain what it means later.

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Lawrence B. Crowell wrote on Nov. 14, 2009 @ 12:58 GMT
Unfortunately the diagram at the top is almost too small to look at, but that contains almost as much of the physics as the equations. For a very low temperature system quantum fluctuations take on a role similar to temperature with thermal fluctuations. Thus the quantum state of a system can adjust it into a different phase, just as a temperature change can adjust a system from one phase to another (say solid to gas or ordinary conductor to superconductor). In this case with electrons the phase transition is between two phases of Fermi electron fluids through quantum critical states of matter.

The math is not that hard. The partition function is over a potential that is quartic, or with the field to the fourth power. This means there are three fixed points for the field, or in more elementary language where the potential function has zero slope. So a particle can sit there with no dynamics. However, these three configurations are like the bottom of a wine bottle, where one point is at the top of a peak. That configuration is not stable there, but is with the other two. This is involved with the three parabolic cylinder functions, which correspond to the three configurations of matter or of fields.

Cheers LC

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Ray Munroe wrote on Nov. 14, 2009 @ 19:24 GMT
Dear Lawrence,

I agree that this is interesting. I started reading the thesis the other day, but got distracted.

My older ideas included Quantum Statistical Grand Unification (in my book). My newer ideas involve "crystalline" lattices, and imply three point Fermion-Boson-Fermion Feynman diagrams (in my "A Case Study" paper). Quartic interactions must arise from boson-boson non-Abelian interactions. It would be cool to tie all of this stuff together. Do you have any ideas?

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 15, 2009 @ 01:59 GMT
If you look at my essay I sketch how this is involved with the cosmological constant. The G_2 holonomy with its basis vectors

e^a = ψ^+(x)γ^aψ(x)

exhibit this sort of physics. The holonomy system is a centralizer with the F_4 (or reduced D_4 and B_4) system which describes gravity. So the two elements of the exceptional Jacobi algebra "track each other." Hence the phase of the G_2, with this underlying fermionic phase is induced on the gravitational part. There is the matter of negative amplitudes, but I will share those later. The transition through quantum criticality changes a massive quasi-particle into a low mass field in another phase of the Landau fluid. In this case it renormalizes the value of the cosmological constant. The end of Ferrante's essay above, "In fact, the asymptotic expansion of each of these functions yields very different behavior: one is amenable to perturbation theory (i.e., coupling constant expansion, known as the symmetric phase); the other one behaves like a soliton; and the last one is known as the broken-symmetric phase," is a good indicator of some of this physics. Take a look at the thesis paper at

http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.2205v1.pdf

It might take me a few days though for me the break this stuff out. I think today I have been coming down with the flu --- it appears that H1N1 may have made a little visit, and I am feeling worse practically by the minute here. This has been a bit rough of late, for I usually go years between colds and stuff, but of late I have been getting these illnesses. I had a cold last September, and it looks as if I will be hammered this coming week as well.

Cheers LC

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Ray Munroe wrote on Nov. 15, 2009 @ 14:11 GMT
Dear Lawrence,

I have read your paper several times and don't know that I really understand more than 60% of it. I have downloaded the thesis you referred to, but it will take some time to read. For a long time, I was confused whether I was dealing with a G2 or an I_7(2), but the G2 reflects the 3-fold "color-like" symmetries of three E8's or a 3-legged fermion-boson-fermion Feynman diagram. Is there any chance that these ideas might also tie in some of my older ideas, such as Quantum Statistical Grand Unification? Our daughter caught the flu (probably the "normal" one, not swine flu or bird flu, but they evolve so quickly) on Halloween night, and she still has a little bit of a cough, but the rest of us have stayed healthy.

Take Care!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 15, 2009 @ 18:29 GMT
I read your HYPERCOLOR paper. The root space of G_2 I attach with your color and then hypercolor diagrams. The Klein X(7) modular curve is interesting and it exhibits some patterns of the octonions, in particular with the Moufang plane. The 7 fold structure is an element of the G_2 group, where the long and short 7-fold roots define the 14 roots for the group. The hypercolor system seems to be exactly this system. In what I am doing with G_2 it is with the supergravity multiplet. The G_2 ---> SU(3) + 3 + bar-3, which is one model for the QCD, which has some parallels with AdS_3 spacetime physics.

I actually don’t feel that badly today. I suppose this is not N1H1, fortunately, but some innocuous virus that came and went pretty quickly. More later

Cheers LC

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Lawrence B. Crowell wrote on Nov. 15, 2009 @ 18:31 GMT
My attachment did not work, so I attach the image file on the root space of g2 and your hypercolor scheme. Cheer LC

attachments: g2_color__roots.JPG

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Ray Munroe wrote on Nov. 16, 2009 @ 01:12 GMT
Dear Lawrence,

I am flattered that you used the hypercolor idea. I think that what has been bothering me about your ideas is Spin(7). Could it be an SO(7,1)xSO(7,1) that simplifies to a Spin(7) + remnants? Perhaps this explains why my model has 8 hyperspace dimensions versus the 7 hyperspace dimensions of M-theory. I understand that G2, Spin(7) and Klein's Chi(7) all have seven-fold symmetries, but so does SO(7,1)xSO(7,1).

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 16, 2009 @ 02:50 GMT
G_2 fixes a basis vector according to a quotient with spin(7). The G_2 fixes a vector basis in S^7 according to the triality condition on vectors V ε J^3(O) and spinors θ ε O, τ:Vxθ_1xθ_2 - - >R. The triality group is spin(8) and a subgroup spin(7) will fix a vector in V and a spinor in θ_1. To fix a vector in spin(7) the transitive action of spin(7) on the 7-sphere with spin(7)/G_2 = S^7 with dimensions

dim(G_2) = dim(spin(7)) - dim(S^7) = 21 - 7 = 14.

The G_2 group in a sense fixes a frame on the octonions, and has features similar to a gauge group. This fixes the basis set on S^7 with g_2 a subset of SO(8).

I have been a bit “sidetracked,” if you want to call it that. I realized something about the structure of Taub-NUT space and the role of time operator, which while it is a measure zero set does define a discrete structure with modularity on a time operator for a string. This has a bit to do with the unitarity of quantum gravity and black hole complementarity.

Cheers LC

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Ray Munroe wrote on Nov. 16, 2009 @ 13:11 GMT
Dear Lawrence,

OK - So Spin(7) is a subgroup of Spin(8), so that an SO(8)xSO(8) -> SO(7,1)xSO(7,1) -> SO(7)xSO(7) decay pattern is a reasonable way to include triality and septality symmetries. I think this is the difference between my 12 dimensional model (with a Spin(8) hyperspace) and an 11 dimensional M-theory (with a Spin(7) hyperspace).

I have been following your conversations with Jason about a time operator.

Have Fun!

Ray Munroe

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Daniel Doro Ferrante wrote on Nov. 16, 2009 @ 15:36 GMT
@Georgina Parry: I never claimed to have solved no one's problem. All i did say was that the work in Physics is to find the appropriate variables to attack a problem; if this "attack" will be hard or not, is a different question.

@Lawrence B. Crowell: I'm sorry it's taken me so long to get back to this discussion, but things have been pretty busy lately. Having said that, i must confess that i'm not quite following this dialogue between you and Ray Munroe, and here's the reason: right in your very first comment, you mention Skyrmions. Usually, these objects have the form outlined in eqs (1) and (2) of What exactly is a Skyrmion?, i.e., they're given by something analogous to
$[\phi^{\dagger}\, \partial_{\mu}\phi , \phi^{\dagger}\, \partial_{\nu}\phi]^2$
, where [equation]\phi[\equation] is valued in an appropriate Lie algebra. This is, quite clearly, a *topological* term. Further, by what i understand, you seem to be proposing that a condensation of Skyrmions could lead to a solution to the Cosmological Constant problem.

My question, then, is the following: how are you relating "Skyrmion symmetry breaking" with what i'm trying to explain? More to the point, what do you understand as "symmetry breaking" for Skyrmions? (Note that this is not necessarily straightforwardly related to superselection rules.)

Cheers.

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Lawrence B. Crowell wrote on Nov. 16, 2009 @ 17:03 GMT
The f_4 = SO(O)+O^3 or contains so(8)+V+θ+bar-θ, which is a triality over three octonions, the vector V and its supersymmetric partners θ+bar-θ. The f_4 constructs a so(8) system with so(O) > so(8), we might think of the so(8) as existing on the 3-sphere with SO(8)xS^3 as 3-way (triality) system.

There are some funny things going on with dimensions here. These are quantum correction codes on the 26 dimensional exceptional space. It is of interest to note that

12 + 22 + 32 + . . . + 242 = 702

and with the addition of 0 we can define an integral form of a null condition

+/-02 + 12 + 22 + 32 + . . . + 242 - 702 = 0

where the plus minus condition defines either Lorentz metric or AdS type of metric. The 1, 2, 3, . . , 24 represent roots of the Leech lattice Λ_{24}, and the additional two terms are from three scalars on an infinite momentum frame in the 27 dimensional Jordan exceptional algebra J^3(O). If one considers supersymmetry there are then restrictions on Virasoro anomalies, which brings the superspace to 11 dimensions.

Cheers LC

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Lawrence B. Crowell wrote on Nov. 16, 2009 @ 17:06 GMT
I made a graphic error. The sequence is supposed to be

1^2 + 2^2 + 3^2 + . . . + 24^2 = 70^2

with the second of these being

+/-0^2 + 1^2 + 2^2 + 3^2 + . . . + 24^2 - 70^2 = 0.

I wrote this in Word, which resulted in this goof up.

Cheers LC

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Daniel Doro Ferrante wrote on Nov. 16, 2009 @ 21:26 GMT
@Lawrence B. Crowell: I apologize if i'm being dense, but i really don't see how your answer addresses my question. :-(

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Lawrence B. Crowell wrote on Nov. 16, 2009 @ 23:53 GMT
Daniel,

Sorry for the confusion. My last post was for Ray. I had the window open for a while and then wrote to him. I just now saw your post. The term you quote there is the Skyrmion action. It applies for a certain Lie algebra, which in the case I am interested in is g_2, which along with f_4 is a centralizer of the Jordan exceptional matrix. The g_2 is a holonomy on S^7 and the F_4 contains the symmetries of M^4. The g_2 basis elements describe a three form on S^7 according to the Hopf fibration. This holonomy is in superspace tied to the spacetime symmetries within F_4.

The g_2 basis elements are what have fermionic or Skymrionic content, which is a soliton. That is sort of where things stand right now. The hypothesis is that the underlying fermionic content exhibits a quantum phase transition, or the "breakdown of a quasi-particle" with a large mass. Right now this is at the frontier of what I am setting up. The theory is cast in a Born-Infeld action. The symmetry breaking I am trying to work up involves the breaking of conformal invariance.

I have to make this a bit brief due to time. I can if you want break this out in greater detail later in the week, or maybe tomorrow. Your paper, of which I have only started to read into, offers up some possible machinery for working up this problem. The upshot is that the large quasi-particle mass is in this context what gives bare cosmological constant, and the physical cosmological constant is due to the transition to what is analogous to the Landau electron fluid.

Cheers LC

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Peter van Gaalen wrote on Nov. 17, 2009 @ 07:00 GMT
Hi Daniel,

I try to understand. A phase transition from one quantum state to another quantum state has to do with interference patterns of probability waves. During a phase transition there must arise some interference patterns that are completely different then the interference patterns before a certain temperature. Like the Einstein-bose condensate.

Every particle has a frequency. At my blog I just showed a derivation that the fine structure constant is the ratio of two different planck constants. (It is dimensionless isn't it? Feel free to proove me wrong)
$\alpha = \dfrac{h_e}{h_g}$
I think this illuminates the Aharonov-Bohm effect in which a charged particle derives an additional phase. So probably an electrical charged particle has two phases. This probably also is true for the other coupling constants. For example colour charge must have a phase of its own. So a particle with different charges has also as many different phases.

In GUTs there are running coupling constants. This means the value of h_e changes at higher temperatures. So the interference patterns will change. Ultimately at a certain temperature all different planck constants will have the same value.

I do not know if and if so, how the different phases interfere with each other, but I think it is very interesting.

Peter van Gaalen

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Ray Munroe wrote on Nov. 17, 2009 @ 13:32 GMT
Dear Lawrence,

Skyrmion makes me think of three quarks comprising a baryon. Are you trying to imply a preon structure?

There are a few "triality" symmetries. In Lisi's model and my model, the first and second dimensions are responsible for color triality. Also in my model, dimensions 11 and 12 are responsible for a generational triality (Lisi's triality).

With your model, there is a G2 2-brane that ties the three E8's together into a triality symmetry. Do we tack this G2 2-brane onto the end and call it dimensions 25 and 26, or is it the "glue" that holds everything together and belongs in the middle? In my model, I wonder if dimensions 13 and 14 are a "Grand" Supersymmetry triality that ties together particles of spin (0, 1/2, 1)? Because you are working with unbroken E8's, your "middle" might be the 9th and 18th dimensions. I have also proposed that this triality may be a 3-legged fermion-boson-fermion Feynman vertex. When you say Skyrmion, you make me think of preons. Which direction do you think this new triality is taking us?

Dear Daniel,

I'm still reading your thesis. I might want to write down a partition function for my K12' and see if that introduces any new physics that I may have overlooked while focusing on symmetries.

In my book, I unified the four fundamental force strengths with Quantum Statistical Grand Unification. It showed that gluons are a Bose-Einstein condensate of the original Grand Unified Mediating (GUM) Boson.

Dear Peter,

It is an interesting idea. I need to think on it. My model has many branes. What if each brane has its own Planck constant, speed of light, etc.?

Have Fun!

Ray Munroe

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Steve Dufourny wrote on Nov. 17, 2009 @ 14:17 GMT
Hi all ,

Dear Peter Van Gaalen,

You say

"Ultimately at a certain temperature all different planck constants will have the same value. "

It is logic in correlation with a kind of Bose Einstein condensate .All is linked with the thermodynamic and of course inter dependant in its spherical laws and rotations .The absolute zero in this line of reasoning is relevant in the rotations of spheres ,it is a personal choice of course .

But if the rotating spheres are linked with the mass thus the energy ,thus the thermodynamic takes all its sense with this simple reality ,physical and pragmatic .The pression ,the volume ,the velocities of rotations ,orbitals and spinals ,the temperature ,the density ,the mass,the chemical link ,biological micro meso macro ....all that is in a specific dynamic due to a specific equation of evolution where the codes of becoming are a reality in its complexification .The increase of mass thus must be inserted with pragmatism too .There the variables of thermodynamic are very relevant if we correlate all that with the topology and "he rotating cosmological and quantum spheres".

It is the same with the synchronization of the rotations ,the gravity and the electromagnetism synchronizes and permits the evolution with the main central code of information in the main central sphere of all gravitational systems and its spherical gravitational waves .The space ,this dark matter has not mass thus hasn't rotation ......the evolution takes all its sense thus fortunaly .

Just a thought

Sincerely

Steve

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Ray Munroe wrote on Nov. 17, 2009 @ 15:20 GMT
Dear Peter,

What if each brane has its own thermodynamics? My Quantum Statistical Grand Unified Theory would expect different coupling strengths at different temperarures, just like the running couplings.

Dear Steve,

Are you and Peter practically neighbors? A couple of hundred kilometers isn't that far.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 17, 2009 @ 18:51 GMT
Skyrmions were advanced as a theory of baryons. The baryon has a topological index which is given by baryon number. The original theory was an SU(3) parton model, and protons (or structures of protons) are soliton solutions which physically are condensates or coherent states (similar to superconducting states or overcomplete states of photons in lasers) of fermions. This has relevance for physics with RHIC quark-gluon plasma states.

It is not too much of a stretch to extend this theory from SU(3) to G_2, for after all SU(3) is the maximal subgroup of G_2, with additional roots 3 and bar-3. This automorphism and centralizer then acts in conjunction with F_4. The F_4 contains the information for gravitation, and if we think of these as axes in the J^3(O) or E_8 then a rotation by G_2 (a local gauge transformation) is accompanied by a "rotation" on the F_4. In the general 26-dimensional Lorentzian spacetime of the bosonic string (also the automorphism of the Fischer-Greiss group), or with fermionic reduction on Virasoro anomalies to 11 dimensions, the condensates on the g_2 then correspond to local regions which might identify as a cosmology. So different regions might then correspond to local condensations of the G_2 basis elements with fermionic content in the superspace.

What Daniel's paper does is to potentially provide a formalism for doing this. The three "basins" for solutions seem to provide the machinery to do this. The solitonic phase, perturbative (renormalization group flow) phase and the broken symmetry phase might well provide a way in which this can be understood. As yet I have to finish digging through the analysis here, and I am not sure how degrees of freedom or selection rules will operate.

Cheers LC

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Georgina Parry wrote on Nov. 17, 2009 @ 19:30 GMT
Daniel Doro Ferrante wrote "@Georgina Parry: I never claimed to have solved no one's problem. All i did say was that the work in Physics is to find the appropriate variables to attack a problem; if this "attack" will be hard or not, is a different question."

I think you misunderstood me. I was just saying, humorously, having read your post and having nothing worthwhile to contribute to this thread, that if as you say " ...what the true job of a physicist is and I think I can state it as finding the fundamental variables that most appropriately describe a physical problem." Then I think I have identified those variables in my own particular considerations. I will of course continue to examine the problems and any arguments against to determine whether this is in fact the best solution or whether it is possible to construct a superior explanatory model. Excuse me, just wanted to clear that up.

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Ray Munroe wrote on Nov. 17, 2009 @ 19:47 GMT
Dear Lawrence,

Are you saying that we may have three E8's with different phases: symmetric, broken-symmetric, and solitonic that are centralized by a G2 2-brane? Does this mean I need to reduce my symmetric theory down to a 10-dimensional E8xG2, so that we have room for broken-symmetric and solitonic solutions? Or does one of these wierd phases (maybe the broken-symmetric?) decompose into a couple of H4's and give my K12'? Does the solitonic phase explain why El Naschie's fractal ideas seem to work?

I only saw the attractors with basins and flows in a couple of paragraphs above - not in the thesis.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 17, 2009 @ 21:04 GMT
In the Jordon exceptional algebra we have the vector V, which is an octonion O ~ E_8, plus the θ_1, θ_2, which correspond to the supersymmetric pairs of V. So the Jordan matrix has three copies of the octonions. A theory with just V is the Jordan J^2(O) ~ V\oplus R, and supersymmetry extends this to the 27 dimensional J^3(O). A light cone frame (infinite momentum frame) places a constraint on the diagonal scalar terms and reduces this to 26 dimensions. Then since supersymmetric fields are written generally as

A - -> A + bar-ξ*ψ + ξ*bar-ψ + ξbar-ξF,

Then by working through the Virasoro center you find the super space is 11 dimensional. The 26 dimensional structure contains the Leech lattice Λ_{24}, which is the “quantum error correction code.”

The important dynamics are with the automorphisms or centralizers. This particular theory is just for the supergravity multiplet. How this results in the decompositions you outline I am not entirely sure of.

Cheers LC

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Daniel Doro Ferrante wrote on Nov. 17, 2009 @ 21:26 GMT
@Peter van Gaalen: *Quantum* Phase Transitions have *nothing* to do with "interference patterns of probability waves". In fact, they have little -- if nothing! -- in common with their more "classical" counterparts.

Further, the Aharanov-Bohm effect is just a baby version of something called _flux compactification_. In this particular case, though, even though there is no compactification per se, you still have a singular point (thus my calling it a "baby version").

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Daniel Doro Ferrante wrote on Nov. 17, 2009 @ 21:33 GMT
@Ray Munroe: The thesis was written for a certain intended audience of specialists; therefore, some results are simply not needed to be spelled out, for they are completely implied by whatever is actually written down. In the case of this thesis, i did take a more "succinct" approach (in the sense above).

For instance, another result which is not written down explicitly is the relation of this work with analytic continuation of the Path Integral. Or the relation between gauge theory monopoles for 1st and 2nd order actions. And so on and so forth.

On the other hand, the text here at FQXi is more "detailed", in the sense that i tried to "spell out" much of what is "missing" in the thesis, in order to bridge certain lines of reasoning.

So, no need to be surprised, this is exactly what was intended and expected. ;-)

Cheers.

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Daniel Doro Ferrante wrote on Nov. 17, 2009 @ 21:40 GMT
@Lawrence B. Crowell: You say,

"The g_2 basis elements are what have fermionic or Skymrionic content, which is a soliton. That is sort of where things stand right now. The hypothesis is that the underlying fermionic content exhibits a quantum phase transition, or the "breakdown of a quasi-particle" with a large mass. Right now this is at the frontier of what I am setting up. The theory is cast in a Born-Infeld action. The symmetry breaking I am trying to work up involves the breaking of conformal invariance."

I'm curious to see this BI-Action and this fermionic-soliton; personally, i've just never seen a Skyrmion built out of fermionic fields.

As for the SUSY stuff (you wrote in a couple of comments above), i must confess that i just could not understand much: would you mind using some of the TeX enabled here at FQXi? :-)

Cheers.

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Ray Munroe wrote on Nov. 17, 2009 @ 22:08 GMT
Dear Daniel,

Can you supply more information about the broken-symmetric phase? A key difference between Lawrence's ideas and mine involve an 8-dimensional E8 octonion splitting into two 4-dimensional H4's. Can such a scenario occur in the broken-symmetric phase?

Thank You!

Ray Munroe

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Daniel Doro Ferrante wrote on Nov. 17, 2009 @ 22:27 GMT
@Ray Munroe: If i could see the Action that both of you are using, than maybe i could get a better understanding of what you two have been saying so far.

But, before we even go down this lane, does E8 break into H4xH4? If so, i would very much like to see the proof.

Cheers.

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Ray Munroe wrote on Nov. 17, 2009 @ 22:56 GMT
Dear Daniel,

Lawrence has a cubic action. He would probably send you his next paper if you ask.

Technically, my K12' is not an E8xH4, but they are related. The only way I can see my ideas overlapping with Lawrence's is if an E8 decomposes into two H4's (or something very similar). If we study the 240 roots of the 8 dimensional E8, we find a symmetry of 240=8x(2x3x5). If we study the symmetries of the 4-dimensional H4 120-cell, we find a symmetry of 120=4x(2x3x5). I don't consider this a proof that E8 decomposes into two H4's because there is confusion with the dual H4 600-cell. This confusion has delayed a Supersymmetric version of my K12', and probably does tie into Lawrence's G2 in an unexpected way.

I like Lawrence's work and would be pleased if our ideas merge, but also understand that they may diverge.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 17, 2009 @ 23:39 GMT
And to make matters worse, I pasted this into the wrong area. I try again.

The basis elements of the G_2 are Skyrmionic and composed of fermions as (I will try the TeX here, but my experience with the editor here has been none so good)

$$e^a~=~{\bar\psi}\gamma^a\psi$$

and for the $A_\mu~=~U\partial_\mu U^{-1}$ the gauge terms are

$$A_\mu~=~{\bar\psi}\partial_\mu\psi.$$

The Skyrmion model was meant to describe mesons of a form like this. These elements are defined on a 7-sphere according to a tangent 2-plane on $C^5$, and these elements satisfy AdS/CFT for and $S^5~\sim~\partial AdS_5$. These elements can be compared to mesons in a way.

I deriving the BI from the exceptional algebra. I will try to post something on that before long.

Cheers LC

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Lawrence B. Crowell wrote on Nov. 17, 2009 @ 23:49 GMT
The basis elements of the G_2 are Skyrmionic and composed of fermions as (I will try the TeX here, but my experience with the editor here has been none so good)

$e^a~=~{\bar\psi}\gamma^a\psi$

and for the
$A_\mu~=~U\partial_\mu U^{-1}$
the gauge terms are

$A_\mu~=~{\bar\psi}\partial_\mu\psi.$

The Skyrmion model was meant to describe mesons of a form like this. These elements are defined on a 7-sphere according to a tangent 2-plane on $C^5$, and these elements satisfy AdS/CFT for and
$S^5~\sim~\partial AdS_5$
. These elements can be compared to mesons in a way.

I am deriving the BI from the exceptional algebra. I will try to post something on that before long.

Cheers LC

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Lawrence B. Crowell wrote on Nov. 18, 2009 @ 00:57 GMT
Ray,

The exceptional algebra has a cubic action, which can be transformed to a BI action --- at least in a limited sense. I am trying to work it in a general setting.

I have to confess I have been this last week sidetracked into a problem involving discrete group actions, time operators with string in Taub-NUT spacetime. There is the Pauli theorem on how a time operator can't in general exist, at least for a discrete and bounded Hamiltonian. Yet in the discrete group structure of Taub-NUT the 1 to infinity map involved with Pauli's theorem is circumvented. There is then a time operator [T, H] = iħ, which exist in a measure epsilon setting. The purpose is to indicate how black hole complementarity has unitary content.

Daniel,

The decomposition of H_8, or the Gosett polytope of roots, into H_4xH_4 is based on the Weyl group over E_8 W(E_8) ~ diag[H_4,H_4] + permutations. H_4 are the 120 and 600 cells in four dimensions, which are Poincare duals in the Gosset poloytope. Underlying all of this is that the universe is a quantum error correction code, which preserves quantum information --- even through the maw of black holes and singularities.

Cheers LC

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Steve Dufourny wrote on Nov. 18, 2009 @ 10:41 GMT
Dear Ray ,

I don't know ,Is he from Holland(Hello Peter)Are you from Holland? ,if yes ,yes indeed it is near Belgium ,some kilometers in fact ,very near .

Best Regards

Steve

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Daniel Doro Ferrante wrote on Nov. 18, 2009 @ 15:46 GMT
@Lawrence B. Crowell: There are a few things that i do not understand from what i've read so far.

(1) "The basis elements of the G_2 are Skyrmionic and composed of fermions (...)": So far as i understand it, Skyrmions are given by the square of the commutator of the momenta. Thus, they are a *scalar* (besides, as if this wasn't enough, it's easy to note that only *scalars* are allowed in the Action, so the Skyrmion *must* be one). Therefore, unless we are not defining Skyrmions differently, there's definitely something funny going on here.

(2) Gauge connection: I am not convinced that what you define as a connection, namely, (noted that i changed the notation to use differential forms)

$A = \bar{\psi}\, \mbox{d}\psi \; ,$

is a /bona fide/ connection. Usually, fermions (sections in the associated Clifford bundle), bosons (sections in the associated Exterior bundle) and gauge bosons (connections, differential forms -- hence my changing the notation above) are independent variables (degrees-of-freedom) in a given model (i.e., Action). What you seem to be implying, with your notation, is that "fermions" and "gauge bosons" are somehow "connected" (as per your definition). In order to do so, at the very least, you need to guarantee the mathematical rigor of your definition, and, as it stands as of now, this does not seem to be correct (as far as i can tell).

(3) "Underlying all of this is that the universe is a quantum error correction code, which preserves quantum information (...)": this is a *hypothesis* and has *never* been *proven* in any way, shape or form. So, we need to be careful with wild speculations like this, otherwise it's pretty easy to fall into an abyss of crackpotism.

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Ray Munroe wrote on Nov. 18, 2009 @ 16:15 GMT
Dear Daniel,

I agree that fermions and bosons are separate degrees of freedom. In my model, they are reciprocal lattices of each other. I probably made this connection most clear in my "Hypercolor" paper that Lawrence and I have discussed on this site. I suspect that Lawrence is working out the supersymmetric aspects of his model. Things get more complicated when we have to add in spin-0 sfermions, spin-1/2 bosinos, spin 3/2 gravitinos and such. He certainly has a legitimate G2 triality symmetry. Skyrmion might be one triality analogy, but I think this is more complex and related to Gravity and/or Supersymmetry with spin-2 gravitons and spin-3/2 gravitinos.

With his Born-Infeld-like action, he may be able to unify Gravity and Electromagnetism in this framework.

Some hypotheses are difficult to prove. Unfortunately, we probably have a generation of theoretical ideas that experimentalists cannot yet confirm or deny. Does that mean that we stop counting degrees of freedom and stop trying to extrapolate towards the next great theory? I won't. It's a dirty job, but someone has to do it. I don't care if they think I'm a "maverick".

I apologise if we have "taken over" your blog site. Lawrence saw some interesting similarities between your ideas and his. I appreciate that you have been so willing to interact with other bloggers.

Have Fun!

Ray Munroe

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Daniel Doro Ferrante wrote on Nov. 18, 2009 @ 16:24 GMT
@Ray Munroe:

I'm just trying to understand what i'm reading here, in this thread -- i have not followed other threads, so i cannot vouch for information given elsewhere. If there's any SUSY anywhere in this discussion, i would like know; besides, in order to define a "well behaved", /bona fide/, super-connection is something extremely non-trivial (and, quite frequently, the source of many problems).

As for the provability of hypothesis, note that there's a lot of "nuance" here. One thing is to say "Fermat's Theorem" and proceed to prove it. Another thing completely different is to make a general statement about the Universe and not even offer a framework (as is the case that i pointed out). Moreover, the subject of "quantum computing" is very well defined and has a framework of itself; if we are trying to import ideas from there and draw analogies for Quantum Gravity, we need to be extremely careful. Personally, before making such a grandiose statement, i would like to see all the proper ingredients from this quantum computing analogy properly mapped into quantum gravity ones. This is the very least a "proposed idea or theory" has to do in order to have any chances of ever taking off. Without this, we're not being 'mavericks', we're just fooling ourselves; and the line that divides both is not so thin as we'd like to imagine.

But, no apologies necessary: this is part of the game: a post comes out and comments ensue. This is how it's supposed to be. :-)

Cheers.

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Anonymous wrote on Nov. 18, 2009 @ 21:29 GMT
Indeed the idea that the universe is governed by a quantum error correction code is a hypothesis. Though I think it may have some basis to it. The black hole complementarity of Sussikind indicates that quantum information is preserved in black holes. This does not as yet provide the mechanism for this. So if quantum information is preserved it might well be that the fundamental basis for...

view entire post

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Lawrence B. Crowell wrote on Nov. 18, 2009 @ 21:30 GMT
This post by "anonymous" is by me LC

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Peter van Gaalen wrote on Nov. 19, 2009 @ 07:21 GMT
@Daniel: "*Quantum* Phase Transitions have *nothing* to do with "interference patterns of probability waves". In fact, they have little -- if nothing! -- in common with their more "classical" counterparts."

Isn't that a bit exaggerated? Or how should I see that? Has it to do with decoherence? According to Brain Green: "Decoherence is a widespread phenomenon that forms a bridge between the quantum physics of the small and the classical of the not-so-small by suppressing quantum interference - that is by diminishing sharply the core difference between quantum and classical probabilities." and: "Although photons and air molecules are too small to have any significant effect on the motion of a big object like this book or a cat, they are able to do something else. They continually 'nudge' the big object's wavefunction, or, in physics-speak, they disturb its coherence." Daniel is that what you mean?

@Daniel: "Further, the Aharanov-Bohm effect is just a baby version of something called _flux compactification_. In this particular case, though, even though there is no compactification per se, you still have a singular point (thus my calling it a "baby version")."

Interesting, I never heard of flux-compactification. "A flux compactification is a particular way to deal with additional dimensions required by string theory. It assumes that the shape of the internal manifold is a Calabi-Yau manifold or generalized Calabi-Yau manifold which is equipped with non-zero values of fluxes, i.e. differential forms that generalize the concept of an electromagnetic field. (wiki)"

Maybe string theories explain the Aharonov-Bohm effect as _flux compactification_ but I wonder if Aharonov and Bohm were also aware of that perspective.

String theory uses a lot of dimensions. Mostly spatial dimensions. I think that there are only three spatial dimensions. I don't see how a dimension like 'electromagnetic flux' can arise from spatial and temporal dimensions alone. But there must be some connection.

"It uses string theory techniques to explain condensed matter phenomena." There I lost you.

Peter

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Lawrence B. Crowell wrote on Nov. 19, 2009 @ 13:15 GMT
An example of such flux compactification is seen with Dirac magnetic monopoles, which I referenced earlier here. These Dirac tails are analogues of F-theory or type IIB strings in M-theory. I posted this here a couple of weeks ago, but it didn't seem to inspire much interest,

Quantum phase here does not involve a complex number for a wave, but rather a thermodynamic state of matter. For a quantum phase of matter the fluctuations at zero temperature, making the term thermodynamics a bit "odd," are quantum mechanical. Quantum fluctuations then are a measure of the degree or order or disorder in a system at T = 0.

Cheers LC

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Ray Munroe wrote on Nov. 19, 2009 @ 13:57 GMT
Dear Peter,

You said "It uses string theory techniques to explain condensed matter phenomena." Lawrence and I are working on the problem from the other angle, using condensed matter ideas to reinforce String Theory. Have you had an opportunity to read our essays? Those essays are both quite mathematical, and appear to be different on the exterior, but have many underlying similarities.

There are only three visible spatial dimensions, but what if Hyperspace never inflated (like Spacetime did) and is thus still very small?

Do you still live in the Netherlands? Steve Dufourny is in Belgium. His background is in Botony.

Dear Lawrence,

I read your "Jordan Exceptional Spacetime/ Black Hole/ SUSY" paper again yesterday. I'm slowly understanding it all. It is very deep from conceptual and mathematical perspectives. To "prove" these ideas via rigorous mathematical detail may consume a lifetime. It seems that Daniel either wants more details, or more proofs.

Have Fun!

Ray Munroe

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Daniel Doro Ferrante wrote on Nov. 19, 2009 @ 15:55 GMT
@Peter van Gaalen:

» Quantum Phases: What i have proposed in the 3 parts of this post is that quantum phases are different solutions to the quantum equations of motion. All of the other comments you made are related to classical phases, and not the ones being treated here.

» Flux Compactification: I did say that Aharanov-Bohm was just a "baby version" of it, right? And i also mentioned the fact that, in this case, the only relevant thing is the singularity. So, this is all you need to connect one with another. And, while it's true that neither Aharanov nor Bohm did not know about flux compactification, this does not stop us from seeing the phenomena under this new light, does it?!

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Daniel Doro Ferrante wrote on Nov. 19, 2009 @ 16:09 GMT
@Lawrence B. Crowell: Seth Lloyd's ideas are nothing but wild speculations at this point. Further, my definition of "rigor" and "speculation", etc, is borrowed from Penrose's in The Road to Reality.

As for the YM Lagrangian that you write down, maybe we have different notations and notions of it, but here's what i understand as such,

$F = dA + A\wedge A \Longrightarrow S_{\mbox{YM}}[A] = \int_{\Sigma} \mbox{tr}(F\wedge *F) \; ;$

where the connection 1-form,
$A$
, is valued in some Lie algebra.

What you wrote above, to my eyes, seem significantly different from this; so, i'm having a hard time understanding what you meant.

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Lawrence B. Crowell wrote on Nov. 19, 2009 @ 17:34 GMT
Consider the manifold M^4xM^7 for supergravity, where M^4 may be any spacetime (M, g). The simplest spacetime is a Minkowski spacetime. The structure defines a four dimensional field theory which for N-Killing spinors is N supersymmetric for a G_2 holonomy on M^7. The exceptional group G_2 is the automorphism group of the octonions o = x_0I + x_ae_a for the basis elements e_a obeying the algebra

$e_ae_b~=~-\delta_{ab}~+~\omega_{abc} e^c$

where the tensor Ω_{abc} is determined by products of three octonionic elements which are G_2 invariant. This is the tensor component of a three-form Ω which is expanded according to elements on the M^7 as

$\Omega~=~{1\over 3!}\omega^{abc}e_a\wedge e_b\wedge e_c$

$=~e_1\wedge e_2\wedge e_3~+~e_4\wedge e_3\wedge e_5~+~e_5\wedge e_1\wedge e_6~+~e_6\wedge e_2\wedge e_4~+~e_4\wedge e_7\wedge e_1~+~e_5\wedge e_7\wedge e_2~+~e_6\wedge e_7\wedge e_3$

$=~e_1\wedge e_2\wedge e_3~+~(1/2)e_i\wedge e_m\wedge J_{i mn}e_n$

so the spin tensor J_{i mn} has the element I = 1, 2, 3 and m, n = 4, 5, 6, 7. This is an aspect of the alternativity of the octonions which define triplets of quaternions as seen in the index i.. The product of the octonionic elements means the product of spin tensors obeys

$J_i\cdot J_j~=~-\delta_{ij}~+~\epsilon_{ijk}J_k.$

By definition the G_2 holoomy means the three form is closed dΩ = 0, and the Hopf fibration S^3 -- > S^7 -- > S^4 induces a symmetry between elements in seven dimensions so d*Ω = 0. In addition for the spin connection σ^{ab}, the projection with the tensor is zero ω_{abc}σ^{ab} = 0. This means that Ω is covariantly constant, which is a condition it being a Killing spinor. This gives a set of first order differential equations for the metric elements, The existence of additional covariantly constant field-form restricts the G_2 holonomy so the Killing spinor equation has more than one solution and the 4 dimensional field theory has extended N > 1 supersymmetry..

Then for this reason the flat connection term on the G_2 means that the YM action is of the form

$S_{YM}~=~\int d^4x\int d^5x\int_0^{z’}{{dz}\over z}\Big[(A_{\mu a}A^{\mu a})^2~+~ (A_{\mu z}A^{\mu z})^2\Big],$

where yesterday I made a mistake and forgot the square on the products of A’s --- sorry. For a non-flat connection on spacetime there will also be a dF^2 term as well.

Cheers LC

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Peter van Gaalen wrote on Nov. 19, 2009 @ 20:27 GMT
Hi Ray,

Ray@: Have you had an opportunity to read our essays? Those essays are both quite mathematical, and appear to be different on the exterior, but have many underlying similarities.

No, the problem is that it's difficult for me to understand. I try but it also cost me lots of time. But I red your essay and parts of your book and I am going to respond because your ideas are interesting. I have some questions.

Ray@: There are only three visible spatial dimensions, but what if Hyperspace never inflated (like Spacetime did) and is thus still very small?

In that case there are more than three spatial dimensions. But what I don't like is a spacetime with one time (imaginair: ict) dimension and 9 or more spatial real dimensions. In math don't have such (hyper)complex numbers. It makes no sense. Why not other dimensions than spatial dimensions?

Ray@: Do you still live in the Netherlands? Steve Dufourny is in Belgium. His background is in Botony.

Yes I still live in the Netherlands (Rotterdam). Belgium is not far away.

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Ray Munroe wrote on Nov. 19, 2009 @ 20:51 GMT
Dear Peter,

My 12 dimensional model might have two "time-like" dimensions and ten "space-like" dimensions. My seventh dimension is both wierd and connected to gravity - it could be an "imaginary time". I'm not certain of this speculation, but it might make sense considering the imaginary content of Quaternions and Octonions. This might also be the primary difference between my 12 dimensional model and 11 dimensional M-Theory. Even Lawrence's 11 dimensional model starts out as an M^4 of Spacetime plus a Spin(8) that decomposes into a Spin(7,1) and eventually Spin(7) of Hyperspace. The seven-fold (septality) symmetries are important, but we need to understand where this 12th dimension disappeared to?

I think Steve is in Mons, about 220 km from you.

I understand the importance of time management. We are preparing for the "Black Friday" retail season. It keeps me busy!

Have Fun!

Ray Munroe

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Steve Dufourny wrote on Nov. 20, 2009 @ 13:14 GMT
Hi all ,

Yes indeed dear Ray ,in dutch we say Bergen .Bergen is about 260 km from Paris and 250 from amsterdam .And I am too about 60 km from Brussels.You know Ray even near my house ,7 km ,google have created a new data center ,and Mirosoft a new center of innovation .The reason is the geographic situation and the system of optic fibre in Saint-Ghislain I think ,an other important point of vue is the geothermical potential ,indeed an interesting water system exists here under us .But my politicians don't understand well the rule of our nature unfortunally .I suppose what google or microsoft them are understood this energy potential .

Thanks

Best Regards

Steve

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Lawrence B. Crowell wrote on Nov. 20, 2009 @ 13:49 GMT
@RRay: The Lorentzian metric in 26 dimensions stems from the root space result that

$1^2~+~2^2~+~3^2~+~\dots~+24^2~=~70^2,$

where there is a bit of abstract algebra behind this. A null direction in 26 dimensions can be defined with this according to

$\pm 0^2~+~1^2~+~2^2~+~3^2~+~\dots~+24^2~-~70^2~=~0.$

The sequence of 1 to 24 are combinations of elements in the Mathieu group M_24, which are embedded in the space of 26 dimensions with a Lorentzian metric. The light cone condition here is specified by a contraint on three scalars in the 27 dimensional Jordan matrix. One can make this spacetime have two time directions by chosing the negative sign on the "0." In a curved spacetime setting this has some AdS structure to it.

How this fits into the model you suggest is a bit uncertain, but the Weyl group for the Leech lattice is diag[H_4,H_4] + permutations. One might then be able to arrange one of the H_4 to be embedded into a 14 dimensional spacetime with one or two time directions and AdS-like structure. Of course this is somewhat speculative at this point.

Cheers LC

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Ray Munroe wrote on Nov. 21, 2009 @ 03:36 GMT
@ Lawrence "One might then be able to arrange one of the H_4 to be embedded into a 14 dimensional spacetime with one or two time directions and AdS-like structure."

As you are aware, my model is 12-dimensional, and similar to E8xH4. Spacetime is dimensions 1 thru 4, AdS is dimensions 5 and 6 (the Hyperflavor-brane), 'imaginary time' is the 7th dimension, the WIMP-Gravity-brane is dimensions 8-10, and the Generation-brane is dimensions 11 and 12. Are you saying that I need a 14-dimensional model similar to E8xH4xG2? I have played with an E14/ K14, but it seems to have too many outrageous symmetries: pentality and septality, and it seems to imply 5 generations of matter - something we have no reason to expect at this stage.

The sum of metric squares is interesting. I remember this from an earlier blog.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 21, 2009 @ 14:04 GMT
The 26 dimensional bosonic strng lies behind this. We must be careful, for the physical basis for the 26-dim string is rather unclear. Yet this connects up with an enormous amount of structure, from Jordan algebras to Borcherd's results on the 26-dim Lorentz symmetry as the automorphism on the Monster group --- look at Conway & Sloane "Sphere Packings ..." and vertex algebras. In the Jordan algebra the three octonions or E_8s are identified as "vectors" plus two supersymmetric pairs (field and their conjugates), which reduces the dimension of the space to 11 dimensions. With the light cone gauge (infinite momentum frame) one dimension is compactified and the system reduced to 10 dimensions.

This of course gets us faced with M-theory. I think the 27 dimensional Jordan algebra is a natural system for working the Banks, Fischler, Shenker, Susskind (BFSS) matrix model in a twistor space format. The BFSS will then contain the underlying 26-dimensional bosonic string as one of the string/M-systems which transforms into the 10-11 dimensional strings, I and II strings of A & B varieties, and the heterotic strings.

How the bosonic string determines uniquely the sorts of expanded root systems you derive is of course difficult to understand. Maybe working this out in 24-26 dimensions, breaking this down by Weyl group on E_8's and then using M-theory might tell us something about the spectrum of elementary particles. There are considerable theoretical results along these lines in the R-R and HS-NS sectors.

There is a lot of structure here, and considering strings as a knot/vortex topological effect (Skyrmions etc) based on some underlying quantum error correction code is being a bit out on a limb. My experience has been that people in high positions can get away with these hypotheses, while people like me get their butts kicked. Yet as Feynman suggested, don't worry too much about what other people think.

Cheers LC

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Steve Dufourny wrote on Nov. 22, 2009 @ 11:45 GMT
Hi all ,

Yet as Feynman suggested, don't worry too much about what other people think.

yes indeed dear Lawrence with of course relativity and universality .If not the road of the individualism becomes the main part of the personal line of reasoning and of course without a pragmatic correlation with the physicality that beomes confusings .The sorting is like a superimposing ,and the universal thinking of course is the driving force of the extrapolations ,reals and not imaginaries ....

Just a thought

Regards

Steve

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Ray Munroe wrote on Nov. 23, 2009 @ 13:24 GMT
Dear Lawrence,

I suspect that your 26 dimensions decompose first into E8xE8xE8xG2, then an E8 (is this a broken-symmetric phase?) splits into H4xH4 so that we have (E8xH4)x(E8xH4xG2), then the first H4 becomes Spacetime, the first E8 becomes Hyperspace, and the (E8xH4xG2) becomes Supersymmetry/ Superspace. If a 27th dimension exists, I'm not sure what happens to it.

Yes - A 14 dimensional component is required. But I think this is Supersymmetry, not Spacetime and Hyperspace, because the Supersymmetric part of the theory must be at least as large as the non-SUSY part of the theory. My interpretation of a 14 dimensional Spacetime plus Hyperspace is that this would imply 5 (or 7 because we introduce a septality symmetry?) generations of fermions.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 23, 2009 @ 20:08 GMT
The G_2 is an automorphism which preserves the Jordan product axb = ab + , where the inner product defines the wedge or cross product

[equation]

and is then a trace map : Im(O) - - >Im(O). Hence with the action of this inner product on another element c is a three form (remember G_2 is defined for 3-forms) so that *c = . This defines the three form on 7-dimensional Riemannian...

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Ray Munroe wrote on Nov. 23, 2009 @ 20:38 GMT
Dear Lawrence,

I also like the beauty of the Clifford algebras, but Cl(8)xCl(8)? Do we really have to go there? I see many of the 14-fold symmetries (which may relate to G2 or Klein's Chi(7)), but have lost any intuitive insight. I was only proposing Cl(4)xCl(5) plus SUSY... It seems that every time I think I'm beginning to understand you, you leave me in the dust again.

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 24, 2009 @ 01:09 GMT
It is the case that CL(16) is a bit big 256^2 = 65536, which is the number of states. However, for the SO(24) group in the bosonic string sector there are a lot of states as it is. For the open string the Regge trajectory predicts states α’M^2 = -1, which are the ground state tachyons. States of the form α’M^2 = 0 are 24 massless polarized vector bosons. From there α’M^2 = 1 states are

$\alpha^i_{-2}|0,k\rangle,~ \alpha^i_{-1}\alpha^j_{-1}|0,k\rangle$

which in SO(N) there are N and (N + 1)N/2 states, where N = 24 gives 24 and 300 states. Now if we go to α’M^2 = 2 we have

$\alpha^i_{-3}|0,k\rangle,~\alpha^i_{-2}\alpha^j_{-1}|0,k\rangle,~ \alpha^i_{-1}\alpha^i_{-1}\alpha^j_{-1}|0,k\rangle$

where by similar counting N, N^2, (N+2)(N+1)N/3! gives 24 + 576 + 2600 = 3200 states. This is the maximum spin, or spin = 2ħ. We then work to include the states of the closed strings with α’M^2 = 4(N-1), with tachyons α’M^2 = -4, then states

$\alpha^i_{-1}{\tilde\alpha}^j_{-1}|0,k\rangle~=~|\Omega^{ij}(k)\rangle$

The symmetric and antisymmetric portions of these two sets of (N – 1)N/2, or 276, states (552 in total) and the trace terms are 24 dilatons --- a total of 576 states.

For the closed string there are 576 states, which is greater than the 256 left and 256 right states we might expect from CL(8). Yet we can impose restrictions on the number of states, by realizing there is some metric condition (gauge-like coordinate condition) on the four dimensional spacetime (10 components per R and L modes) which reduces this to 256! Yet all we have are the states of the graviton and an associated gauge-like field. We don’t have a supergravity multiplet. So to include the superpartners of the graviton (type IIB strings) we need all that open string stuff, and we need to work in the associated 10 and 11 dimensional superspace.

Cheers LC

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Lawrence B. Crowell wrote on Nov. 24, 2009 @ 01:12 GMT
I have no idea why these equations turned out this way, but just "wrap around" to read them On the last the a^{ij} should just be Omega^{ij}. I wrote this on another edictor and I don't know why that happened.

LC

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Ray Munroe wrote on Nov. 24, 2009 @ 13:27 GMT
Dear Lawrence,

Cl(8) has some interesting characteristics. It contains rank 7 (7 dimensional) objects with a 14-fold symmetry that could allow a 7-dimensional Hyperspace and its 7-dimensional SUSY Superspace.

I think this ties into SU(29), which is a rank 28 (28 dimensional) Lie Algebra that I have used before (in my book and my hypercolor paper). SU(29) is interesting because: SU(29)=840=7!/3! which seems to be the maximal lattice kissing number in 12 dimensions. In this way, it ties into Steve Dufourny's ideas (kissing spheres), Emile Grgin's ideas (laminated lattices), your ideas (error-correcting codes), and my 12 dimensions.

Within my model, these 28 dimensions might be (E8xH4xG2)x(E8xH4xG2), where the extra G2 arises from the fractal nature of Klein's Chi(7) (in my "A Case Study" paper). The extra G2 term gives my theory more 'balance' between SUSY and non-SUSY theories, but may imply extra generations - I need to think on this.

Have Fun!

Ray Munroe

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Ray Munroe wrote on Nov. 24, 2009 @ 17:01 GMT
Dear Lawrence,

Should we use 24, 26, 27 or 28 dimensions?

In my model, the 4th dimension is time, the 7th dimension might be 'imaginary time', and we have SUSY 'mirrors' of these dimensions in the 13th and 21st dimensions. Does the light cone boundary condition and related boundary conditions (for 'imaginary time', 'SUSY time' and 'SUSY imaginary time') reduce the 28 dimensions down to a 24 dimensional metric trick?

$1^2 +2^2 +3^2 +...+24^2 = 70^2$

Have Fun!

Ray Munroe

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Lawrence B. Crowell wrote on Nov. 24, 2009 @ 19:19 GMT
There is a different strategy we have. You are primarily focused on root space descriptions of states, while I am primarily interested in the geometric content of things. In the case of differential geometry geodesic motion is determined by the space of the tangent bundle or fibre, which does have a Lie algebraic content.

The group G_2 fixes a vector basis in S^7, for V \in J^2(O), and the spinors θ bar-θ in O_1 and O_2 in J^3(O), according to a triality condition. The triality group in spin(8) and the subgroup spin(7) fix these vectors on according to a transitive action in spin(7) as spin(7)/G_2 = S^7. This has the dimension

Dim(G_2) = sim(spin(7) – dim(S^7) = 21 – 7 = 14.

This then defines a frame on the ocotonions and is similar to a gauge condition. On the algebraic level the g_2 is contained in spin(8). The map spin(8) - -> so(O) is then a double covering from a 14 dimensional space to 28 dimensional and

$so(O) = g_2\oplus Im_r(O)\oplus Im_l(O)$

where g_2 is 14 dim and so(O) is 28, and the right/left multiplication of the imaginary part of the octonions are then 7 dimensional. Further so(7) ~=so(Im(O)), and g_2 is the set of rotations on the imaginary octonions in 21 dimensions. This means that SO(O) ~ aut(O)(+)exp(ad_{Im(O)}, where the automorphism group is G_2.

In this sense there is a 28-dim group structure, which is SO(O) ~ spin(8).

SU(28) is rank 28, but 378 dimensional. Though the rank could be the span of some 28 dimensional space. In this case the SU(28) would be the group action on a 28 dimensional space.

The Weyl group of the E_8 is a diagonal matrix form with H_4xH_4. H_4 is the Coxeter-Dynkin group for the 120/600 cell or the 120 icosians. The Clifford algebra CL_{7} is a system with 128 elements, which embeds the icosians, or the icosians are 120 elements of the CL_7. So the H_4 defines an algebraic system in 7 dimensions.

Cheers LC

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Lawrence B. Crowell wrote on Nov. 24, 2009 @ 19:23 GMT
erratum: I said that SU(28) was 378 dimensional, but I did the formula for SO(28). Dim(SU(28)) = 783.

Cheers LC

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Ray Munroe wrote on Nov. 24, 2009 @ 20:00 GMT
Dear Lawrence,

I am suspicious of similarities between a Cl(7) of order 128 and an H4 of order 124. H4 is rank 4 and has significant 2-fold, 3-fold, and 5-fold symmetries. Cl(7) has some rank 5 components and a significant 7-fold symmetry. Embed them in a larger algebra, and it might make sense. By themselves, they seem to be oil and water.

Have Fun!

Ray Munroe

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Ray Munroe wrote on Nov. 24, 2009 @ 20:07 GMT
I realized you said SU(28) and meant SO(28). The order of SU(28) is 840=7!/3! and is the lattice kissing number in 12 dimensions.

Ray

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Lawrence B. Crowell wrote on Nov. 24, 2009 @ 23:44 GMT
The H_4 has the Schlafli symbol {5,3,3} with its dodecahedral symmetry. The spin(7) is a B_3 group which has a Coxeter Dynkin or Schlafli symbol {4,3,3}. So I suppose there is some sort of problem here, and my suggestion at the end of this morning's post is incorrect.

The Klein X(7) is a modular system. The discrete group for the projective Fano plane PSL(2,7) has 168. In a graded system this involves V, θ_1, θ_2 (vector plus spinor & conjugate), this is 3x168 elements, and for right and left acting imaginary spinorial O (Im_r(O), Im_l(O)), this extends the number to 4x168 = 672 elements due to θ_1 and θ_2, and including the vector V there are 5x168 = 840 elements. For the spinorial terms the action of the G_2

$so(O) = g_2\oplus Im_r(O)\oplus Im_l(O)$

this add 14 elements which is the 672 + 14 = 686 elements of the X(7). If these elements are assigned to be Lie algebraic then the 5x168 plus the 14 from G_2 will give the SU(29)xG_2.

Cheers LC

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Ray Munroe wrote on Nov. 28, 2009 @ 15:12 GMT
Dear Lawrence,

I've been playing around with H4 symmetries, and think it might go something like this:

H4 -> SO(8) bosons x Spin(10) fermions + (3 + 3-bar) right-handed neutrinos

-> SU(5) Georgi-Glashow boson GUT + 4-plet 'Higgs' scalar field + 3 generations matter-anti-matter, so that 124 = 24 + 4 + 2x3x15 + 3 + 3.

The first level of decomposition, SO(8) x Spin(10) of rank 4 + 10 seems to imply a 14-dimensional space to me. However, I was able to combine all of these algebras into my 12-dimensional K12' ~ E8 x H4.

What do you think? I need a project. My wife's grandmother passed away yesterday - I need something to focus on.

Have Fun!

Ray

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Lawrence B. Crowell wrote on Nov. 29, 2009 @ 03:35 GMT
The {5,3,3} for H_4 is *-----*---*---* which appears to be be decomposible into SU(4), or a 5-fold quiver (I_2(5)) of SU(4) valued quaternions. This is then a tessellation of a hyperbolic space, such as the AdS spacetime with group SU(2,2). Now there is a duality here in conforaml field theory which means this tessellation defines an SU(4) valued field theory. The other alternative is a decomposition to SU(5), which would correspond to the Georgi-Glashow decomposition above.

Cheers LC

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Ray Munroe wrote on Nov. 30, 2009 @ 01:15 GMT
Dear Lawrence,

I also see the symmetries involving SU(4)~SO(6), Spin(6) and SO(10). It seems to imply more equivalent dimensions (six different 3-D tetrahedra) with only the pure rotational (or pure translational) tetrahedral group symmetries (the limitation of an SU(4) versus an SU(5)).

I should be back by Tuesday afternoon.

Have Fun!

Ray

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Ray Munroe wrote on Nov. 30, 2009 @ 13:53 GMT
Dear Lawrence,

I goofed my last posting. Six SU(4)~SO(6) tetrahedra (18=3x6 dimensions) only fill Spin(10) (90=6x15 order). It takes 8 tetrahedra (24=3x8 dimensions) to fill H4 (120=8x15 roots). I think one tetrahedron is a 3-D extension of "hypercolor" (in my hypercolor paper), another is Hyperflavor (in my book), and the other tetrahedra probably involve generations and SUSY - I need to think on these. Perhaps they are centralized by a G2 or an I_2(7). I like G2 because of your Skyrmion ideas and because it may represent a 3-legged Feynman vertex. I like I_2(7) because it reflects the symmetries of Klein's Chi(7).

Today is my grandmother-in-law's funeral and lots of drive time. I'll have time to think about these ideas, but not to write them down.

Have Fun!

Ray

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Lawrence B. Crowell wrote on Nov. 30, 2009 @ 18:54 GMT
The triality in J^3(O) includes an E_8 matrix of vectors and two spinor matrices. These do correspond to a Feynman diagram where a vector (boson) decays into a spinor and its conjugate. The J^3(O) is R + J^2(O) + θ + bar-θ, and the vector term V ~ J^2(O) decays into V --> ψ + bar-ψ. This is a manifestation of the automorphism of G_2.

Cheers LC

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Lawrence B. Crowell wrote on Dec. 3, 2009 @ 00:16 GMT
The SU(4) has the 4-irrep and the adjoint 15-bar representation. So a QCD-like or CFT theory that is SU(4) has the 4-irrep on a tetrahedra of roots. IN your comments above the "15" above might then be the adjoint rep of SU(4). The 3 in the 3x8 I presume represents the dimension of the space of roots, on a tetrahron.

Cheers LC

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Ray Munroe wrote on Dec. 3, 2009 @ 02:03 GMT
Dear Lawrence,

I think we are on the same page...

My book (and maybe another paper or two) explains the symmetries of the tetrahedron as 3 basis vectors (in normal space, this would be x, y, and z) plus 12 translation vectors (using my 4-color language, we have red-anti-green, red-anti-blue, red-anti-white, green-anti-red, green-anti-blue, green-anti-white, blue-anti-red, blue-anti-green, blue-anti-white, white-anti-red, white-anti-green, and white-anti-blue - or the possible "color combinations" of gluons and X and Y bosons). In my models, SU(4)=15=12+3, and the number of basis dimensions is then our number of tetrahedra (a maximum of eight?) times 3 basis dimensions (x, y, z) per tetrahedron 24=3x8. This is slightly more complicated than QCD because we have the equivalent of 4 colors per tetrahedron. I still think I need an SU(3) triangular lattice to explain 3 generations - a tetrahedron implies 4 generations - we have no reason to assume or expect this at this point in the game.

H4 could be the overall symmetry of Spacetime. This makes H4 relevant in a bottom-up theory, but I agree that the Leech lattice (or something similar or larger like the Monster group) is probably a more relevant in a top-down theory.

So H4 is an interesting middle-state between 24/26/28 dimensions and these tetrahedra of various kinds of charges such as color, flavor, etc.

I'm still thinking about it. And still trying to pull together models of H4->SU(4)'s or H4->SU(5)'s.

Have Fun!

Ray

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Lawrence B. Crowell wrote on Dec. 4, 2009 @ 03:49 GMT
The relationship between the SU(4) and the H_4 is that the AdS group is SU(2,2), or SU(4) in a Euclideanized form, and H_4 tessellates a hyperbolic space, or the AdS. So the H_4 is a system of quivers, holding 8 roots or states, for the SU(4) [equivalently the SU(2,2)]. The icosian quaternion group is then some 8-folded system on the roots of SU(4).

Cheers LC

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Ray Munroe wrote on Dec. 4, 2009 @ 13:35 GMT
Dear Lawrence,

I agree, and I think I have about 3 or 4 of the 8 folds figured out. I have not yet figured out the SUSY contributions.

SU(2,2) decomposes as 3+6+6=15 which is exactly the same symmetries as the tetrahedron with 3 overall spatial basis dimensions (x,y,z), 3 FCC crystalline basis vectors and their opposites, and 3 alternate FCC crystalline basis vectors and their opposites.

Have Fun!

Ray

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Ray Munroe wrote on Dec. 4, 2009 @ 16:58 GMT
Dear Lawrence,

Certainly, SU(4)~SU(2,2) could be relevant. What do you think about any possibilities for SU(4)~G2xU(1)?

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Lawrence B. Crowell wrote on Dec. 4, 2009 @ 21:07 GMT
The G_2 group is contained in SO(8) which of course contains SU(4). Further the quotient of spin(7) with G_2 defines the 7-sphere. So this quotient fixes a vector basis in spin(7) which is the 7-sphere. The nest of inclusions from there is that SO(6) is contained in SO(7) and of course SO(6) ~ SU(4). So there is an overlap between G2 and SU(4), but there is no direct decomposition between G_2 and SU(4).

The quotient F_4/B_4, for B_4 ~ so(9), determines a short exact sequence between spin(9) and OP^2. So there is an SO(8) embedded in this. The relationship between the G_2 and F_4 actions (local gauge-like transformations) is the two are "anchored together" as automorphisms of J^3(O) and as centralizers of the E_8. How this determines the action of SU(4), or the AdS U(2,2), is not immediately apparent to me. The relative actions on SO(8) between F_4 and G_2 is through the triality action of G_2 --- G_2 gives 3 “copies” of SO(8). So the action of the two may involve some equivalent or homomorphism between the subgroup SU(4). Maybe some thought or work is needed on that.

Cheers LC

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Ray Munroe wrote on Dec. 5, 2009 @ 17:36 GMT
Dear Lawrence,

Analyzing the H4=120=24x5 perspective seems to lead to a Georgi-Glashow model, whereas the H4=120=15x8 option seems to lead to a Pati-Salam model. Georgi-Glashow seems more universal to me because a single SU(5) contains all of the tetrahedral symmetries whereas a single SU(4) only contains half of the tetrahedral symmetries. However, SU(4)xSU(4) contains all of the tetrahedral symmetries plus some.

The SO(8) may be a tetrahedral 4-plet plus its 24 nearest-neighbors (of an FCC lattice).

At least I have one football team in a championship game. Hook 'em Horns!

Ray

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Lawrence B. Crowell wrote on Dec. 6, 2009 @ 02:52 GMT
I will have to think a bit about this. I am wondering what this might have to do with tetrahedra in multi-dimensional spaces with dim >= 3 and j-functions.

Cheers LC

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Ray Munroe wrote on Dec. 6, 2009 @ 03:48 GMT
Multiple dimensions lead to multiple tetrahedra, which break into multiple branes. I haven't worked on the j-functions.

Ray

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Lawrence B. Crowell wrote on Dec. 8, 2009 @ 17:50 GMT
Higher dimensional tetrahedral have the 10j-Klein functional behaviors, which have modular structure. I think modularity is a crucial aspect of any quantum gravity or cosmology. This is a part of why I got into this little problem of time with the Taub-NUT spacetime. It has a discrete subgroup structure on the Minkowski spacetime in any local region, where a time operator can be defined --- here on a measure ε set or subspace of the Hilbert space. This connects with the S-duality modular function and magnetic monopoles. In doing this I am trying to set up physical motivations for that sort of structure. When it comes to tetrahedral, root vectors for elementary particles and so forth, this would mean that elementary particles exhibit a spectrum which is parallel the modular structure of spacetime.

Cheers LC

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Ray Munroe wrote on Dec. 8, 2009 @ 18:22 GMT
Dear Lawrence,

I was just looking at Baez, Christensen and Egan (0208010) again. It looks like the 10j symbols arise from the 4-simplex. The 4-simplex may be related to a Georgi-Glashow SU(5) GUT, and the tetrahedron (3-simplex) may be related to a Pati-Salam SU(4) GUT. SU(4) contains the translational symmetries of the tetrahedron, whereas SU(5) contains both translational and rotational tetrahedral symmetries. I spent the last few days trying to understand how my ideas might mesh with Pati-Salam. I sketched it out graphically, but now wonder if it is unnecessary - would you like to see it? My essay presented how my ideas include the 4-simplex and Georgi-Glashow.

Have Fun!

Ray

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Lawrence B. Crowell wrote on Dec. 9, 2009 @ 01:44 GMT
I too have read this some as well. The first diagram I attach is from Baez, Christensen and Egan (0208010). These are the 10j symbols for the 4-dimensional tetrahedron. The pentagonal symmetry I find interesting. In particular the pent symmetry is an element of the H_4 octahedrachoron which tessellates hyperbolic spaces. The H_3 tessellation I present in the second attachment indicates this sort of tessellation structure. The structure here is SO(4) ~ SU(2)xSU(2) ~ A_3. So we have a system which is an H_4 with an internal symmetry given by the 10j symbols or SO(4). So this is the case where

H_4: *--5--*---*---*

SO(4): *---*---*

The H_4 is already an SO(4) with an additional "pentagonal" I_2(5) symmetry. The two SO(4)'s exist in an SO(8).

Cheers LC

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Ray Munroe wrote on Dec. 9, 2009 @ 02:57 GMT
So you are using 120=2x6x10, whereas I have been looking at 120=24x5 and 120=15x8. SO(8) is important to the tetrahedron, but SU(4) and SU(5) are also important to the tetrahedron, and SU(5) is also important to the 4-simplex. I'll e-mail you my latest ideas on an SU(4) Pati-Salam-like model tomorrow.

Ray

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Lawrence B. Crowell wrote on Dec. 10, 2009 @ 01:47 GMT
The great logician Kurt Godel escaped Austria after Anschluss by Nazi Germany. He came to Princeton and studied the Constitution to become a citizen of the US and told Einstein that he found a couple of logical errors in it. Einstein then remarked it was best to “shaddap” about that, because he might get deported. The Constitution and other founding documents are amongst the great political ideas of history. They are not something to be worshipped, and of course on the other hand not to be treaded upon. The Declaration of Independence does have an interesting little bit in it, where the grievances of King George are spelled out. Amongst them are, “He has excited domestic insurrections amongst us, and has endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages, whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions.” These were highly exaggerated, and in fact the English settlers were a continual problem for the native tribes as they tried to press on and steal their land.

I don’t have time to go blow by blow on things. If you have had foreign travel and better working experience you come to see the world a bit differently. This is particularly if you have been in nations where the heavy footprint of the US has been present. I do think that a healthy culture, society and nation is one which is willing to admit its problems, and all have problems, and discuss these matters in reasonable ways.

Cheers LC

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Lawrence B. Crowell wrote on Dec. 10, 2009 @ 02:03 GMT
One of the problems is there are 453,060 irreducible representations of E_8, which include arrangements of various subgroups. This is an enormous amount of information, which was solved by computing the Kazhdan-Lusztig-Vogan polynomials for the large block of the split real form of E_8. Physics might ultimately involve some superposition of these various irreps, where what we observe locally is just one of them. So it is very difficult to determine with certainly which of these perspectives on the subgroup SU(4) or SU(2,2) are to be taken --- maybe both?

Cheers LC

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Ray Munroe wrote on Feb. 18, 2010 @ 14:04 GMT
Dear Lawrence,

How about if we move our discussion to this thread? It is relevant to Quantum Critical points and degrees of freedom.

On the 2180 blog thread, Lawrence said "The M2-brane is dual to the 5-brane, called black brane. I have been doing analyses on a related issue with the quantum phase structure, a quantum critical point, of the stretched horizon and its relationship to the structure of elementary particles. Zamolodchikov demonstrated how a mirror potential in a scattering matrix for massive particles leads to an E_8 (8,1) representation for 8 "mesons." A BPS black hole near the extremal condition I found has this S-matrix structure. The ratio of m_1/m_2 is the golden ratio

m_1/m_2 = (1 + sqrt(5))/2.

The 8-irrep of these particles is an E_6xSU(3) rep of E_8 which is

(8, 1) + (1, 78) + (3, 27) + (3-bar, 27-bar),

which embeds into the 78-dimensional E_6 form of J^3(O) and the basic 27-dimensional real form of the J^3(O). If have been burning on this since last December. This (8, 1) part of the representation is given by the permutations of the (1, -1, 0) elements, which physically is equivalent to the Ising spin model with the Hamiltonian

H = Σ_i [S^z_iS^z_{i+1} + hS^x_i].

Here S^z_i the spin up and down operators for the (1, -1) portion and the S^x_i spin direction along the equator of the Bloch sphere corresponding to the 0 direction.

A. Zamolodchikov is one of those Russian geniuses who worked out the renormalization group flow for c = ½ conformal field theories. This result by Zamolodchikov is an emergent symmetry with conformal fields. I am writing this up and can send it to you ASAP.

BTW, I think we should shift discussions away from this warp drive 2180 page. I am running at 7-meg and this page still takes a bit of time to load.

Cheers LC"

Lawrence, The golden ratio arises out of pentagonal symmetries. The pentagon represents 4-D Spacetime, and this ties into your ideas and Coldea et al's Ising spin research.

Have Fun!

Ray

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Lawrence B. Crowell wrote on Feb. 18, 2010 @ 19:05 GMT
The E_8 has the system [3,5,8] and the Leech lattice the Steiner system [5, 8, 24], so the pentagonal or icosahedral symmetry is contained in both. The two H_4s which emerges from E_8 in the Weyl group is tessellated by 120/600 cells. These are dual polytopes which are 4-dim generalizations of the octahedron and icosahedron.

What I have found is the physics of black holes has a relationship with the structure of elementary particles. The emergent E_8 at T --> 0 and the extremal condition is a quantum critical point, where the symmetry persists across the UV to IR range.

It is best to shift these discussions from the 2180 blog site, which is getting "packed."

Cheers LC

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Ray Munroe wrote on Feb. 18, 2010 @ 20:14 GMT
Dear Lawrence,

Interesting. Are these Black Hole phenomena equivalent to fundamental particles or solitons/quasi-particles? Or does an S duality relate Black Hole phenomena back to low-temperature magnetic phenomena? Steve D has spinning spheres, which reminds me of circulating magnetic field lines.

Have Fun!

Ray

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Jason Wolfe wrote on Feb. 19, 2010 @ 01:11 GMT
I will shift my hyper-drive contributions to this blog, as well.

I have stated that FTL propulsion could be achieved by causing the spaceship to emerge from within a hyper-space shell; it emerges into the physical universe for supplies, etc. Then, to travel FTL, the spaceship is withdrawn into the hyper-space shell. There may be a better way to do this.

The engines, part of the structural integrity, sensors, and other stuff are built within the hyperspace dimension itself, using hyper-space particles/building material. The part of the spaceship that has to exist in space-time (cargo bay, cockpit, sleeping quarters, bathrooms, etc...) are built as a function unit. It is the function unit that is (a) drawn into the hyper-space spaceship or (b) extended into our universe from the hyper-space spaceship. Tethering the space-time functional unit to the hyper-space spaceship is a somewhat easier technical challenge.

Don't ask me how to built a rocket ship out of hyper-space materials, I have no idea.

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Lawrence B. Crowell wrote on Feb. 19, 2010 @ 02:50 GMT
This is a BPS type of result. However, I don't go so far as to say elementary particles are black holes of some very small mass. The black hole is by the holographic principle covered by strings on a membrane a string length above the event horizon. This is according to a stationary observer of course. If you fall into the black hole you notice nothing crossing the horizon. Further if you let the gravitational coupling g_s --> 0 the strings on the event horizon are "liberated" so to speak.

This does related to quasi-particles. The quantum critical point is determined by T --> 0 and the BPS charge near extremal condition where

r_+/- = m +/- sqrt{m^2 - q^4}

are horizons which merge at m = q. The magnetic field is H ~ q close to the horizon and this condition at T --> 0 is the quantum critical point. For q > m the horizon disappears and the singularity is "naked." This is similar to heavy fermion state where the mass m* is divergent, and the quantum critical point renormalizes this huge mass in quasiparticles. The chemical potential for the black hole is a summation over intensive and extensive variables which transform under a set of symmetries which give the Legendre transformations which keep the first law fixed. A singularity of this sort determines a phase change in the system. So the phase at T = 0^+ is determined by quantum fluctuations with Et/ħ ~ E/kT and so the Euclideanized time is t = ħ/kT. This establishes the degrees of freedom for the phase transition across the critical point. The degrees of freedom are governed by the (8,1) part of the SU(3)xE_6 representation.

Cheers LC

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