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  • Post #41

Dear Lawrence Crowell,

Since you are a black hole theorist, let's discuss the very similar objects - so-called holes in space-time (ST-holes). I have a deep suspicion that both objects may have the same nature.

Space-time holes appear in quantized theories of space-time. Imagine virtual quanta of space-time or an atom of space. This quanta appears and disappears continuously. If quanta of space-time disappear, it creates a hole in space-time or a vacant place in space-time. A hole is filled very quickly by environment. Since the speed of motion of environment is limited by the speed of light therefore the life-time of holes is non-zero. The holes are real objects and we can detect one using its properties. Inside of hole the abstraction and duration properties tends to zero due to a hole collapses quickly. Therefore if we increase the concentration of holes, the distance between every two points must decrease. In the limiting case when space consists of holes only the distance between every two points are equal to zero. Thus holes are accompanied always by time dilation and length contraction effects. Therefore we can detect the source of holes by detecting time dilation and length contraction effects. Thus I need two atomic clocks only to detect the source of holes. There is suspicion that some nuclear processes are able to create holes and I hope to detect one. What you think about it?

Sincerely, Leshan

There are wormholes, which are similar to black holes. The difference is that at the horizon there is a thin shell of mass-energy or a quantum field which induces a curvature "jump" that is hyperbolic or defocusing. This is a Lanscoz junction. Gravity tends to focus spacetime paths (geodesics) if the spacetime geometry obeys the Hawking-Penrose energy conditions of positivity. This jump violates that which creates a multiply connected topology. The defocued rays have to diverge somewhere, and they do so around an identical opening in spacetime.

I have written a number of posts about these types of spacetimes. They tend to be pathological. One way of looking at this pathology is to think of the event horizon as a sort of capacitor. If I put a current source to the horizon and measure the impedance of the horizon I will find it is related to the impedance of free space ~ 440 Ohms. If I put this exotic field on the horizon and change the membrane configuration to make a wormhole then the impedence of free space is adjusted to a negative value. So from an electrical circuit perspective it would be a sort of "negative resistor," if I put a certain current on the "device" then from Ohm's law E = IR, I get a negative voltage drop across this "source." This is pathological in a way. I have used other examples to illustrate such pathology, such as with thermodynamics.

The wormhole might form a basis for Planck scale physics or so called quantum foam. Of course string theory is meant to valence the world from this chaos. The string "covers" the chaos of such physics on a scale smaller than the Hagedorn temperature or string length L = sqrt{8π}L_p. The work I am doing with illustrating how string physics is based on quantum error correction codes physics might be understandable on a scale about half this length. Beneath these scales as one approaches the Planck length the sort of chaos you mention might in fact take place, but this is a sort of troublesome area we really want nature to valence from observable physics.

Cheers LC

In reference to the above on G_2 group and the conformal approach to holographic principle:

The G_2 group acts on 7-manifolds as a system of three-forms. The G_2 group is then an holonomy with R^3 bundles over a quaterionic space. The G_2 group is a determined by space C^5, where for every point a ∈ C^5, let the 2-plane π_a exist in the tangent space T_aC^5. This means the metrics for the 7-manifolds are determined by five parameters plus two abelian isometries. These isometries have fixed points on the G_2 manifold, which are two combinations of Killing vectors with codimension 4. In general we may consider the manifold M^4ﾃ--M^7, 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 = -δ_{ab} + ω_{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

Ω = (1/3!)ω_{abc}e_a∧e_b∧e_c

= e_1∧e_2∧e3 + e_4∧e_3∧e_5 + e_5∧e_1∧e_6 + e_6∧e_2∧e_4 + e4∧e_7∧e_1 + e_5∧e_7∧e_2 + e_6∧e_7∧e_3

= e_1∧e_2∧e_3 + (1/2)e_i∧e_m∧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*J_j = -δ_{ij} + ε_{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.

The Killing spinor we have derived is related to the conformal curvature equation

ĝ^{ab}D_a∂_bρ = (1/6)Řρ.

I presented the other day. I use ρ here for the conformal factor so as to avoid confusion with the definition of the 3-form. The diffeomorphic plus Weyl transform a string metric is

δg_{ab} = -(nabla_aδσ_b + nabla_bδσ_a - g_{ab} nabla_cδσ^c) + (2δω - nabla* δσ)g_{ab}.

The moduli are defined as variations δ'g_{ab} which are orthogonal to the diff+Weyl variations. This leaves the conformal Killing equation

nabla_aδσ_b + nabla_bδσ_a - g_{ab} nabla_cδσ^c = 0.

The orthogonal variation which defines the moduli may be defined with the string world sheet action. The result is that ∂_bρ = K_b is a Killing vector, and a conformal Killing vector as well. We then let the conformal factor determine maps Ω:M^4 - -> M'^4 on the above spacetime so that the total action in n = 10 or 11 dimensions

L = (1/k^2)∫d^nx sqrt{g}(R - Σ_pdα_p∧d*α_p)

remains invariant. The result is that the Killing isometries act on the quaternionic space.

There are two such Killing vector which act on the quaternionic space

K = k_1∂_t + k_2∂_σ,

Which defines the two dimensional plane embedded in the tangent space of C^5. This two-plane defines the isometry of the G_2 manifold, as does the Killing vector. The Killing vector is then a particular Pfaffian-form with dK = 0, formed from a linear combination of 3-forms which act on the M^7. The Killing vectors then define fixed points, which are due to the intersection of 6-dimensional surfaces, D-branes. The boundary between intersection D6-branes then determine the occurrence of D5-branes. The holonomic action of G_2 is then set by dK = 0, or equivalently above dΩ = 0.

More later --- Cheers LC

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  • Post #44

This is to reference some post on other FQXI blog sites on quantions.

I have been giving the matter of quantions some study. I am not entirely decided about their status as yet. My sense is they are an interlinking between two complex number or quaternions in a way which defines norms differently. This might have something to do with S-matrix. So I will outline some aspects of S-matrix theory and black hole complementarity, and then try to make possible links to quantions.

The holographic principle and black hole complementarity are generalizations of the S-matrix. Susskind's treatment of strings falling onto a black hole according to a distant observer treats the S-matrix on a domain which is causally defined on an infinite domain of support according to the tortoise version of the radial Schwarzschild coordinate

r* = r - 2m ln|r - 2m|

The S-matrix requires an infinitely extended domain by which fields are causally related, which is "manufactured" by this coordinate. In these coordinates the string exhibits a range of strange behavior, which I am not going to review again in great detail. Yet the string ends up covering the black hole horizon and is frozen their according to this distant observer. To an infalling observer on a commoving frame with the string none of this is the case, but rather the string enters the black holes with no apparent change and then exhibits tidal forces of an extreme nature near the interior of singularity. The string is a form of S-matrix theory, and the two cases reflect the existence of two S-matrices, each according to state space elements which are incommensurate with each other, or according to noncommutative operators. This is one way of looking at the so called black hole complementarity principle. There is then a superposition of the string in these two bases of states, and for this reason the distant observer may see the string frozen above the event horizon and also "burned up" by Hawking radiation made of quanta scattered from the string according to the infalling observer's frame.

The ordered S-matrix defines each vertex, or particle, and its neighbor. In a linear chain a general state is an S-matrix channel of the form

|φ> = |p_1, . . . , p_i, . . . , p_j , . . . , p_n>

This state or S-matrix channel is related to but distinction from the channel

|φ'> = |p_1, . . . , p_j, . . . , p_i, . . . , p_n>

The particles or vertices p_i and p_j have exchanged their neighbors, which means some "relationship" structure to the amplitude has been fundamentally changed. The S-matrix is written according to S = 1 - 2πiT, so two states or channels |p_1, . . ., p_n> and |q_1, . . . , q_n> are related to each other by the S-matrix as

(p_1, . . . , p_n|Sjq_1, . . . , q_n> = (p1, . . . , p_n|(1 - 2πiT)|q_1, . . . , q_n>

= (p_1, . . . , p_n|q_1, . . . , q_n> - 2πi(p_1, . . . , p_n|T|q_1, . . . , q_n>:

For the < | the in channel and | > as the out channel p_n and q_1 are neighbors, and neighbors through the T-matrix. This eliminates an open vertex in the chain. The vertices or particles p_1 and q_n are the open elements in the chain and defines an "anchor" for the chain, and are thus defined as neighbors in this manner.

A four point function and the transition matrix defined by vertex operators T = V(p_1)ΔV(p_3) will contruct the Euler-beta function for coherent states of the S-matrix. This is the connection of course between string theory and the old bootstrap or S-matrix theory. Now for two S-matrices, which pertain to the different domains of causality on a black hole this theory is made more difficult. The S-matrix is a braiding operation of sorts between elements of a quantum group G. So we might model this as a commutator structure (braiding) between two elements a and b \in G. So we might denote this as ab --- ba. Now let us assume the states we observe are super-positions of incommensurate states involving two quantum groups G and G'. We will then have a structure of the sorts (ab)c --- a(bc), that exist in an associahedron I_2(5) with a homotopy structure. This homotopy then connects to a K-theoretic field theory, which I discuss in my paper

http://www.fqxi.org/community/forum/topic/494.

I will not belabor this part of the things, until later or somebody takes an active interest in what I am suggesting here.

The black hole complementarity principle. The complementarity is an odd structure, for Hawking radiation is due to a Bogoliubov transformation between basis elements. In this setting the theory of spacetime is classical and the fields scatter off the black hole or spacetime with an event horizon. The response of the black hole or spacetime is a metric back reaction, which is a classical response to a quantum scattering. Yet black hole complementarity has demonstrated that quantum information is preserved for the case of a BZT black hole in an anti-de Sitter spacetime. So a connection between the quantum principles of unitarity (or maybe more generally modularity) and a classical field theory which exhibits thermal physics (black hole entropy and Bekenstein bounds etc) exists within this AdS/CFT setting. Yet we do not as yet understand how quantum information is preserved. We just know that it is.

So the quantumal rules of Grgin seem to segue into the picture here. The permitted multiplication rules

(fαg)αh (gαh)αf (hαf)αg = 0

gα(fσh) = (gαf)σh fσ(gαh)

(fσg)σh − fσ(gσh) = agα(hαf)

Connects the Jordan exceptional algebra to a quantum algebraic system. The associator is then by the homotopy equivalence mapped to a quantum group as a system of permutations (related permutahedra) with one set of norms determined by the underlying permutative rules or associahedra and the other by standard rules of complex conjugation in quantum mechanics. So the associator is [f, g, h]σ = agα(hαf) which induces the map between the octonions and the quantion group. This seems like an interesting problem to develop.

There are so called massive gravitons in N = 8 supersymmetry. These are of course issues with some of these problems. Where in your paper, or the case study 3 paper do you construct massive gravitons?

Cheers LC

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  • Post #45

i have ben losing my postings on your site because of Internet discoonections at my end. it seems you are lucky not to have some comments from me. i do appreciate your mathematical skill to work out the Physics and suggesting some innovative steps too. Somehow i get a recolllection of Einstein when he made remarks about the quantum mechanics being not a permanent theory for understanding Physics. he somehow seem to believe that the whole truth can not rest in probabilities. What about intellectual logic in the process of evolvement of any science. The present impasse in Physics is more due to lack of frsesh ideas and concepts. Once certain concepts get evolved and prove their success, one forgets that the same may be limiting us elsewhere. The best theory in Physics is one that involves the broadest of considerations encompassing as many phenomena as possible.

Presently i agree that we are able to understand the black holes through combining quantum and gravitational aspects. There has however been a recent exptal. report that has measured radiations coming out of one such object. Thus, we only are able to improve relative understandings in Physics and the finality is certainly not reached ever.

In my essay on this forum, i have hinted about an intense collaboration between physical and live sciences, towards a possible understanding about the human mind itsef. Everyone agrees that it is the basis for conceptualisation in Physics. This approach of evolving new Physics appears far superior to just relying on more mathematical jugglary. After all, maths. provides just a tool in Physics like the experimental data. The real thing is conceptualization in a better and better manner. Consciousness gets directly involved with the operation of the human mind. The total knowledge should be our aim and if it involves collaboration between humanity and sciences ( Physics is just one here)one should certainly give it a try.

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  • Post #46

I think that quantum mechanics is probably spot on, at least without gravitation. The think people need to realize is that quantum amplitudes obey perfectly deterministic wave equations. The modulus square of these amplitudes are interpreted as probabilities and according to the Born rule within the context of measurements. For basic QM the wave functions are complex numbers, determined by a Fourier series of such numbers, and a measurement is a destructive action which shifts this to a real number. In this perspective we might ask not whether quantum mechanics is fundamental, but rather why it is there exists a classical or macroscopic (non-quantal) world.

If you read some of my posts here, as well as my essay, the issue is ultimately how black hole complementarity preserves quantum bits. The AdS/CFT result and Susskind's analysis of BTZ black holes in AdS illustrate that quantum information is preserved. The next question is how. I will not claim to have solved this issue, but I think I have some suggestions for this and how this is connected to the problem of the cosmological constant.

Cheers LC

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  • Post #47

Dear Lawrence,

I read your paper and learned a great deal from it because I have not been following black hole research. I find your writing both clear and enjoyable.

This post is to keep the promise I made yesterday, but I have no questions.

Best regards, Emile.

Emile,

The key point here is black hole complementarity involves two causal domains for the S-matrix. S-mtarix theory requires there to be causal domain of support without boundary. This is why Susskiind works with the tortoise coordinate for the xternal observer, where

r* = r - 2m ln|1 - 2m/r|

The causal domain for the observer on a commoving frame with an infalling string is different. The dstant exterior observer observes the time dilation effect on the infalling string and further how it covers the black hole horizon. Conversely the commoving observer witnesses none of this and records nothing particularly different from a string on a flat spacetime until the center of the black hole is reached. So the two observers witness the S-matrix according to quantum mechanically incommensurate observables.

Some features of quantions might aid in this if it can be worked for Jordan algebras. The metrical distance has Lorentzian features to it, which might provide a way of working how a string (or any general quantum field theoretic configuration) can exist in a superposed state with the interior and exterior basis of observables for the two S-matrix configurations.

Cheers, LC

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  • Post #49

Thank you very much .

It's very relevant in fact ,thus we can considered two systems or more thus two spheres systems like for the pulsars .

If the taxonomy ,the correct taxonomy is correlated with the spherical multipoles ,it becomes very relevant .

The moment in my opinion and the mass are correlated with the specific number and the specific entanglement,

with the rotations thus the multipoles are so numerous ,

but we can class the fields too ,spherical more the gravitational waves .

Lawrence I see too the gravitational radiation correlated ,could you tell me more about these correlations Electro magne/Gravit radiations.

Sometimes I say me what several hypothesis are possible aout this oscillation,universal .in the quantum universe....

First like pulsars two systems are necessary ,second a changement of sense ,an instant,of the quantum spheres,three an oscillation of the spheres on a kind of arc of the spherical systems ,or a pure link with rotating spheres.It's difficult these questions about the frequences ,these oscillations ,these periodics .

10^-16 with the gamma ...RX .10^-10..UV 10^-8...our perception in the visible....after the IR 10^-4 .... micro waves 10^-2 ...after we continue towards the radio waves in the meter and after towards the big waves lenght ....all this system is limited I think and many parameters can permit to increase the perception .If the taxonomy is correlated with the rotations ,a max velocity of rot for the ultim main central quantum spheres ,and a min for the ultim cosmological universal sphere which don't turn or near 0 ,I prefer a big decimal after zero than the zero which don't exist .

We can link with the mass ,the rotations ,the gravity ,the EM ,if the senses ,angles ,volumes ,evolution are considered .

For the gravitational waves thus the classment will permit to accelerate the perception .

The light is really fascinating ,what is for you Lawrence its rule in fact ,why the light is linear ,and if the velocity of rotation of the main centyral sphere coul be more than this linear velocity ??? if I link all with mv ...thus the mass of Earth for exemple and its velocity of rotations around itself thus with for exemple a electron ,what is the results ,Ray had calculated that some months ago .Thus it lacks something in the equation ,probably the evolution point of bue and if the two thus photons were in two senses of rot one linear thus and the other balanced in our Universe ,local .

Thus in fact the light becomes mass with the time evolution but it's complex all that ,I becomes crazy hihihi

In all case thanks to take your time to answer me so often ,it's nice and furthermore I learn .

What do you think about the light dear Larence ,I am curious .

Best Regards

Steve

I must confess I sometimes have trouble trying to figure out what you are saying here. You argue things are related to spheres in some way. The Lie algebra which has spherical geometry are the orthogonal rotation groups SO(n). These do get deformed in certain ways though. Also for relativity and AdS spacetimes there are SO(m, n) which have hyperbolic content.

Cheers LC

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  • Post #51

My Sept 28 comments did not interest you to respond. may be i did n't enquire anything specific to your text. Being an experimentalist to the core, i feel fascinated by the conceptual pisture rather than the mathemetical applications to work the possible solution. i happen to note on Internet a news item on the daily galaxy website, that some radiation coming out of a distant black hole, some 12 billion years away, was detected against the normal expectation. Does black hole size has something to do with such a measurement if reliable. From what little i know, black hole is the last stage of a dead star/galaxy, as it starts attracting external mass and radiation on account of its intense gravitational field coupled with extremely high mass density too. To understand their behavior some attempts have also been made to introduce quantum gravitational approach. What are your expert opinion on these aspects?

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  • Post #52

I think I responded with my statement about quantum mechanics being exact, or nearly so, up to maybe quantum gravity.

My sense about mathematics is that it is an important way one works physics. I have an MS degree in mathematics, but don't consider myself to be a professional mathematician. IN what you say you indicate a greater interest in conceptual pictures. Yet mathematics are just those picture making elements. I don't advance the idea that mathematics is of greater importance than physical ideas. I think the two in a way can work together. A decent repertoire of mathematical knowledge helps in conceptualizing and formalizing things.

As for the distant black hole, this is rather interesting. There is this period called the reionization period about 400 million years after the big bang. About 400 thousand years into the universe the radiation dominated phase ended and luminous matter (ignoring dark matter) consisteded mainly of hydrogen, a little He and so forth. Gravitational clumping eventually gave rise to the first stars, which were strange population III stars that burned bright and exploded into supernova. These presumably gave rise to black holes and expelled matter somewhat richer in heavier elements. The generation of black holes is what is of interest. It appears as if these may have rapidly grown by coalescence or accretion of matter. This image is of a black hole of galactic proportions which existed about 1 billion years after the big bang.

This does feed into a question on why the early universe had such low entroy. It clearly was lower entropy than today, and it could not have been at some maximum entropy as well. A high entropy initial universe might have just generated lots of black holes to start with.

Cheers LC

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  • Post #53

Hi ,

Lawrence ,I just try to convince you about the correct architecture .

For the Blach Holes ,there too it's a question of rotating spheres.

Like I said ,the Lie Algebra aren't sufficient ,too mathematical without real physicalities and reals numbers .This kind of symmetry is false because all quantum spheres are specifics in their volume,thus the symmetry is very very different .

Like I said with Florin ,you can see my writings ,a balance with the imaginaries and reals must be made .The complexification with naturals numbers from a multiplication of prime numbers must be physical .

If not the infinity is not encircled in fact and thus the real number too .

I like read your extrapolations ,be sure but unfortunally the physicality and its fundamenatals aren't the main piece of your extrapolations ,too much complexs implies a complexification of the mind .I think .

It's probably the reason why I don't understand your words

you say

"A high entropy initial universe might have just generated lots of black holes to start with."

I don't encircle this meaning ,could you develop please?

ps you dislike really my little and poor spheres hihihi ??

Best Regards

Steve

The highest entropy a lump of mass can have is when it is packed into a black hole. So an early universe with lots of black holes would have a higher entropy. Yet this does not appear to be the state of the early universe. It started out with an unusually low entropy.

Infinity is not really a number. It is more a set, or a label given to a set with some unber of elements which are not finitely countable.

Cheers LC

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  • Post #55

Yes indeed but the prime are limited and the naturals ,products of the multiplication of primes are unlimited in the multiplication of the human and its mind ,like the complexification .

It's just this difference what I say between math and physic .A physical system without limit hasn't any thermodynamics laws in my opinion .

If the pression ,the volume ,the density ,the temperature ,the mass ,the energy ,the evolution point of vue and its increasing of mass ,the probable specific dynamic of our Universe ......all that must be coordonnated ,with a real topology in evolution .There without limit ,it's impossible .

Some fundamentals like the mass and the limits are essentials No ?

Best Regards

Steve

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  • Post #56

Dear Lawrence,

i read your response to my posting appreciatively. However, the first stars formed only around 400 million years after Big Bang. The dead star to have reached Dark hole status could not have been such a coalescing star as you picture. i suspect that we do not know yet all about the dark holes. There may well be variety amongst them that differ in behavior. Cosmology is an open field thus far and we also need to keep our minds wide open for fresh ideas. I for one do not feel convinced about quantum gravity as it is not showing any signs of unification with the other three interaction fields. The latter three have no problems with quantum mechanics but i do find gravity requires much more understanding before it may be unified with Quantum mechanics. Further, as per Einstein often stated belief that Q.M. may be replaced by another theory as he did not like that physical processes are entirely governed by randomness, as logic does have a place in the design that our Universe has evoleved. If it has been random then many other evolutionary pictures could have come in. Thus multiverses idea for our universe may not be ruled out completely. Multi dimensionality will also cause problems for applying quantum mechanics to gravity. The latter may have components that conform and do not conform to multi-dimensional and multi-verses ideas.

There is this dark age period after the end of the radiation dominated phase of the universe. It largely ended around 400 million years later with the gravitational accretion of matter into the earliest stars. These population III stars are not entirely understood. Pure hydrogen has a low opacity, and os it is not well understood how they maintained a nuclear fusion in a thermal-hydrodynamic equilibrium. These stars did result in two things. They produced a huge amount of radition which reionized the universe. The other is that in their death they produced the first large black holes in the universe. These appear to have grown by feeding off matter which coalesced into the earliest quasars or galaxies.

Quantum gravitation requires some new ways of thinking about quantum mechanics. The approach I take is by looking at the how the basis for the S-matrix is incommensurate for the two classes of observers with respect to a black hole. The distant observer who remains outside the black hole and the observer who falls into the black hole respectively observe different stringy physics. This is because the S-matrix, the simplest example being the Veneziano amplitude and the Euler β function which results, is a flat spacetime version of the string/M holography.

Cheers LC

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  • Post #58

Dear Lawrence,

Mathematics is one of the cornerstones of physics.

Regarding the use of mathematics in physics, I used math in my own ideas. I am not opposed to Emile and Florin's use of quantions - I simply have not yet been convinced that we cannot accomplish similar things (even though I know that the mathematical structures are different) with Pauli and Dirac matrices, but I scored both of their papers well, and remain open-minded regarding their ideas. I truly would like to see a quantion theory with interactions.

If I understand quantions properly, they are an effectively two dimensional algebra in competition with the Pauli sigma matrices. Just as we can build twistors (an equivalent to Dirac gamma matrices) out of pairs of Pauli spinors, we should be able to build relevant 4-D structures out of pairs of quantions. That should be relevant for 4-D spacetime.

BUT, we are also dealing with an AdS 2-D M2-brane. Is the rule of algebra in these dimensions based on quantions or on Pauli matrices? We also anticipate anyons in these bizarre dimensions. Would one algebra be more likely to yield this feature than the other?

K12' has a G_2 of color buried deep within (see Lisi's paper or Ref [11]). Regarding the G_2 holonomy, Gordon Kane also referred to this mathematical structure. I was originally distracted by the similarities between K12' and Klein's Chi(7), and thought I was working with an I_2(7) seven/fourteen-fold symmetry. But now I realize that I may be working with a Lambda_{10}, and NOT a Chi(7). If it is a Lambda_{10} (Conway & Sloane "Sphere Packings, Lattices and Groups" - Laminated Lattices), then it would have the same five/ten-fold symmetries as the G_2 holonomy. I think this five-fold symmetry is important to Supersymmetry, where we might expect five fundamental particle multiplets: scalar bosons (spin-0), matter fermions (spin 1/2), vector bosons (spin-1), gravitino fermions (spin-3/2), and tensor bosons (spin-2).

What are your thoughts?

Ray Munroe

Quantions are a sort of two-way map from complex valued coordinates, where one way determine a sum of bar-z_i*z_i as a quantum probability, and the other map which determines a metric distance. This is a bi-fibration system, one from the Hilbert space to the projective Hilbert space and the other to a metric space, If the complex coordinates are spinor valued (which quantions appear implicitly to be) then the two maps can well enough be defined. It is not a wrong or bad idea, but honestly I don't find that it has the power Grgin and other seem to think. This idea is pretty closely related to twisters, due to the bi-fibration structure. To be honest given the relative coverage given to twistors vs quantions I would recommend considering twistors. Witten a few years ago illustrated some aspects of twistor valued D-branes, which has put some wind back in Penrose's sails.

The G_2 holonomy is given by the tangent plane on C^5 with the action of a 2-plane. This determines G_2 as having 7 complex dimensions, or with two representations in 7 and 14 dimensions. I have a fairly long piece above dated Sep. 25, 2009. The G_2 is an automorphism of the Jordan exceptional algebra, and it is a holonomy over the three octonions in the matrix. The largest subgroup of G_2 is the SU(3), which suggests this structure is related to an AdS_3 ~ QCD type of model. I would say that the I_2(7) symmetry or the related OP^2 (Moufang plane) structure is more likely on tack here. The roots of the G_2 are related within this system to the 7-elements of the Fano or Moufang planes.

The quantions seem to be a way of representing a set of states with a bilinear map, which in curved spacetime leads to nonunitary equivalent fields according to Bogoliubov elements. This then appears to be really a holonomic system of some type, which for an extended group might result in a description of quantum gravity, or field in curved spacetime. Under a trace of elements it should then reduce I think to thermalized fields such as Unruh or Hawking radiation.

I will try to write more on this later. One reason for posting essays here is to increase connections with various people. Christian Corda and myself have an interesting exchange on the physics of detecting gravitational waves. So I ahve a number of new irons in the fire here.

Cheers LC

Greetings Lawrence,

I have started reading your paper and will likely return for more soon. I read through to the end of Exceptional Nature, then skimmed ahead looking for familiar concepts and conclusions. I'll need to get back to it, having reached my limit. I have several questions and comments already, though, so here I am looking for clarification. Bear with me if I ask a stupid question or re-state something obvious. At this point it looks like you are making a strong case for the utility of the Jordan exceptional algebra, but I haven't fully grasped your meaning yet.

First some foundations. I find that whenever invoking the Holographic principle, I am wise to mention and cite 't Hooft's paper on dimensional reduction arXiv:gr-qc/9310026. I'm looking forward to seeing him speak next month at FFP10 about black hole/information questions. And wasn't it Maldacena's contribution that we could move either up or down dimensions, at a dimensional boundary, with the same generalization? Next; are the Black Holes embedded in AdS space-time the same as the BTZ Black Holes Witten talks about in arXiv:0706.3359?

It seems as though your formulation's proof hinges on the closed string becoming an open string at the black hole's event horizon, so I'll start there. It seems as though a tidally distended string could have a time retarded portion and a portion in advancing time, to keep the loop open, making the stretched loop into a pair of facing cobordisms, as the 'string' is really a flux tube. Is the fact that it is flattened against the event horizon (making it 2-d plus 1 for time) the key here? Are the tidal forces of the Weyl curvature the factor which causes it to stretch and shear? Are we accepting on faith that it 'must' become 2-d conformal in order to pass through the event horizon?

I may believe you are on the right track, and enjoy seeing if your analysis bears out when we see more quantum Black Holes in the accelerator experiments, but I don't claim to understand why your formalism should work yet.

More Later,

Jonathan J. Dickau