I have given you essay a read through, which means I have not yet read it a second time for greater detail and content.

The quantum path integral is a measure over the distribution of a quantum field or particle. It assigns amplitudes to each path, which in the large N limit converges to the classical variational method.

The connection between quantum mechanics and octonions is not completely clear. The associator (ab)c - a(bc) that is not zero is not as well founded according to operators as noncommutative structures are. Further, the physical meaning is not as clear. I think octonions are really a system of quaternions (7 of them) which are related to each other by a general duality principle. This duality principle may then be expressed by the associator.

Cheers LC

Dear Lawrence,

thanks for your answer. Yes it is not an easy problem to consider exotic 4-manifolds. Actually from the differential topological point of view, two non-diffeomorphic 4-manifolds are distinct. Therefore you have to sum over these possibilities in the path integral and for each class the measure remains the same. (see arXiv:1112.4882 and arXiv:1003.5506)

Your idea about the splitting of the 11-manifold (I assume a compactification?) looks interesting. The meaning of the E8 in the intersection forms of 4-manifolds and its relation to the corresponding Lie group is mysterious for me too.Currently I have no idea to bridge this gap. A possible way is the theory of calibrated manifolds. Every oriented 4-manifolds embes into a 7-manifold and one can choose a G2 structre on the 7-manifold. Then the 4-manifold is an associated submanifold of this calibrated geometry. The deformation theory has a large overlap with Seiberg-Witten theory (the way to describe exotic 4-manifolds).

Another connection is via singularity theory (Arnolds approach) by using the ADE singularities. The E_8 singularity is directly related to the E_8 4-manifold.

I agree that spacetime has no foam structure. In particular, I like your argument that there are only one of each type of elementary particle. I resolves a conundrum in this theory. (so I have to go more deeply in your essay and the corresponding papers) If we started to describe matter by exotic smoothness we used the Casson handle and obtain similar results like in our (now published) paper arXiv:1006.2230. But we always got one type of a particle for each type. Maybe you understand the reason and I will read it more carefully. In principle, in the current version of the paper we have the same problem: we obtain the fermions as knot complements and the bosons as torus bundles. There are three types of torus bundles related to the usual groups U(1), SU(2) and SU(3). Currently we conjecture that gravity is a sphere bundle (which will be explain the universality). Interestingly, there are interesting connections between the Anosov torus bundle (represneting the SU(3) gluons) and the sphere bundle which I omit. Gravity as the "square" of a gauge theory is very intersting for me in this context.

I have also to understand your duality more fully. I remember back on lectures of Fadeev about Yangians (currently I dust my notice of the lectures). The deformation theory of Lie algebras is also part of our description of exotic smoothness. We obtain the deformation in a natural way using codimension-1 foliations. Then we obtain a relation to skein spaces (used to define R matrices for quantum groups).

So, our approaches converge in some sense, I will read your essay more carefully

Best wishes

Torsten

Dear Torsten,

Thanks for the reply. I will try to read your papers on this in the near future. I also need to review matters of the Atiyah-Singer index, Seiberg-Witten theory, Freedman- Uhlenbeck work on moduli at singular points and the rest. Back in the late 1990s I was better spun up with these matters.

The one thing which I think needs to be considered is that spacetime is hyperbolic, and all of this algebraic geometry machinery is set up for elliptic complexes. We might of course Euclideanize spacetime by considering τ = it. We then have -dt^2 = dτ^2 and we patch over the problems. This in effect deforms the moduli space so that sequences of gauge equivalent connections converge, say as a Cauchy sequence. With out this trick the moduli space is not Hausdorff and we do not have universal convergence conditions.

An 11-dimensional spacetime, 10 space plus 1 time, decomposes into the M^{3,1} plus M^7. Poincare duality on the total space tells us that homological data on the 4 dim part is equivalent to the data on the 7-dim part. Of course this may not necessarily have all the data, where homotopy tends to contain more. However, if we were to run with this the exotic data for smoothness might be contained in the 7-dimensional part. Of course at lower energy these spaces become compactified. In the 10 dimension supergravity theory the space of compactification is a Ricc flat 6 dimensional space. A canonical example is the 5-torus. A more potentially realistic theory is K3xK3. The 7 dimensional space in the 11 dimensional theory embeds the 6-manifold.

The first exception al group G_2 fixes a basis in a 7-sphere, as vectors in J^2(O). This consists of the vectors V and two spinors S1 and S2. This fixes a vector in spin(7) on the 7-sphere with spin(7)/G_2 ~ S^7. G_2 fixes a frame for the octonions or E_8 and acts as a gauge group. In addition

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

The complexified version of G_2 (G_2xC) is seen from the double covering so(O) ~ so(8). The inclusion of of the algebra g_2 into so(O) maps a 14 dim space into 28 dimensions of so(8).

There then seems to be some possible relationship between the G_2 ~ Aut(E_8) and the 28 cyclic group for 7 distinct exotic 7-spheres of Milnor. I also think this G_2 as a gauge action plays a possible role in the holographic reduction of 10-dim supergravity. The physics boosted to the "infinite momentum frame," or sometimes called the light cone condition or gauge, reduces the theory to so(9) ~ B_4. The G_2 plays a special role with the next complex group F_4, where F_4 = cent_{E8}(G_2), and the two groups are relatively abelian. The F_4 group gives

F_4/B_4:1 --- > spin(9) --- >F_{52/16} --- > OP^2

Which is sequence from the B_4 to the projective Fano plane.

Enough of the mathematics for now. It is curious that in your work you found only one particle. What I argue from physical grounds in one of my references is that a D-brane that is highly boosted will exhibit finer grained dynamics, as seen with Feynman's wee partons. This means the number of degrees of freedom on a D-brane increases. The highly boosted D-brane contains then holographic information that is becoming redundantly represented. It does not make physics sense for the number of real degrees of freedom to increase. Instead there is only the appearance of an increase. So I argue by ansatz that a particle exists as only one fundamental states, but that holography induces multiple configuration variables representations of that particle. It is then astounding that you have found a situation where there can only exist one of each type of particle.

Cheers LC

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  • Post #79

I would like to correct above mention summary of elementary particles:

Fermions:

12(6 quarks+3 leptons+3 neutrino).The Generations as a manifestation next cosmological epoch.

Bosons:

4(1 gluon+3 vector(2W+1Z)+1photon).Gluons hasn't color because Pauli Exclusion Principle not valid in 2D space.

See http://fqxi.org/community/forum/topic/1444

The Present time the Universe has:

Fermions;

3 stable(proton,electron,neutrino),1 neutron (non-stable)

Boson:

1 photon.No stable vector mesons,no free quarks,no free gluon.

See my old essay http://www.fqxi.org/community/forum/topic/946

The gauge boson for QCD, termed a gluon, have a chromo (color charge that is one color plus an anticolor. The QCD charges or colors are label red green blue, which in pairs from the root space of SU(3). This is a set of combinations of these three colors, or 8 in total. The root vectors are

v1 = (rb-bar br-bar).sqrt{2},

v2 = i(rb-bar - br-bar).sqrt{2},

v3 = (rg-bar gr-bar).sqrt{2},

v4 = i(rg-bar - gr-bar).sqrt{2},

v5 = (bg-bar gb-bar).sqrt{2},

v6 = i(bg-bar - gb-bar).sqrt{2},

v7 = (rr-bar - bb-bar)/sqrt{2}

v8 = (rr-bar gg-bar - 2bb-bar)/sqrt{2}

where r-bar means the complex conjugate of r times γ^0. Guons then have a pair of colors, which exchange those colors with the colors associated with quarks.

Cheers LC

Dear Lawrence,

interesting math, in particular the special thinks about the E_8 and G_2. I have to study the projective Fano plane, an interesting relation.

Yes, the appearance of a single particle was also a surprise for me. Maybe I have to understand more of your work.

Very interesting ideas, thanks a lot for your time.

Torsten

Hi Lawrence,

I read through your essay, but have not returned to it yet - to read for detail. But I've noted some of your comments, and wanted to add one or two of my own. First off; I saw your EJTP paper on "Counting States in Spacetime" which you posted on Rick Lockyer's essay site, and I note several points of overlap with the following paper by Frank Potter.

Our Mathematical Universe: I

Second; as I understand it octonions can indeed be represented as a system of 7 quaternions, but then the quaternion variables must be resolved in a definite order or sequence, or handled in a consistent way, as the effect of each term is cumulative (as with procedural steps or process stages). I think Rick uses the term ensemble multiplication.

But this is not quite the same as saying that the 'octonions are really a system of quaternions.' Maybe O is more fundamental than H, as Rick asserts. But perhaps saying octonions can be treated as an ordered or nested system of quaternions would work, though.

Regards,

Jonathan

    In a response to Jonathan Dickau I make greater mention of these matters. I also make a bit of a pitch for your essay.

    Cheers LC

    Hi Jonathan,

    Thanks for the paper. In looking at it I see many things which are in my notes and which I have in other papers and the book "Sphere Packing, Lattices and Codes" by Conway and Sloane.

    The graininess of spacetime is something which I think only comes about with the measurement of black hole states. As I indicated on Giovanni Amelino-Camelia's essay blog site there is an uncertainty principle,

    ΔrΔt ~ (2Għ)/c^4 = L^2_{Planck}/c.

    which is commensurate with equation 1 on Giovanni's paper . Spacetime appears grainy depending upon the type of measurement one performs. In the case of a quantum black hole a measurement involves spatial and temporal coordinates in a null congruency called an event horizon. If one makes another type of measurement spacetime is then as smooth as grease on an ice skating ring. The measurements of delay times for different wave lengths from very distant gamma ray burstars indicate that space is smooth down to a scale 10^{-50}cm --- far smaller than the Planck scale. This then ties in with some interesting work by Torsten Asselmeyer-Maluga on the role of exotic four dimensional space in quantum gravity. These are homeomorphic spaces that are not diffeomorphic. In 11 dimensions the 7-dimensional is dual to the 4-dimensional space. The exotic 7-spheres found by Milnor are simpler, with only 7-distinct non-diffeomorphic forms, rather than an infinite number.

    The octonions are a system of 7 quaternions. The exotic system in 7-dimensions I think might be connected to the automorphism G_2 in E_8 or SO(O). This would then connect with a physical meaning of octonions and nonassociativity in physics. The Polyakov path integral

    Z[A] = ∫δD[ψ]/diff(ψ) Ae^{-iS[ψ]}

    "mods out" diffeomorphism or equivalently gauge changes on a moduli. Yet with exotic spaces this definition becomes strange. However, if there are 7 quaternions which are related to each other by nonassociative products (ab)c - a(bc) =! 0, then the measure can maybe be realized according to associators δD[ψ]/diff(ψ).

    I discussed octonions a bit with Lockyer, but he seemed a bit put off. As I see it, and from some experience, presenting a gauge theory with nonassociative brackets and stuff falls pretty flat, I am not necessarily saying this is wrong, but doing that sort of work has a way of getting people to present their backside to you. I think the role of nonassociators is best advanced by other means so that in the future they may simply be too convincing to ignore.

    Cheers LC

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    • Post #85

    Lawrence,

    Sorry you were offended by my calling you out for posting on my essay blog without the common courtesy of having read the essay first. I only meant to inform you that you might possibly find some perspective on your question about how Octonion Algebra relates to physical reality since it was the thesis of my essay. Thanks for reading it later. I am curious about your characterization that it is just a gauge theory using associators. The Lorentz gauge mention was simply to demonstrate a point of commonality between 4D and Octonion presentations of Electrodynamics, that's it. Hardly a cornerstone of the presentation. I never once mentioned the associator, and frankly have never used non-associative brackets in any mathematical description. Octonion Algebra does indeed present a non-zero associator because it is a non-associative algebra. It MUST be so in order to be a normed composition algebra, hence a division algebra. Without this non-associativity and the remainder of O structure, it would be impossible for the algebraic invariances to match up the math to what we can measure or detect, and algebraic variances to give us clues on the math for what is hidden from us but none the less in play.

    Rick

    Thanks Lawrence,

    That nicely spells out where you are coming from. Glad you enjoyed the Potter paper, also. I've not looked at Giovanni's essay yet, but a quick read through of Torsten's paper has made it a 'must read' for the insights he shares. I am certainly not put off by your comments or Rick's and have found a lot of fascinating insights on the forum - even in the points of dispute.

    I am glad the back and forth has kept everybody thinking. More fun lies ahead!

    all the best,

    Jonathan

    Jonathan,

    First off I have not gotten around to reding your paper yet. It is taking me some time to get to them all.

    Torsten's work is pretty hard stuff. The differential geometry of exotic spheres runs pretty deep. I studied this for my masters in mathematics. It has been a while since I have thought much about that. It did occur to me that exotic spherse might have something to do with quantum gravity.

    I try to get as many people with their theoretical ideas and results together, because it is not likely that any of us b ourselves will come to the "big picture."

    Cheers LC

    In indicated to Giovanni Amelino-Camelia there should be some connection between the theory κ-Minkowski spacetimes and the boost system he advances with twistor theory. The connection to twistor theory is I think not hard to see. The boost operator P_μ that acts on [x_i, x_0] = ilx_i such that

    P_μ > [x_i, x_0] = il P_μ > x_i

    The coordinates (x_j, x_0) we write in spinor form

    x_j = σ_j^{aa'}ω_{aa'}

    x_0 = σ_0^{aa'}ω_{aa'},

    where ω_{aa'} = ξ_a ω_{a'} ξ_{a'}ω_a. This commutator has the form

    [x_i, x_0] = σ_j^{aa'}σ_0^{bb'}[ω_{aa'}, ω_{bb'}]

    = iC^{cc'}_{aa'bb'} σ_j^{aa'} σ_0^{bb'} ω_{aa'}

    = i|C| σ_j^{aa'}ω_{aa'}

    where the magnitude of the structure matrix is |C| = l. In general this may be written for

    x_j = σ_j^{aa'}ω_{aa'}

    x_0 = σ_0^{aa'}ω_{aa'} iq_{aa'}π^{aa'},

    where the commutator [ω_{aa'}, π^{bb'}] = iδ_a^bδ_{a'}^{b'} and the general form of the commutator is then

    [x_i, x_0] = i|C| σ_j^{aa'}ω_{aa'} iσ_j^{aa'}q_{bb'}[ω_{aa'}, π^{bb''}

    [x_i, x_0] = ilσ_j^{aa'}ω_{aa'} - σ_j^{aa'}q_{aa'}.

    The boost operation B = 1 a^l_jP^j on the commutator [x_i, x_0] is then equivalent to the commutation between spinors [ω_a, ω'_b] for ω'_b = ω_b iq_{bb'}π^{b'},

    [ω_a, ω'_b] = [ω_a, ω_b] iq_{bb'}[ω_a , π^{b'}]

    = C^c_{ab} ω_c iq_{ab}.

    This could be explored more deeply. Ed Witten demonstrated the "twistor revolution" in string theory. If twistors are connected to κ-Minkowski spacetime there might then be a link between string theory and LQG and other "edgelink" type of quantum gravity theories. This would be potentially interesting, for this might serve to correct the difficulties with each of these.

    Cheers LC

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    • Post #89

    Lawrence,

    You wrote: "Einstein changed Newton's laws by adjusting the first and third laws, motivated by the locality of electromagnetic fields predicted by Maxwell's equations."

    Einstein did not adjust anything - he just introduced two postulates the second of which was false. In 1887 the Michelson-Morley experiment had refuted the light postulate and had confirmed the variable speed of light predicted by Newton's emission theory of light. It is time for you, Lawrence, to stop claiming that Banesh Hoffmann, John Norton and John Stachel are wrong:

    http://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/dp/0486406768

    "Relativity and Its Roots" By Banesh Hoffmann: "Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether."

    http://www.aip.org/history/einstein/essay-einstein-relativity.htm

    John Stachel: "An emission theory is perfectly compatible with the relativity principle. Thus, the M-M experiment presented no problem; nor is stellar abberration difficult to explain on this basis."

    http://www.philoscience.unibe.ch/documents/kursarchiv/SS07/Norton.pdf

    John Norton: "These efforts were long misled by an exaggeration of the importance of one experiment, the Michelson-Morley experiment, even though Einstein later had trouble recalling if he even knew of the experiment prior to his 1905 paper. This one experiment, in isolation, has little force. Its null result happened to be fully compatible with Newton's own emission theory of light. Located in the context of late 19th century electrodynamics when ether-based, wave theories of light predominated, however, it presented a serious problem that exercised the greatest theoretician of the day."

    http://philsci-archive.pitt.edu/1743/2/Norton.pdf

    John Norton: "In addition to his work as editor of the Einstein papers in finding source material, Stachel assembled the many small clues that reveal Einstein's serious consideration of an emission theory of light; and he gave us the crucial insight that Einstein regarded the Michelson-Morley experiment as evidence for the principle of relativity, whereas later writers almost universally use it as support for the light postulate of special relativity. Even today, this point needs emphasis. The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."

    Pentcho Valev pvalev@yahoo.com

    I am not sure why you decided to make your life's work to discredit relativity. You keep posting the same thing over and over, with the same references.

    The invariance of the interval, equivalently the constancy of the speed of light, means in addition to the three rotations of space there are three Lorentz boosts. The physics of this has been tested literally thousands of times in many different ways. The empirical support for relativity is simply overwhelming. You are not going to find many people here who are well grounded in physics who agree with you.

    Cheers LC

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      • Post #91

      Hello Lawrence and Mr.Danoyan,

      You know Lawrence.When I am not parano, I see the convergences with strings and the 3D.

      So I am discussing,:) The light permits to compse all the colors.The angles indeed are relevant.I saw this idea from Mr Dicarlo on the hread of Mr Barbour.

      If the angles and the volumes are inserted with the correct quantum finite number, it becomes very relevant for our correct 3D architecture, the sphere and its spheres. The combinations are very numerous (rotations spinal,rotations orbital,volumes, serie finite !!!,linear velocity, sense of rotation differenciating m and hv.It permits to unify the gravitation with the 3 other foundamental forces.).

      Lawrence I am persuaded that we can create a 3D holographic Sphere and its spheres, cosmologic and quantic. If we consider that the space and the mass and the light are the same at a kind of zero absolute.So if the quantum number is finite and precise.So it implies a real relevance when we insert the rotations and motions more the volumes and the angles. The puzzle is simple and complex. It is relevant to consider that the cosmological number is the same. This serie is so universal. The fractalization in a pure road of primes number seems very relevant with the main central sphere, the most important volume, the 1.

      The QCD can be optimized in fact Lawrence. Perhaps that the volumes are still very relevant considering the main light from the main central sphere.

      I think that the oscillations can be correlated with rotations and the QM. I see the light turning at the maximum but in the opposite sense than this gravitation in evolution. If the space is also an quantum entanglement.So it is interesting to see its velocities of rotations.and the sense also.In the logic the lattices between spheres disappear in the perfect contact.And if the main central sphere is the most important volume.So it is interesting to see how this space can be checked.In my line of reasoning, the space between sphere can imply so a contraction of this space, like witha vaccuum, but of course two points are necessary, an arrival and a departure of course.It is relevant because we can decrease the space between cosmological spheres.If the arrival point has an other solution ,it is relevant. The second relevance of this line of reasoning is that the mass can be changed in light, so we move at c.The third relevance is that we can decrease with my model,the internal clocks, so the rotations of sphers, so the duration. Now if we check these 3 quantum systems.So we can 1 decrease the space between two spheres.2 we can go at c.(we reencode the mass at the arrival point) and 3 we can decrease our internal duration, so we utilize less of time during the travel. It is the principle of future teleportation. It is there that the volumes of spheres are essential for the stability of informations during the reencoding.

      It is very relevant at my humble opinion.

      Best Regards

      • [deleted]

      • Post #92

      Lawrence,

      Roger Schlafly wrote in his site:

      "Pentcho, you are right that the emission theory was the only known explanation [of the null result of the Michelson-Morley experiment] in 1887..."

      Is Roger right? Also, Lawrence, you used to claim that John Norton is wrong when he says that:

      http://philsci-archive.pitt.edu/1743/2/Norton.pdf

      John Norton: "The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."

      Do you still believe Norton is wrong, Lawrence?

      Pentcho Valev

      I have not yet read Schlafly's essay or read the posts on his blog. I am not particularly interested in revisiting old stuff like this. Whether one can interpret the M-M experiment in different ways is of little interest to me. Lorentz interpreted the result as due to a length contraction that nullified the effect of the putative aether. Einstein was apparently not aware of the M-M experiment at all. Which ever is the case with interpreting the M-M experiment it is not relevant. Special relativity has been tested by many dozens of other types of experiments repeated many thousands of times. I am not sure why anybody would want to take up the cause of trying to overturn relativity this way. There were people up to the early 19th century who wanted to overturn Newton as well.

      Cheers LC

        • [deleted]

        • Post #94

        The fact is that, in 1887, Newton's emission theory stating that the speed of light varies in accordance with the equation c'=c+v (v is the speed of the light source relative to the observer) was the ONLY existing theory capable of explaining the null result of the Michelson-Morley experiment.

        You find this fact unimportant and accordingly occupy the top of the community rating list. I find this fact extremely important and am at the bottom. Simple isn't it?

        Pentcho Valev