Unitarity, Locality and Spacetime Geometry: Foundations That Are Not Foundations by Lawrence B Crowell

Chris,

There is more to this, which I could not break out due to length limitations. The gauge symmetries are Yangians, or enveloping algebras. These have a duality, where the gauge symmetry in one representation is dual to another without spacetime configuration variables.

Eckard,

I argue for a massive reduction in the number of degrees of freedom. In fact if the universe has quantum states given by E_8xE_8, it means the universe has only 496 fundamental degrees of freedom, or in its supersymmetric extension 512 = 2^8.. In the Leech lattice Λ_{24}there are 4096 weights, due to the theta function representation over 3 E_8 groups, and Λ_{24} is the automorphism of the Conway group Co_1 with rank 8,315,553,613,086,720,000. The full automorphism over the Fischer-Griess group is of rank 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, which is huge. Yet in this total extended picture the number of real degrees of freedom is only 4096.

The actual number of elementary particles is then very small, but they have multiple representations in configuration variables. The configuration variables are a system of entanglements, or holographic projections, which give the appearance of a large number of particles.

I don't think the fundamental issues with physics lie with the foundations of mathematics. I might be wrong of course, but I really do not think mathematics has been on some fools errand for the last 150 years or more.

Cheers LC

I read your paper a week ago with the idea of reading it again with greater attention to detail and your references. I just reread your paper, but unfortunately not in great detail, so I have yet to dig into your paper at great length. I have to confess I have read a pretty small minority of the paper on this essay website.

I went through the Atiyah, Donaldson, Freedman work on exotic four manifolds some years ago. I thought there were certain prospects for a quantum description from this. The difficulty I see with this is that manifolds which are homeomorphic but not diffeomorphic leave a big question on how one defines a Polyakov measure in a path integral

∫(D[g, ψ]/diff(g, ψ)) exp(iS)

where one "mods out" diffeomorphisms or gauge dependencies. The thought occurred to me that in 11-dimensions the dual to four dimensional spacetime is a 7-dimensional space. In that case there are these 28 distinct differentiable structures Milnor demonstrated to exist. I think by doing this the really tough problem with Donaldson's theorem might be transformed to a much more tractable problem. The Cartan matrix for the E_8 is the same as the matrix associated with Donaldson's theorem. The 28 differential structures of the 7-manifold I have pondered have some relationship to the complex G_2, the automorphism of E_8.

Physically spacetime will never be observed to have a foamy or grainy structure. The reason is simple. If I am right there are only one of each type of elementary particle. The multiplicity of elementary particles exists because they are holographic projections onto configuration variables. The configuration variables are simply a measure of how an electron here is entangled with another "there," whether there means an electron in a nearby transmission line, or the degenerate gas in a white dwarf or anywhere in the universe. The same holds for a photon, down quark and so forth. So any UV particle, say a photon, it may "feel" noncommutative geometry more than an IR photon, but due to their entanglement this effect is cancelled out. In effect the extreme IR boson from Hawking-Gibbon radiation is equivalent to an extreme UV boson, and so the apparent fluctuations at the UV scale are removed.

The physical effect of the emergence I propose is with quantum information exterior and interior to a black hole. There exists a duality between the two data sets, and if we were to develop a Planck energy accelerator (which we will not do) then scattering amplitudes should reflect this fact. We do however have a possible window into this with gravity as the "square" of gauge theory. Gluon scattering amplitudes should carry this information as well. This may then be accessible to LHC types of experiments.

I will read your paper in greater detail in the near future, for it is one of the better ones I have seen submitted. It might take me a week or so to make more detailed comments.

Cheers LC

Dear Giovanni,

I just started reading Relative locality in a quantum spacetime and the pregeometry of _-Minkowski http://arxiv.org/pdf/1206.3805v1.pd. You seem to be pointing to a similar end. Noncommutative geometry and Hopf algebras are a main tool in the work with Yangians. I will write more when I complete reading your paper.

Equation 1 is interesting, for it proposes a noncommutative relationship between time and the spatial coordinates. This in my opinion harkens back to an old argument by Bohr. In 1930 there was a famous Solvay conference where Einstein and Bohr sparred over the reality of quantum mechanics. Einstein was convinced of reality and locality and argued staunchly for an incompleteness of quantum mechanics. Quantum theory could only be made complete if there are some hidden variables that underlay the probabilistic, nonlocal quirky aspects of quantum mechanics. At the 1930 Solvay conference Einstein proposed an interesting thought experiment. Einstein considered a device which consisted of a box with a door in one of its walls controlled by a clock. The box contains radiation, similar to a high-Q cavity in laser optics. The door opens for some brief period of time t, which is known to the experimenter. The loss of one photon with energy E = ħω reduces the mass of the box-clock system by m = E/c^2, which is weighed. Einstein argued that knowledge of t and the change in weight provides an arbitrarily accurate measurement of both energy and time which may violate the Heisenberg uncertainty principle ΔEΔt ~ ħ.

Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock's new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is 1 - 2GM/rc^2, where a displacement by some δr means the change in the metric term is ~ (GM/c^2r^2)δr. Hence the clock's time intervals T is measured to change by a factor

T --> T sqrt{(1 - 2GM/c^2)δr/r^2} ~ T(1 - GMδr/r^2c^2),

so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the Δp ~ ħΔr < TgΔm, where g = GM/r^2. Similarly the uncertainty in time is found as Δ T = (Tg/c^2)δr. From this ΔT > ħ/Δmc^2 is obtained and the Heisenberg uncertainty relation ΔTΔE > ħ. This demands a Fourier transformation between position and momentum, as well as time and energy.

Consider an example with the Schwarzschild metric terms. The metric change is then ~ 1x10^{-12}m^{-1}δr, which for δr = 10^{-3}m is around 10^{-15}. Thus for a open door time interval of 10^{-2}sec, the time uncertainty is around Δ t ~ 10^{-17}sec. The uncertainty in the energy is further ħΔω, where by Fourier reasoning Δω ~ 10^{17}. Hence the Heisenberg uncertainty is ΔEΔt ~ ħ.

This argument by Bohr is one of those things which I find myself re-reading. This argument by Bohr is in my opinion on of these spectacular brilliant events in physics.

This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein's box as a black hole with mass m. The quantum periodicity of this black hole is given by some multiple of Planck masses. For a black hole of integer number n of Planck masses the time it takes a photon to travel across the event horizon is t ~ Gm/c^3 = nT_p, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is

ΔT ~ Tg/c(δr - GM/c^2),

as measured by a distant observer. Similarly the change in the energy is given by E_2/E_1 = sqrt{(1 - 2M/r_1)/(1 - 2M/r_2)}, which gives an energy uncertainty of

ΔE ~ (ħ/T_1)g/c^2(δr - GM/c^2)^{-1}.

Consequently the Heisenberg uncertainty principle still holds ΔEΔT ~ ħ. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies Δω ~ sqrt{c^5/Għ}, which is the Planck frequency.

The uncertainty in the ΔE ~ ħ/Δ t does have a funny situation, where if the energy is Δ E is larger than the Planck mass there is the occurrence of an event horizon. The horizon has a radius R ~ 2GΔE/c^4, which is the uncertainty in the radial position R = Δr associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have

ΔrΔt ~ (2Għ)/c^4 = L^2_{Planck}/c.

So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.

Cheers LC

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 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

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

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