It From Bit is Undecidable by Lawrence B Crowell

The situation pertains when there is an uncertainty fluctuation of event horizons. This is a possible window into quantum gravity. The physical states in the S-matrix channel are entangled states, with entanglements across the horizon. If the event horizon is classical then everything is nicely associative. A fluctuation in the horizon results in this uncertainty in associating states interior and exterior to the black hole/

The graviton has quantum numbers equivalent to an entangled pair of QCD gluons, or a bi-gluon system, that is neutral with respect to color charge. In that sense this form of S-matrix does connect in formalism with the old S-matrix theory or the so called "bootstrap." In fact string theory is really that unitary bootstrap theory in another guise.

I hope this answers your question.

Cheers LC

I think spacetime is continuous or granular depending upon which sort of measurement you make. The Planck scale is just the minimal length scale which can contain a qubit of information. Space or spacetime on a smaller scale can not contain a qubit with any certainty. An experiment that involves the transmission of information over a vast time distance has no uncertainty with respect to the existence of it information or qubit content due to quantum gravity. As a result spacetime appears perfectly smooth. An experiment on the other hand which attempts to localize a very small region of space, say at high energy, will then register a different perspective on spacetime. In that setting it may appear highly choatic and granular.

The predictions of string theory involve cosmology and black holes. In a stringy universe there are some observable consequences, in particular with respect to multiple vacuum nucleations or bubbles that we call a "universe." Signatures of this may be imprinted in the fine grain detail in the CMB anisotropy. The graviton has quantum numbers equivalent to an entangled pair of QCD gluons, or a bi-gluon system, that is neutral with respect to color charge. This means that plasmas of gluons can be formally equivalent to a black hole. In a funny way it is a black hole. So holographic physics should in principle be testable in heavy ion collisions. These are admittedly rather indirect or oblique sorts of tests, and even if confirmations of string predictions are found the whole string-M/theory enterprise is likely to remain on rather unsteady ground.

The real problem is that we appear to be reaching a sort of ceiling of technological possibilities. One of these is of course directly testing Planck scale physics. Other limits seem to be with high density energy sources such as fusion, that seems as remote as ever, the future of manned spacetravel appears in trouble, and there are slow downs with Moore's law with computers. We also need to consider the pile up of huge problems like global warming. We may be approaching the end of the foundations of physics because of our limitations.

Cheers LC

I am slowly making my way through the essays. I have not yet gotten to yours. Are you proposing this experiment in your essay? I will try to get to yours in the next few days. I can only read one of these papers in a day, so it is a bit slow going.

Cheers LC

I just pulled up your essay and gave it a quick reading. It is not terribly long. It appears that you are asking whether certain properties are physical or geometrical. At this time I don't have an assessment of your work. I will need to read a bit more.

Cheers LC

The relationship between events in the universe and causality was noted by David Hume to not have a strict logical relationship. As I indicate in a footnote Godel's second theorem according to modal operations is a form of Hume's argument about causality =/= logic.

I sort of have to make this a bit brief due to other things I have to attend to now. I will say that the issue of clocks with a spatial separation and synchronization is a subject of considerable interest. This involves Cauchy data on spatial surfaces and how to integrate the Einstein field equations.

Cheers LC

Hi Peter,

I am true to my usual trend falling behind in reading papers on this list. I just did a scan of your paper. You do reference Godel's paper on prepositional logic. The main point of my work is that any Lambda-calculus or Turing machine approach to the structure of a causal system is bound to be incomplete. I do get a sense in reading the first couple of pages of your essay that you are leading into something similar.

I probably will not get to reading papers much until this weekend. I'll post my observations when I do. As I said I am falling behind, and I notice another lot of papers showed up on the list today.

Cheers LC

Hi Torsten,

I remember reading an article back in the 1990s about how the classification of exotic R^4s was not enumerable, which had connections to Godel's theorem.

The exotic R4 structure has its origin in the Casson handles as pointed out by Freeman. A thickened disk D^2 --- > D^2xR^2 can produce various structures, which by the self duality of four dimensions leads to these strange conclusions. In scanning your paper I see you invoke Casson handles. The number of such structures by h-cobordism turns out to be infinite, which as I say above, I remember this to be nonenumerable. This result was proven by one of the big mavens in this area, Atiyah, Freeman, Taubes, ... ?

The one element of this is that the e8 Cartan matrix as the eigenvalued system for an E8 manifold, an exotic R4. It has been a while since I have studied these matters, but as I remember this tells us how to tie 3-manifolds in 7 dimensions in the Hopf fibration S^3 --- > S^7 --- > S^4. The dual to this structure are 4-manifolds. The 7 manifold this knotting is performed is in the heterotic S^7 --- > S^{15} ---- > S^8, and the e8 Cartan matrix gives the eigenvalues for the 7-space.

The interesting thing about the E8 is that the 8-dimensional space is equivalent to the group in a lattice construction; the root-weight space is ~ the space itself. The E8 manifolds of Freeman are I think embedded in the set of possible 8-spaces. This suggests a duality between the smooth manifold in 4-dim and a discrete or noncommutative manifold in a quantum sense.

Physically this seems evident from data obtained so far. Measurements of the dispersion of light from extremely distant sources invalidate a discrete structure to spacetime. This tells us that a measurement of spacetime structure by measurement of photons that traverse a large distance give no signature of grainy structure. Yet a lattice perspective of spacetime with the Grosset polytope and the 120-polytope of quaternions in 4-dim would suggest a noncommutative geometry. However, if the lattice is equivalent to the space, then this smooth structure is dual to a grainy picture of spacetime. This structure should emerge in an extremely high energy experiment that probes small regions, rather than testing across vast distances.

Cheers LC

The icosian or 120-cell has two quaternions with length (1/2)(1 +/- sqrt{5}) where the plus one has length 1.618..., which is the golden mean. In fact these quaternions define something called the golden field in a Galois ring. This is related to the Fibonacci sequence.

Cheers LC

You state that the quantum universe is unknowable. I would say there is some limit to how much we can know about it. This limit is due to the cut-off in measurable physics at the Planck or string scale. As one considers scales beneath the string length and then beyond the Planck scale spacetime folds up onto itself in such ways that quantum fluctuations result in closed timelike curves and things that are "paradoxical." This is probably a domain that is fundamentally unobservable.

A rather simple argument can illustrate how this cut-off on the extremely small scale manifests itself on a larger and I think potentially a cosmological scale. The amplitude computed in a path integral is a summation over 3-metrics g

Z = ∫D[g]e^{iS(g)},

where a standard method is to Wick rotate the phase e^{iS(g)} --- > e^{S(g)}. This is a way to get attenuation of high frequency modes, and it is a "bit of a cheat," though at the end one must recover the i = sqrt{-1} and "undo the damage" for the most part. This phase then becomes e^{-GM^2}, which illustrates how the action and entropy are interchangeable. The integral measure is the size of the phase space of the system ~ exp(S). The amplitude is then on the order

Z ~ e^{S}e^{-S} = 1.

This holds universally no matter how large the black hole is. A black hole is a sort of theoretical laboratory for the universe at large, where the universe has a cosmological horizon at r = sqrt{3/Λ}. The implication is there is a limit to what we can possibly observe about the foundations of the universe, which probably touch on the amount of quantum information available with respect to quantum gravity/cosmology.

We have of course two different quantities. The volume of the phase space is equal to the exponential of the entanglement entropy of the system, while the e^{-S} is exponential of the thermal entropy. The amount of information is S_{th} - S_{ent}, so this amplitude is not going to be exactly one. There is some "kernel" to the black hole which corresponds to an elementary unit of information. This means that quantum information is ultimately conserved, and that the number of degrees of freedom for a black hole in spacetime is a constant, regardless of the size of the black hole. It also means that the universe as a whole (here thinking of a toy universe with just a black hole) has a finite limit to its domain of observability.

The application of modal logic is a sort of "boilerplate" to examine causality and locality. Further, my considerations are quantum field theory instead of quantum mechanics. QFT involves operators which act on a Fock space to describe quantum states or QM. So this is an underlying physics. Your work seems to illustrate the "traces" of this sort of underlying QFT in matters of CHSH nonlocality.

The paper by Dzhunushaliev looks interesting. Your work with G2(2), which I am presuming is a split form of G2, focuses on the automorphism of the E8 or octonions. The F4 is a stabilizer of E8 (constant under G2 action). The automorphism on E8 defines an invariant interval on C^4 as a twistor space. This in higher forms, say on the magic square can construct generalizations on H^2 and then O, within the CxO, HxO and OxO hierarchy of the magic square. Generalizing the H^2 twistor space to octonions gives O^2, and scattering amplitudes are functions on copies of OP^1, subject to SL(2,O) = Spin(9,1) transformations. Embedding O^2 in O^3, gives OP^1 as a line in OP^2 and SL(2,O)=Spin(9,1) becomes the subgroup of SL(3,O) = E6(-26), consisting of transformations (collineations) that fix a point in OP^2.

The J^3(O) or O^3 has connections to SO(2,1) and I think this rather erudite stuff connects to anyons on a 2-space plus time constructions. All of this I think is some sort of superselection rule on this sort of theory. The question might then be whether your idea about [0, 1, ∞] as the dessin d'enfant is some sort of category theory on the superselection rules according to curves on Ricci flat spaces, such as (K_3)^2 for E_6 twistor theory.

Cheers LC