Lawrence,

Finally got to yours. Once you accepted Bohr axioms and followed the 'more standard' path I agree your analysis and conclusions are rather inevitable, not admitting EPR. Possibly rather couched in algebra for an essay, but certainly comprehensively analysed.

You know, as did Bell, I consider one of Bohrs assumption wrong, identifying pair morphology which omits Poincares 2nd momentum so leading to mysterious 'quantum spin', ironically caused by trying to avoid any assumptions! QM would then reduce to What Wheeler believed; "..built from some undecidability of an elementary system."

Thanks for that quote which I hadn't seen. I also didn't know Wheeler had been thrown out of Godel's office twice! But not in classical reality I assume.

That undecidability by the way is simply the question as you stand on Earths equator; "Are you spinning clockwise or anticlockwise"? Or orthogonally at the pole; "Are you moving left or right"? Perhaps both as confounding to logic as QM itself!?

Very best

Peter

Thanks Lawrence,

I think 'stance' is about 'beliefs', which I eschew. Let me ask you this;

Let's say Bohr came up with TWO options for interpreting QM's data set;

FIRSTLY a classical mechanistic sequence of orthogonal 'curl' AND linear OAM momenta with rotational vector additions, orthogonal, and uncertain at the changeovers.

SECONDLY what we have now; one as a weird 'quantum spin' state, so a string of other irrational or non causal effects EPR rejected.

Which would any intelligent physicist have been likely to go for? Bear in mind John Wheeler anticipated exactly the first option & John Bell firmly agreed.

I simply identify the sequence achieving that. It seems to me only embedded flawed beliefs, following the flock or poor understanding cause most to chose the 2nd.

Do you REALLY believe that's not possibly the case?

Peter

Quantum mechanics is a wave mechanics that is as I say perfectly deterministic. It is once there is a so-called collapse, or a transfer of quantum phase out of the system on a time scale t

Carrot sign cut off post

Quantum mechanics is a wave mechanics that is as I say perfectly deterministic. It is once there is a so-called collapse, or a transfer of quantum phase out of the system on a time scale t ltlt 1/ОЅ, for ОЅ the fundamental frequency of the quantum system, that things get a bit odd. There have been various attempts to rescue this situation, where hidden variables are one putative approach.

As my post above indicates QM as an L^2 system is dual to general relativity, which with its metric structure is also L^2 measure. Another dual system is then L^1, which is pure classical probability theory and lim_{qв†'в€ћ}L^q systems which are completely classical deterministic systems. These can be ordinary classical mechanics or a Turing machine or some other type of system. This is one thing that makes gravitation as a classical system different from a standard classical system. A part of that is that time, which is conjugate to energy in a Fourier sense, is not treated as a coordinate variable in standard classical mechanics. Quantum mechanics also does not treat time as an operator. If it did then energy, as the generator of time, would not have discrete spectra and would not be bounded below. So, there is really a rather unknown issue involved with the nature of time here.

The standard reduction of a wave function is one where from a probability perspective the quantum amplitude probabilities are reduced to a classical probability with L^2 в†' L^1. Then correspondingly the physical properties of a quantum state that is stable under environment quantum noise means L^2 в†' lim_{qв†'в€ћ}L^q, so to give classical systems. This is a form of Zurek's einselection of quantum states. The problem is that quantum L^2 systems are unitary while classical systems are symplectic. The only instance where these two happen concurrently is for a two-state system. The overlap of such states has both a Riemannian metric geometry and the symplectic geometry of classical mechanics. The Riemannian geometry corresponds to the Fubini-Study metric of quantum mechanics. For a large number of two-state systems in an overcomplete state, or a form of laser coherent state, the condensate of so many states over-rides this.

Spacetime is likely an epiphenomenology of large N entanglement of states, in a way similar to coherent laser states of light. This may be a bridge between QM with spacetime and the above rotation.

I think a part of this has to do with topological distinctions between different quantum phase structures or entanglements. These topologically distinct quantum phases can't be evolved into each other by unitary evolution, such as the Schrödinger equation, and yet descriptions of systems with wave collapse violate this. The main point I advance is these different quantum phases have different p-adic realizations of their fractal IST sets. The result of Matiyasevich illustrates how solutions to different p-adic elements of a set are local and not extendable. This is equivalent to saying there is no global method for solving all Diophantine equations. This knocks down Hilbert's 10th problem.

The net effect then is the outcome of a quantum measurement has not causal or what might be called computable basis. Quantum outcomes occur for no underlying reason at all.

Now, this might be a bit odd. It not just something that would rankle Einstein, but it also means that all quantum interpretations are not determined. Quantum interpretations are a set of creative ideas meant to entertain the human mind and not something intrinsic to nature. This is whether one works with many worlds, or Bohr's Copenhagen, or Bohm beables, Qubism and the rest of these. In effect QM faces us with the existentialist idea of ontological incompleteness

Lawrence. All you did there was start from Bohr's hidden assumptions about OAM, in which case that WHOLE non causal system can't be avoided.

Take 3 steps back to my 'FIRSTLY' scenario above, where we have the TWO CLASSICAL momenta Poincare found making up OAM.

Quantum spin would then never have been required!. A&B reversing setting simply changes their OWN findings, so we can sweep ALL the nonsensical interpretations away! none of it was ever needed. The 'wavefunction' is modified at each interaction, as Zeilinger confirs experimentally. The experimental data is rationally explained, with 'uncertainty' merely of rotational direction at the equator, and linear momentum at the poles.

Is that really entirely invisible to you?

Peter

Things just are not this way. Quantum spin is something forced on us if nothing else. The attempts to find an electric dipole moment have of late found the electron has a radial electric field down to 10^{-29}cm. This is close to the Planck scale, 1.6ﾃ--10^{-33}cm, which is the smallest scale one can identify a unit of information. However, before going to that scale the hypothesis that the electron is a point-like particle without a classical radius dates back to the late 1920s. There is the classical radius of an electron, which can be computed fairly easily, but it was found the electron did not have this scale dimension.

Your argumentation will doubtless continue, which is seen by people who argue for various alt-science. Physics involved with showing some underlying classical framework to quantum mechanics is about as dead as a doornail. I see no prospects for any realistic physics with this sort of thing. This continued argumentation is what might be called "shaving down a point," which is the endless attempt to find some tiny loophole in an otherwise reasonable argument. This was pioneered in many ways by the creationists who argue for a biblical account for the nature of biology.

Lawrence Crowell re-uploaded the file Crowell_fqxi_2020.pdf for the essay entitled "Undecidability of States and Epistemic Horizons as Quantum Gravity" on 2020-04-04 22:00:34 UTC.

    20 days later

    I am not sure whether it is worth trying here, but I cured a word tangle in the abstract, I also include below some additional study and work that goes along with this.

    I recently re-read the paper The Page curve of Hawking radiation from semiclassical geometry by Almheiri, Mahajan1, Maldacena, and Zhao arXiv:1908.10996v1. There is no paper of late that I have read so many times as this. This paper relies heavily upon the idea of quantum extremal surfaces and this paper Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, Engelhardt and Wall arXiv:1408.3203v3 . This physics is the relationship between spacetime and quantum horizons. The metric g_{ab} = g0_{ab} + g^{Д§^ВЅ}_{ab} + g^Д§_{ab} + .... is expanded in units of the Planck length в„"_p = в€љGД§/c^3 в†' O(Д§^ВЅ). This is the logic of MTW with g = g0 + в„"_p/L + ... . This leads to a description of quantum states with an entanglement between exterior and interior states.

    It occurred to me this could be seen in some simple ways. For the Kerr-Newman metric

    ds^2 = (1 - 2m/r + Q^2/r^2)dt^2 - (1 - 2m/r + Q^2/r^2)^{-1}dr^2 - r^2dО©^2

    The near horizon condition for an accelerated observer above the horizon is found to be

    ds^2 ≈ (r/m)^2dt^2 - (m/r)^{-2}dr^2 - r^2dΩ^2

    and curiously if one derives a similar metric for the near singularity condition r = 0 the metric is

    ds^2 ≈ (Q/r)^2dt^2 - (Q/r)^{-2}dr^2 - r^2dΩ^2.

    The two metrics are interestingly dual with respect to mass and the charge, or equivalently with angular momentum. Both metrics are AdS_2Г--S^2 or the anti-de Sitter spacetime in 2 dimensions, and in a BPS setting indicate a duality between the mass variable and quantum numbers associated with gauge fields. The extremal black hole with m = Q is the extremal black hole, and Carroll, Johnson and Randall showed this condition leads to the discontinuous map of the spacelike trapping region to AdS_2Г--S^2.

    This dualism suggests that quantum states in the deep interior of the BH, on so called islands, near the singularity are dual or even entangled to states near the horizon or in the exterior. Conformal patches on the two AdS_2 spacetimes may correspond to each other, and the corresponding CFT_1, with AdS_2 ≈ CFT_1, are dual chains similar to a Haldane chain. The boundary of AdS_n is a timelike region, which means AdS space is not globally hyperbolic due to the existence of a timelike boundary at infinity. However, if there are boundary conditions at infinity there are then causal properties are still fine assuming that there are boundary conditions at infinity Hence a conformal patch in AdS_3 may have a boundary with CFT_2 that shares boundary conditions equivalent to the AdS_2. One way to think of this is that CFT_2 can define gauge-like gravity that has the same DoFs of bulk gravity in AdS_2. In this way the AMMZ argument for a higher dimensional space ties the two CFT_1 chains together in an entanglement.

    The AdS_n black hole correspondence identifies holographic content of the event horizon with the CFT_{n-1} corresponding to the AdS_n the BH is embedding within. The AdS_3 corresponds to the BTZ black hole. The CFT_2 in a gauge-like gravity theory can describe quantum gravitation for the AdS_2. The area law S = /4в„"_p^2 + O(Д§) quantum corrections is then "constructed" through this correspondence between AdS_2 and AdS_3. The "area" in the AdS_2 case is a 0-dimensional, and in the AdS_3 as with the BTZ BH it is a circle. Hence for the AdS_2 the theory is a scalar field theory.

    The construction is then a form of flag manifold. The AdS_n = O(n,2)/O(n,1), where O(n,2) is the isometry group of the space. The AdS spacetime is a Stiefel manifold and a form of flag manifold with

    F^N_{d1,d2,...,dm}(в„‚) = SL(N, в„‚)/P^N_{d1,d2,...,dm}= U(N)/[U(k1)Г--U(k2)Г--...Г--U(kr)Г--U(kr+1)],

    for k1 = d1, kr+1 = N, k_{n+1} = d_{n+1} - d_n. The real valued form of this flag manifold is

    F^N_{d1,d2,...,dm}(в„ќ) = SL(N, в„ќ)/P^N_{d1,d2,...,dm}= O(N)/[O(k1)Г--O(k2)Г--...Г--O(kr)Г--O(kr+1)].

    It is clear that AdS_n = F^{n,2}_{n,1}(в„ќ). The flag F^N_1(в„‚) is в„‚P^{N-1} = U(N)/[U(1)Г--U(N-1)]. Similarly, the real valued version is F^N_1(в„ќ) = в„ќP^{N-1} = O(N)/[O(1)Г--O(N-1)], where O(1) is a trivial group and в„ќP^{N-1} = O(N)/O(N-1) , where S^2 = O(3)/O(2). In the split form we then have

    в„ќP^{N-1,2} = O(N,2)/O(N,1),

    And AdS_n = в„ќP^{n-1,2}. The flag manifold F^4_1(в„‚) = в„‚P^3 = U(4)/[U(1)Г--U(3)] gives twistor space. In the split form

    F^{2,2}_1(в„‚) = в„‚P^{2,1} = U(2,2)/[U(1)Г--U(2,1)],

    this is according to the isometry group of AdS_5.

    The flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)Г--U(2)] is the Grassmanian space G_{4,2}(в„‚). This is identified as a spacetime, where dS and AdS spacetimes are defined according to a line element

    A = t^2 В± u^2 - x^2 - y^2 - z^2,

    with u^2 de Sitter and -a^2 anti-de Sitter. The light cone for A = 0 separates the AdS and dS. The flag F^{2,2}_{2,1} = U(2,2)/[U(1)Г--U(1)] with the bi-fibration

    This is a follow on with this. I am not sure if it is worth pressing on with this, but these are additional studies and developments.

    I recently re-read the paper The Page curve of Hawking radiation from semiclassical geometry by Almheiri, Mahajan1, Maldacena, and Zhao arXiv:1908.10996v1. There is no paper of late that I have read so many times as this. This paper relies heavily upon the idea of quantum extremal surfaces and this paper Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, Engelhardt and Wall arXiv:1408.3203v3 . This physics is the relationship between spacetime and quantum horizons. The metric g_{ab} = g0_{ab} + g^{Д§^ВЅ}_{ab} + g^Д§_{ab} + .... is expanded in units of the Planck length в„"_p = в€љGД§/c^3 в†' O(Д§^ВЅ). This is the logic of MTW with g = g0 + в„"_p/L + ... . This leads to a description of quantum states with an entanglement between exterior and interior states.

    It occurred to me this could be seen in some simple ways. For the Kerr-Newman metric

    ds^2 = (1 - 2m/r + Q^2/r^2)dt^2 - (1 - 2m/r + Q^2/r^2)^{-1}dr^2 - r^2dО©^2

    The near horizon condition for an accelerated observer above the horizon is found to be

    ds^2 ≈ (r/m)^2dt^2 - (m/r)^{-2}dr^2 - r^2dΩ^2

    and curiously if one derives a similar metric for the near singularity condition r = 0 the metric is

    ds^2 ≈ (Q/r)^2dt^2 - (Q/r)^{-2}dr^2 - r^2dΩ^2.

    The two metrics are interestingly dual with respect to mass and the charge, or equivalently with angular momentum. Both metrics are AdS_2Г--S^2 or the anti-de Sitter spacetime in 2 dimensions, and in a BPS setting indicate a duality between the mass variable and quantum numbers associated with gauge fields. The extremal black hole with m = Q is the extremal black hole, and Carroll, Johnson and Randall showed this condition leads to the discontinuous map of the spacelike trapping region to AdS_2Г--S^2.

    This dualism suggests that quantum states in the deep interior of the BH, on so called islands, near the singularity are dual or even entangled to states near the horizon or in the exterior. Conformal patches on the two AdS_2 spacetimes may correspond to each other, and the corresponding CFT_1, with AdS_2 ≈ CFT_1, are dual chains similar to a Haldane chain. The boundary of AdS_n is a timelike region, which means AdS space is not globally hyperbolic due to the existence of a timelike boundary at infinity. However, if there are boundary conditions at infinity there are then causal properties are still fine assuming that there are boundary conditions at infinity Hence a conformal patch in AdS_3 may have a boundary with CFT_2 that shares boundary conditions equivalent to the AdS_2. One way to think of this is that CFT_2 can define gauge-like gravity that has the same DoFs of bulk gravity in AdS_2. In this way the AMMZ argument for a higher dimensional space ties the two CFT_1 chains together in an entanglement.

    The AdS_n black hole correspondence identifies holographic content of the event horizon with the CFT_{n-1} corresponding to the AdS_n the BH is embedding within. The AdS_3 corresponds to the BTZ black hole. The CFT_2 in a gauge-like gravity theory can describe quantum gravitation for the AdS_2. The area law S = /4в„"_p^2 + O(Д§) quantum corrections is then "constructed" through this correspondence between AdS_2 and AdS_3. The "area" in the AdS_2 case is a 0-dimensional, and in the AdS_3 as with the BTZ BH it is a circle. Hence for the AdS_2 the theory is a scalar field theory.

    The construction is then a form of flag manifold. The AdS_n = O(n,2)/O(n,1), where O(n,2) is the isometry group of the space. The AdS spacetime is a Stiefel manifold and a form of flag manifold with

    F^N_{d1,d2,...,dm}(в„‚) = SL(N, в„‚)/P^N_{d1,d2,...,dm}= U(N)/[U(k1)Г--U(k2)Г--...Г--U(kr)Г--U(kr+1)],

    for k1 = d1, kr+1 = N, k_{n+1} = d_{n+1} - d_n. The real valued form of this flag manifold is

    F^N_{d1,d2,...,dm}(в„ќ) = SL(N, в„ќ)/P^N_{d1,d2,...,dm}= O(N)/[O(k1)Г--O(k2)Г--...Г--O(kr)Г--O(kr+1)].

    It is clear that AdS_n = F^{n,2}_{n,1}(в„ќ). The flag F^N_1(в„‚) is в„‚P^{N-1} = U(N)/[U(1)Г--U(N-1)]. Similarly, the real valued version is F^N_1(в„ќ) = в„ќP^{N-1} = O(N)/[O(1)Г--O(N-1)], where O(1) is a trivial group and в„ќP^{N-1} = O(N)/O(N-1) , where S^2 = O(3)/O(2). In the split form we then have

    в„ќP^{N-1,2} = O(N,2)/O(N,1),

    And AdS_n = в„ќP^{n-1,2}. The flag manifold F^4_1(в„‚) = в„‚P^3 = U(4)/[U(1)Г--U(3)] gives twistor space. In the split form

    F^{2,2}_1(в„‚) = в„‚P^{2,1} = U(2,2)/[U(1)Г--U(2,1)],

    this is according to the isometry group of AdS_5.

    The flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)Г--U(2)] is the Grassmanian space G_{4,2}(в„‚). This is identified as a spacetime, where dS and AdS spacetimes are defined according to a line element

    A = t^2 В± u^2 - x^2 - y^2 - z^2,

    with u^2 de Sitter and -a^2 anti-de Sitter. The light cone for A = 0 separates the AdS and dS. The flag F^{2,2}_{2,1} = U(2,2)/[U(1)Г--U(1)] with the bi-fibration

    в„‚P^{2,1} в†ђ F^{2,2}_{2,1} в†' G_{4,2}(в„‚) вЉѓ dS_5 в€Є AdS_5.

    These spaces are Kähler manifolds, where complex projective space is the Fubini-Study metric for projective Hilbert space. A simple entanglement geometry is of the form

    в„‚^2вЉ--в„‚^2вЉ--... вЉ--в„‚^2/SL(2, в„‚)Г--SL(2, в„‚)Г--... Г--SL(2, в„‚).

    where the flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)] is similar to в„‚^2вЉ--в„‚^2/[SL(2, в„‚)Г--SL(2, в„‚)], but with a topological difference between в„‚^{вЉ--2} and SL(4, в„‚) ≈ U(2,2). U(2,2) has roots and weights with a Weyl chamber, while в„‚вЉ--в„‚ does not. The twistor bi-fibration is a projective map on a form of quantum entanglement.

    This then concerns the emergence of space or spacetime from quantum entanglements. The geometry of spacetime, or G_{4,2}(в„‚) in 6 complexified dimensions is equivalent to a form of tripartite entanglement. Of course, we do not expect spacetime to be built up from just three states, but rather a spectrum of such states or a condensate of identical states. This returns us to the duality between the near singularity metric and the near horizon singularity. The near singularity metric is determined by gauge charges, while the near horizon condition is determined by mass. This duality between gauge charges and gravitation is a form of this equivalency between quantum mechanics and general relativity.

    Lawrence, Yes, 'quantum spin' is forced on us, but by our initial assumptions, as Bell suggested. It's not the 'shaving down' or 'tiny loophole' you suggest but a root & branch reconstruction. But as I suspected the 'new' orthogonal momentum in OAM is indeed invisible to you. No matter, and as you well know you're in the great majority. But then so are flocks of sheep of course!

    For me science is about deriving from scratch not following beliefs, but it seems ever less so as the solution is falsifiable. My first finalist essay 10yrs ago suggested we may have had adequate advancement of intellectual evolution to see the "imaginative solution that will amaze" (J.Bell "Speakable..)" by 2020. I always was an optimist!

    Very Best

    Peter

    5 days later

    quote

    Wheeler with his question, "Why the quantum," pondered whether quantum mechanics was built from

    more elementary nuts and bolts. In the old Oxford symposium book on quantum gravitation Wheeler

    speculated quanta were built from some undecidability of an elementary system [18]. Wheeler was

    thrown out of G¨odel's office for asking this question, and for anyone who has pondered this and deigned

    to mention this, it is often greeted with disapproval. Szangolies presents arguments for undecidability by

    considering an elementary model and a Cantor diagonalization [6]. This leads to a form of information

    barrier or epistemic horizon. In this paper it is argued that different entanglements are obstructed away

    from each other. This obstruction is fundamentally the same as the epistemic horizon is what keeps two

    entanglement types topologically separate. In what is presented here the door is further opened

    end of quote

    This is in my reasons why I gave you a solid 10 - I concur with your viewpoint

    Thanks, and my essay does take off in a different direction from Szanglolies, but is framed around his idea.

    I have yet to read many essays. I have been saddled with this Covid-19 for over 2 weeks. There is a lot of fatigue with this. It has pneumonia-like symptoms, though comparatively mild. The biggest issue now is just the fatigued feeling and having to sleep 12 hours a day.

    Anyway I will make a real effort to start reading essays and yours first.

    Cheers LC

      12 days later

      I tried to post this last week and it would not let me. I try again.

      I recently re-read the paper The Page curve of Hawking radiation from semiclassical geometry by Almheiri, Mahajan1, Maldacena, and Zhao arXiv:1908.10996v1. There is no paper of late that I have read so many times as this. This paper relies heavily upon the idea of quantum extremal surfaces and this paper Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, Engelhardt and Wall arXiv:1408.3203v3 . This physics is the relationship between spacetime and quantum horizons. The metric g_{ab} = g0_{ab} + g^{Д§^ВЅ}_{ab} + g^Д§_{ab} + .... is expanded in units of the Planck length в„"_p = в€љGД§/c^3 в†' O(Д§^ВЅ). This is the logic of MTW with g = g0 + в„"_p/L + ... . This leads to a description of quantum states with an entanglement between exterior and interior states.

      It occurred to me this could be seen in some simple ways. For the Kerr-Newman metric

      ds^2 = (1 - 2m/r + Q^2/r^2)dt^2 - (1 - 2m/r + Q^2/r^2)^{-1}dr^2 - r^2dО©^2

      The near horizon condition for an accelerated observer above the horizon is found to be

      ds^2 ≈ (r/m)^2dt^2 - (m/r)^{-2}dr^2 - r^2dΩ^2

      and curiously if one derives a similar metric for the near singularity condition r = 0 the metric is

      ds^2 ≈ (Q/r)^2dt^2 - (Q/r)^{-2}dr^2 - r^2dΩ^2.

      The two metrics are interestingly dual with respect to mass and the charge, or equivalently with angular momentum. Both metrics are AdS_2Г--S^2 or the anti-de Sitter spacetime in 2 dimensions, and in a BPS setting indicate a duality between the mass variable and quantum numbers associated with gauge fields. The extremal black hole with m = Q is the extremal black hole, and Carroll, Johnson and Randall showed this condition leads to the discontinuous map of the spacelike trapping region to AdS_2Г--S^2.

      This dualism suggests that quantum states in the deep interior of the BH, on so called islands, near the singularity are dual or even entangled to states near the horizon or in the exterior. Conformal patches on the two AdS_2 spacetimes may correspond to each other, and the corresponding CFT_1, with AdS_2 ≈ CFT_1, are dual chains similar to a Haldane chain. The boundary of AdS_n is a timelike region, which means AdS space is not globally hyperbolic due to the existence of a timelike boundary at infinity. However, if there are boundary conditions at infinity there are then causal properties are still fine assuming that there are boundary conditions at infinity Hence a conformal patch in AdS_3 may have a boundary with CFT_2 that shares boundary conditions equivalent to the AdS_2. One way to think of this is that CFT_2 can define gauge-like gravity that has the same DoFs of bulk gravity in AdS_2. In this way the AMMZ argument for a higher dimensional space ties the two CFT_1 chains together in an entanglement.

      The AdS_n black hole correspondence identifies holographic content of the event horizon with the CFT_{n-1} corresponding to the AdS_n the BH is embedding within. The AdS_3 corresponds to the BTZ black hole. The CFT_2 in a gauge-like gravity theory can describe quantum gravitation for the AdS_2. The area law S = /4в„"_p^2 + O(Д§) quantum corrections is then "constructed" through this correspondence between AdS_2 and AdS_3. The "area" in the AdS_2 case is a 0-dimensional, and in the AdS_3 as with the BTZ BH it is a circle. Hence for the AdS_2 the theory is a scalar field theory.

      The construction is then a form of flag manifold. The AdS_n = O(n,2)/O(n,1), where O(n,2) is the isometry group of the space. The AdS spacetime is a Stiefel manifold and a form of flag manifold with

      F^N_{d1,d2,...,dm}(в„‚) = SL(N, в„‚)/P^N_{d1,d2,...,dm}= U(N)/[U(k1)Г--U(k2)Г--...Г--U(kr)Г--U(kr+1)],

      for k1 = d1, kr+1 = N, k_{n+1} = d_{n+1} - d_n. The real valued form of this flag manifold is

      F^N_{d1,d2,...,dm}(в„ќ) = SL(N, в„ќ)/P^N_{d1,d2,...,dm}= O(N)/[O(k1)Г--O(k2)Г--...Г--O(kr)Г--O(kr+1)].

      It is clear that AdS_n = F^{n,2}_{n,1}(в„ќ). The flag F^N_1(в„‚) is в„‚P^{N-1} = U(N)/[U(1)Г--U(N-1)]. Similarly, the real valued version is F^N_1(в„ќ) = в„ќP^{N-1} = O(N)/[O(1)Г--O(N-1)], where O(1) is a trivial group and в„ќP^{N-1} = O(N)/O(N-1) , where S^2 = O(3)/O(2). In the split form we then have

      в„ќP^{N-1,2} = O(N,2)/O(N,1),

      And AdS_n = в„ќP^{n-1,2}. The flag manifold F^4_1(в„‚) = в„‚P^3 = U(4)/[U(1)Г--U(3)] gives twistor space. In the split form

      F^{2,2}_1(в„‚) = в„‚P^{2,1} = U(2,2)/[U(1)Г--U(2,1)],

      this is according to the isometry group of AdS_5.

      The flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)Г--U(2)] is the Grassmanian space G_{4,2}(в„‚). This is identified as a spacetime, where dS and AdS spacetimes are defined according to a line element

      A = t^2 В± u^2 - x^2 - y^2 - z^2,

      with u^2 de Sitter and -a^2 anti-de Sitter. The light cone for A = 0 separates the AdS and dS. The flag F^{2,2}_{2,1} = U(2,2)/[U(1)Г--U(1)] with the bi-fibration

      в„‚P^{2,1} в†ђ F^{2,2}_{2,1} в†' G_{4,2}(в„‚) вЉѓ dS_5 в€Є AdS_5.

      These spaces are Kähler manifolds, where complex projective space is the Fubini-Study metric for projective Hilbert space. A simple entanglement geometry is of the form

      в„‚^2вЉ--в„‚^2вЉ--... вЉ--в„‚^2/SL(2, в„‚)Г--SL(2, в„‚)Г--... Г--SL(2, в„‚).

      where the flag manifold F^{2,2}_2(в„‚) = U(2,2)/[U(2)Г--U(2)] is similar to в„‚^2вЉ--в„‚^2/[SL(2, в„‚)Г--SL(2, в„‚)], but with a topological difference between в„‚^{вЉ--2} and SL(4, в„‚) ≈ U(2,2). U(2,2) has roots and weights with a Weyl chamber, while в„‚вЉ--в„‚ does not. The twistor bi-fibration is a projective map on a form of quantum entanglement.

      This then concerns the emergence of space or spacetime from quantum entanglements. The geometry of spacetime, or G_{4,2}(в„‚) in 6 complexified dimensions is equivalent to a form of tripartite entanglement. Of course, we do not expect spacetime to be built up from just three states, but rather a spectrum of such states or a condensate of identical states. This returns us to the duality between the near singularity metric and the near horizon singularity. The near singularity metric is determined by gauge charges, while the near horizon condition is determined by mass. This duality between gauge charges and gravitation is a form of this equivalency between quantum mechanics and general relativity.

        Dear Lawrence,

        i had a similar illness two years ago during the last essay contest. It was the influenza-virus and even after being cured i needed much more sleep. Without that additional sleep i think my that fatigue would have turned into cronical fatique. So please regenerate yourself properly.

        I read your essay several times and i like that you try to get the bigger picture on QM. I also liked what you have to say about p-adic numbers, locality and decidability.

        Take care and best wishes

        Stefan

        Lawrence,

        Welcome back.

        Quite a scholarly piece that penetrates the underlying quantum world and entanglement quite well to prove prediction limits in our classical world. I wonder if you have seen the studies that seem to bridge the two worlds: https://phys.org/news/2018-11-quantum-criticality-superconductivity.html. I cite them in my essay re bridging efforts in overcoming limits. Your topological obstruction theory is quite interesting, at least what I can understand of it.

        Jim Hoover

          Lawrence,

          Mine is your 6th rating. I say this because there is someone who gives 1s to a number of us w/o comments.

          Jim Hoover

          DEARS READERS,

          LAWRENCE CROWELL CANNOT CURRENTLY POST IN FQXI WEBSITE. THUS, HE ASKED ME TO POST THE FOLLOWING MESSAGE FOR ALL OF YOU:

          "TO ALL WHO POST ON MY SITE. FOR SOME REASON I CAN'T ACCESS THE BLOG WRITING. AS A RESULT DON'T BE PUT OFF IF I CAN'T RESPOND. SUPPOSEDLY THE FQXI TECH FOLKS ARE GOING TO FIX THIS.

          THANKS LC"

          Lawrence,

          The problem could be your browser. I use google and have only had trouble posting ratings since yesterday.

          Jim Hoover

          Hi Dr Cromwell. I am particularly excited by the 3-tangle turple in Quantum mechanics.maybe it means from the default two observers in the Michelson morley double experiment,what we interpret as qm may be the virtual "third picture/observation" borne from the Human brain supposition of dual state of matter in the quantum world. I have a simple piece on bias out of the paradox in my essay https://fqxi.org/community/forum/topic/3525.you may kindly review/rate on the simple diagrams,any input will be appreciated. thanks .All the best in the essay.