Undecidability of States and Epistemic Horizons as Quantum Gravity by Lawrence B. Crowell

Todd Brun found [ https://arxiv.org/pdf/gr-qc/0209061v1.pdf ] that P = NP is true for closed timelike curves. This is a short, readable and decent paper. The extension to all PSPACE and undecidable propositions is of course difficult to prove explicitly. However, a spacetime that permits CTCs will present Cauchy horizons, and in principle an observer can in a finite time verify whether a Turing machine halts or does not halt, even if the proper time of that TM is infinite. This is of course an in principle argument.

It is potentially interesting in the context of P = NP vs P в‰  NP whether this result really does mean this is undecidable proposition. P = NP appears true in a spacetime with CTCs, such as AdS or wormholes and so forth. We have no knowledge whether P = NP can hold in our more normal dS-like spacetime with positive vacuum energy.

Aaronson and Watrous found [ https://arxiv.org/pdf/0808.2669.pdf ] that classical systems on closed timelike curves can perform some BQP algorithms quantum computers execute. This emboldens my hypothesis that quantum physics and spacetime physics are categorically equivalent, or that spacetime is an epiphenomenon of large N-entanglements. On the top of page 5 of this paper is an interesting diagram. This illustrates a register or memory system with two parts, one part for a spacetime such as what we observe with open timelike curves and another with part with closed timelike curves. Aaronson and Watrous argue the Deutsche self-consistency condition on CTCs should hold and that a quantum wave corresponding to the causality respecting must also constructively interfere with the CTC wave function. The argument then is it is possible to emulate all PSPACE this way.

Spacetimes such as Gödel's universe, the global metric on AdS or the timelike interior of a Kerr black hole have CTCs. BTW, it appears that Gödel had some mental obsession with closed loopy systems! The question might be raised, what is the separation or distinction between causal respecting CR and closed timelike curve CTC spaces? The diagram on the AW paper suggests there is some quantum wave interference between a wave function associated with the CR and CTC spacetimes. The de Sitter and anti-de Sitter spaces respectively fit as a single sheet hyperboloid surrounding a light cone and two hyperboloids bounded within the conical openings. These meet at I^{±∞}, which means they share the same quantum information as defined by the AdS/CFT correspondence of Maldacena et al. . In the setting of holography we have something similar, and there are arguments of AdS black hole correspondence as well.

This according to A. Almheiri1, R. Mahajan, J. Maldacena, and Y. Zhao (AMMZ) [ arXiv:1908.10996v1 [hep-th] ] also has some correspondence with the interior state of a black hole. This paper is rather qualitative and speculative. The idea is the interior of a black hole has "islands" of states defined by a dimension difference of one. We might compare this to how the Reisner-Nordstrom metric, and by extension the Kerr metric, has a near horizon condition for an accelerated observer equivalent to AdS_2Г--S^2. The AdS globally has CTCs, and locally we consider conformal patches that restrict away from CTCs and respects CR. What I am working on now is to illustrate how the AMMZ islands correspond to local AdS regions or conformal patches. This would imply event horizons or boundaries imposes restrictions away from a complete correspondence. This is in line with my FQXi paper on topological restrictions between entanglement types and their correspondence with Szangolies' concept of the epistemic horizon.

The issue with P = NP vs P ≠ NP is then still open. As I approach this with p-adic numbers and the Gödel undecidability of these sets, a complex number version corresponds to problems in algebraic geometry. Mulmuley has devoted much work on the algebraic geometry of computation. This leads to interesting issues with the Riemann ζ-function.

LC

Hi Jonathan,

Actually that was a general note, However thanks for the kind words.

I too have only read a handful of papers here. I have been pretty hard at work on this. I have not had much time to read papers or communicate with authors. I am now off work, as we are now into "social distancing." So we have to endure this exercise in Camus' existential angst in his novel The Plague with the expectation this will lessen the impact.

Stay safe and away from crowds. This virus seems to take older people down a lot. I am not worried about myself much, but I have some concerns with others. My mother is over 90 and I concerns there.

Cheers LC

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

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.

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.

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

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.