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

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.

I am back in action now. I can now write and vote.

LC

The 3-tangle is a case with GHZ and W entanglements. These are separated by a topological obstruction. This just means there is no unitary basis by which one can evolve into the other. This is at least as a closed system. These entanglements can lose quantum phase to a reservoir of quantum states, or quantum noise, which mean that ignoring those states, say tracing them out, gives the appearance of the same state. This is then a coarse grained perspective of decoherence.

I will take a look at your page ASAP. I have been dealing with this Covid19 and it has made me quite fatigued and I have difficulty doing as many things as I used to just 6 weeks ago. I have made a self-promise to really read as many papers here as possible, but by evening time I find myself too tired out.

Cheers LC

All I did was to change about 4 words in the abstract. There was a bad misstatement in there. In fact I think you pointed it out.

I am largely done with the flu-like symptoms, and have been for several weeks. The one problem is the fatigue is still a bit persistent. I am sleeping more, but not up to the 12hour/day a month ago.

Cheers LC

The equation e^{iПЂ} = -1 is called the Euler equation. It also leads to complex number. Consider the natural logarithm of a negative number ln(-x). In the field of real numbers we are stuck but we can use the rule of logarithms ln(ab) = ln(a) + ln(b). So, ln(-x) = ln(x) + ln(-1) and we can at least peel off the part we don't understand. As a result, with Euler equation write

ln(-x) = ln(x) + ln(e^{iПЂ}) = ln(x) + iПЂ

due to the inverse function of the exponent e^{iПЂ}. We can also use e^{iПЂ/2} = I to evaluate the logarithm of an imaginary number ln(ix) = ln(x) + iПЂ/2.

Quantum mechanics can be thought of as a map of classical mechanics to the quantum according to the Poisson bracket formalism. This bracket {q, p} = (∂q/∂q)(∂p/∂p) - (∂q/∂p)(∂p/∂q) = 1 is replaced with the commutator {q, p} в†' i/Д§[q, p], where these are now operators. The commutator is then equal to i = в€љ(-1). So, in that sense quantum mechanics does have this funny relationship with imaginary numbers.

The wave function in QM is complex valued. A complex number is z = x + iy. The magnitude of this number is computed with its conjugate z* = x - iy and so zz* = x^2 + y^2. Then with a little trigonometry we have z = zz*(cosОё + isinОё) = zz*e^{iОё}. A polar form of a wave function is of this form.