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

Jochen,

The crux centers around the p-adic solutions of Diophantine equations. My paper references Palmer's work, in fact a paper by Hossenfelder and Palmer [ S. Hossenfelder, T. N. Palmer, "Rethinking Superdeterminism," https://arxiv.org/abs/1912.06462v1 ] with the fractal sets in state space. It is a nifty idea in one sense, though I doubt the superdeterminism concept it is meant to uphold. Palmer's notion of incompleteness seems wrong to me, as a fractal as a recursively enumerable set of completely computable. It is the complement of ER sets that are not computable or have some undecidable aspect to them. These sets are forms of Cantor sets, and to define a field or metric geometry one has to use p-adic theory. Incompleteness comes into the picture with the theorem proven by Matiyasevich, which then means the topological obstructions between different entanglement types is a form of G├Âdel incompleteness. In effect the fractal geometry with a p-adic metric has its complement in the remainder of the p-adic set that is undecidable. The other p-adic sets have separate solutions or algorithms not isomorphic.

I did not go through the "nuts and bolts" work you have done with an explicit Cantor diagonalization of n experiments with k state with f(k,n) as you did. This approach you have taken is to explicitly argue for how this epistemic horizon is a form of mathematical incompleteness. I make the statement this approach is equivalent to the incompleteness of the solution theory for Diophantine equations as p-adic sets. Given the time here of course I have not had the time to seriously work that out. This would of course be a worthy question to pursue. I think though this is correct.

My approach it meant to be more of a practical way this sort of theory could be employed in questions over quantum measurement, decoherence and quantum gravitation. Quantum gravitation or Hawking radiation have much the same implication with quantum decoherence. All processes conserve probability so Tr(¤ü) = sum_i p_i = 1 is conserved. Decoherence changes Tr(¤ü^n) for n > 1, In particular Tr(¤ü^2) is changed. However, this adjustment occurs because this trace with the square of density matrix elements with ¤ü_{ij}^2 for i Ôëá j are removed for this density matrix pertaining to the state. If we consider the total density matrix ¤ü = ¤ü_sÔè--¤ü_r, s = system and r = reservoir, this change in Tr(¤ü^2) occurs for ¤ü_s = Tr_r(¤ü). The entropy measure S = -Tr(¤ü log(¤ü)) by Taylor series involves all powers of the density matrix. So this change reflects a loss of information accounting. I say accounting because what happens is there is a trace of the entire density matrix when the experimenter is concerned entirely with their system

Thanks for the thumb's up on this. This is even though I made an embarrassing tautological language mistake in the abstract due to editing. I have read you essay here, as well as you paper from 2018. I just have not gotten around to commenting yet.

Cheers LC

With spins you can have entanglements that are singlet or triplet for spins anti-aligned or aligned. In fact you can have a superposition of all fur Bell states. With bosons the exchange is positive or with Z cyclicity for n particles. With fermions you have Z_2 for two fermions, indicating a different topology from bosons, and for N fermions this is generalized with Slater determinants.

The connection with gravitation implies there is some generalization of what is meant by entanglement.The equivalency of quantum entanglement with measure of spacetime is some generalization of how entangled and separable states are correlated with each other. This might also be seen in how quantum gravitation as an UV physics is dual to gauge fields at lower energy (smaller mass) or IR scale:

Quantum gravitation at UV = quantum fields at IR,

which is one way of writing the Einstein field equations.

LC

If one works with classical physics of course there is no entanglement. Entanglement is a representation of topological difference between quantum and classical mechanics. The N-tangle that separates entropy configurations with N states in different entanglements is a topological obstruction that does not occur in classical physics.

LC

If you read my paper, you can see I discuss a fair amount of material on the geometric aspects of quantum mechanics and entanglement. The references I include are from authors who have contributed to these developments. The most elementary form of this is the Fubini-Study metric.

Quantum mechanics is curious in that it is completely deterministic as a wave dynamics, but the amplitudes of the wave give probabilities that occur in measurement or decoherence spontaneously. There is no established idea of a causal process that gives any outcome of a quantum measurement. Hence quantum mechanics has this strange sort of duality between what is deterministic and what is stochastic. QM as a wave dynamics is deterministic, and yet unobserved as such. However, this determines probability distributions and the actual observed outcome is purely stochastic.

If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ВЅ as an L^2 measure theory. It then has a corresponding q = ВЅ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q в†' в€ћ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's Game of life or classical mechanics. A quantum measurement is a transition between p = ВЅ for QM and в€ћ for classicality or 1 for classical probability on a fundamental level.

What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel's theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.

I am aware of your stance on these things. I personally think this is something that Cervantes wrote about. However, if you are determined to stay on your steed Rosinante and pursue your quest then you have the freedom to do so.

Cheers LC

This implies some relationship between CHSH polytopes and Kirwan polytopes. I am not exactly sure how that will work. To be honest I seems to at least tangentially have something to do with the Born rule. The CHSH polytope pertains to conditional probabilities with entanglements and the Kirwan polytope with eigenvalues of entanglements.

Cheers LC

Updated comment: The idea of superdeterminism is really a statement of nonlocality. The idea that no two points or events in the universe are completely independent is a nonlocality expected of quantum gravity. Anything approaching a black hole is never seen to cross the event horizon, but at the same time Hawking radiation occurs. This means that quantum states do not have a unique location in space, but rather have a nonlocality that is probably a salient feature of quantum gravitation.

Palmer and Hossenfelder place this in the context of hidden variables, where an average over these gives standard QM result. This has a Gaussian or standard statistical distribution that on average removes this dependency between regions of spacetime. This means the statistical independence of establishing initial states and the subsequent measurements are not entirely secure. Palmer places this in the setting of undecidability, but on a version of undecidability that is somewhat controversial. This is the Blum, Shub, and Smale (BSS) concept of undecidability of fractals. I work within a more standard idea, where the complement of fractal sets are undecidable. This leads into the undecidable nature of Diophantine sets and p-adic numbers.

Cheers LC

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