Spacetime weave - Bit as the connection between Its or the informational content of spacetime by Torsten Asselmeyer-Maluga

I read the first few pages of your essay with some care. It is interesting that you discuss the issue of quantum measurement. This touches on the issue of contextuality in QM. The Kochen-Specker theorem proves there is not context in QM for any quantum measurement. The observer is free to choose the orientation of their SG apparatus, which means choosing a basis in the Hilbert space of the system. Since any basis is freely given by any unitary transformation there is no QM prescription for a basis of choice. General relativity has a similar concept with covariance, and gauge theory is also similar. The measurement problem boils down to how it is that a quantum system is reduced to a certain eigenvalue, and in addition how it is that the basis for that eigenvalue is "chosen." In a Bohr or Copenhagen context this seems to suggest there are operating rules of nature outside of the QM, call it "classicality," that perform this role. In an MWI context there is still some auxiliary postulate or physical axiom involved with how it is the world eigen-branches into the many worlds.

This is not necessarily an act of consciousness. First off the splitting is perfectly random, and randomness may well have its fundamental meaning within quantum mechanics. The outcomes of measurements just updates a Bayesian prior on the nature of the world, and the information obtained is a measure of the Chaitan-Kolmogoroff entropy.

A measurement involves the use of energy. The Stern-Gerlach experiment imposes a divergent magnetic field in order to split the spin of electrons according to a certain z-orientation or basis. Energy is a funny thing, because its conjugate variable is time. There is no time operator that acts on a basis |t> so that T|t> = t|t>. The problem of course is that there would exist a unitary operator U = e^{-iεT} that continuously evolves the energy ε and energy spectra could not be discrete nor can it be bounded below. In a related manner with Fourier transforms we do not have negative frequencies, or negative energy, and integrate Fourier sums of ωfrom [0,∞) which differs from the position and momentum variables that are integrated from (-∞, ∞). Also position and momentum have classical correspondence with Poisson brackets in classical mechanics, while energy and time do not. Quantum measurements seem to require both time and energy. Energy must be applied to define a basis, and the succession of measurements, say of p and then x is done in a tensed fashion, and of course gives a different result than a measurement of x and then p.

I notice paper goes into the nature of time. The discussion appears similar to what you did last year. I will try to think about this, for time in general relativity is a really strange concept. The ADM approach to general relativity results only in the constraints NH = 0 and N_iH^i = 0. In a quantum setting with momentum metric variable π^{ij} = -iδ/δg_{ij} the Hamiltonian constraint results in the equation HΨ[g] = 0, which is related to the Schrodinger equation

HΨ[g] = -i∂Ψ[g]/∂t

But the time variation part is zero. In a general spacetime, say think of a spherical universe or an infinite open one with uniform distribution of mass-energy, there is no natural boundary from which one can integrate over a field to evaluate mass-energy inside; this would be a GR form of Gauss' law. I will have to ponder your ideas about time within this setting. If one is not able to define mass-energy, then correspondingly the definition of time is difficult as well.

Cheers LC

Hi Torsten,

I really appreciate that you took your time to read and comment. And I must apologize that my question was too hung up in my own ideas, and I really have to go deeper into your line of thought. (Which probably means that I have to learn and understand topology - now I wonder how I can squeeze that in an already tight schedule... :) )

A million thanks - on several levels

Kjetil

Torsten, I feel that I have learnt about an interesting new perspective from your essay. I did not really know what the fuss with smooth structures was about before. Now I understand it a little better.

The idea that physics is derived from topology is an appealing one nut it depends on whether there is any non-trivial topological structure at small scales in space-time. I think physicists have gone to and fro with this idea. In the last few years I think that the boring flat topology has been winning out but with the new Susskind/Maldacena insight that entanglement is related to wormholes, we could see things swing back to non-trivial topologies. In that case the maths of smooth-structures should be a big topic of interest.

My own approach is to start from an algebraic structure and try to derive geometry as emergent. In a sense it is the opposite of your approach, but the real meat is in the relationships between algebra and geometry and relationships go both ways.

Dr. Asselmeyor-Maluga

I am a self-taut (thinking makes me tense) realist. May I please make a comment about your essay? In my essay BITTERS, I contend that reality is unique, once.

Writing about "bits" you stated "The sequence is an expression of the dynamics (of the timed motion of bits) and for a given position in the sequence we know the unique precursor and successor." (of)

Respectfully, the only way we could suspect that the position of any precursor and successor bit placement was unique would be if the bit was unique. Each real bit is unique once and because each real bit is unique once, it cannot travel sequentially. It can only travel uniquely once. Only abstract bits can travel sequentially because they are not unique.

I finally got to read your paper with a fair amount of care to detail. I have not scored it yet. I read a hard copy last night and was not on line.

The following comes to mind with this. Given the four manifold M^4 a subregion D^2xT^2 is removed and the complement or dual of D^2xKxS^1 in S^3xS^1 is surgically inserted. It is common to think of spacetime as M^4 = M^3xR. So the manifold constructed from the knot K is

[math]

M_k =((M^3\setminus D^2\times S^1)\times S^1) \cup_{T^3} ((S^3\setminus(D^2\times K))\times S^1).

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On the left the R^1 in M^4 = M^3xR is replaced by S^1, and we can think of the S^1 as a periodic cycle with a real number line as a covering. Think of a wheel rolling on the real number line, or a spiral covering of a circle. In this setting the crux of the matter involves replacing a circle S^1 with a knot K. Physically this avoids topologies with circular time or closed timelike loops such as the Godel universe.

This substitution is then a type of cobordism. We think of there being a "tube" connecting a circle as a boundary at one end and the knot at the other end. This results in "crossings" or caustics of the tube, which suggest this image is viewed completely in higher dimensions. I attach an image of a situation where the knot is a trefoil. This is an interesting way to do cobordism. The boundaries of this space are a circle and the trefoil knot, and the relationship between these two is given by the Jones polynomial. The Jones' polynomial is a Skein relationship for a knot. The function W(C) = exp( i∫A•dx) is the Wilson line or loop integral for the valuation of a gauge connection. The expectation value is the path integral

[math]

\langle W(C)\rangle = \int D[g,A]W(C)e^{-iS}

[/math]

The element α = 1 - 2πi/kN, for N = mode number and k = momentum vector, and z = -2πi/k. Clearly then α^{-1} = 1/(1 - 2πi/kN). For k very large α^{-1} =~ 1 + 2πi/kN. The Skein relationship is then

[math]

\langle W(L^+)\rangle -\langle W(L^-)\rangle = 4\pi i/kN \Big(\langle W(L^+)\rangle + \langle W(L^-)\rangle\Big) = 2\pi i/k\langle W(L^0)\rangle.

[/math]

The trefoil is then under this polynomial equal to the circle plus two circles in a link, which is the Hopf link in the S^1 --- > S^3 ---- > S^2 series.

This enters into path integrals as

[math]

Z[g] = \int D[g] e^{iW(C)}e^{-iS[g]}.

[/math]

The cobordism then reflects a thin sandwich, to use Wheeler's terminology in Misner Thorne and Wheeler "Gravitation." The thin sandwich has a spatial surface of Cauchy initial data as the bottom slice of bread and Cauchy data on the top slice of bread or spatial surface. At the top slice the data corresponds to S^3\D^2xS^1, that is filled out into M^4\D^2xT^2, and the top slice is S^3\KxS^1. The action at the top and bottom of the thin sandwich is evaluated on the two topologies. This thin sandwich is in a sense "thick" if we think of the lapse function or diffeomorphism (homomorphism) connecting the two slices as R, and this is much larger than 2πr of the circles S^1 which "thickens" the spatial surfaces S^3\D^2xS^1 and S^3\KxS^1 into finely thin spacetimes. The action on the bottom is

[math]

S = \int_{M\setminus D^2\times T^2}dtd^3x \sqrt{-g}R + \int_{D^2\times T^2}dtd^3x \sqrt{-g}R,

[/math]

and at the top is is

[math]

S = \int_{(M\setminus T^2\times S^1)\times S_1)}dtd^3x \sqrt{-g}R + \int_{(T^2\times S^1)\times S_1)}dtd^3x sqrt{-g}R

[/math]

and the Willson loop is the ingredient that dictates the behavior of the

[math]

\delta S = \Big( \int_{(T^2\times S^1)\times S_1)}~~~ -~~~~ \int_{D^2\times T^2}\Big)dtd^3x \sqrt{-g}R.

[/math]

The knot topology or quantum group then dictates the quantum amplitude for the transition between the two configurations.

I don't know whether this connection to knots is strictly correspondent with the infinite number of "exotic" structures. It makes sense that one can have a fusion of knots in an arbitrary set of configurations. There is also the infinitely recursive knot-like topologies that Spivak discusses in his "Differential Geometry" books.

Cheers LCAttachment #1: knotcorb.PNG

Hi torsten,

I wrote a longer post on some of this. The Hamilton constraint or the Wheeler-DeWitt equation amounts to the problem that lapse functions or diffeomorphisms between spatial surfaces do not define time. I might be wrong, but I have thought this seems to be an aspect of the inability to define a coordinate atlas on 4-dim spaces that is diffeomorphic to all others. I expand on this in greater detail below.

Cheers LC

Dear Sir,

How do you say that: "Information refers to an inherent property concerning the amount of uncertainty for a physical system." Information can lead to decrease in uncertainty. But it cannot tell us about the degree or amount of uncertainty. Whether we can have "ALL" information about a system, is doubtful. You cannot apply Heisenberg's principle to information.

Wheeler's definition of "It" as "apparatus-elicited answers to yes-or-no questions, binary choices, bits" has to be read with "registering of equipment-evoked responses". The binary unit, or bit, is a message representing one of two choices: 1 or 0 - on or off - yes or no. The 'on's are coded (written in programming language) with 1 and the 'off's with 0. By themselves 1 or 0 does not mean anything. Related to a context, 1 signals some concept representing information about materials objects exists in that context and 0 means it does not exist. Thus, except signaling the agreement or non-agreement with something predefined (i.e., a concept), binary has no other use. Thus, "It" stands for the information content or the concept about something, which is the "Bits". Information is always about something, say, some material, but it not the material itself. There is no need to bring in several weird concepts to deny this simple truth.

The state is the direct expression of information, provided it is measured (perceived as such) by a conscious agent to collapse to a fixed state at a given time. Otherwise it may evolve in time independently on its own, but would be meaningless to the observer (superposition of all possible states). Hence "It implies the Bit" does not hold because "It" stands for the information content or concept of an object as distinct from the object proper - "Bits". It may report the state of the quarks, leptons or bosons, but it is not the same as the quarks, leptons or bosons in that state. These two are distinctly different.

If "every 'it' - every particle, every field of force, even the space-time continuum itself - derives its function, its meaning, its very existence entirely - even if in some contexts indirectly - from the apparatus-elicited answers to yes-or-no questions, binary choices, bits", then the existence of space-time continuum itself should be derived from the "apparatus-elicited answers to yes-or-no questions, binary choices, bits". In other words, the existence of space-time is due to information. Then how can space-time "contain information", which makes the existence of information dependent on space-time? The only logical interpretation is, both exist independently, but inseparably linked as observable and result of observation - matter and its property. You also admit it in your conclusion No.1. In that case, the whole paragraph is redundant and amounts to name dropping only.

The rest of your essay follows a similar pattern. It would have been better to have presented a cogent analysis of real observables without aimlessly pulling in various directions and importing unrelated hypotheses to spread the cult of incomprehensibility.

Regards,

basudeba

Unfortunately time is a bit narrow right now. I will try to expand on this in greater depth maybe this evening or tomorrow. Ed Witten sees a great foundation to the knot polynomial approach to path integrals. The occurrence of knot topology suggests a Chern-Simons type of theory, where there are underlying cocycle conditions for a L_{cs} = A/\dA (1/3)A/\A/\A with a quantum interpretation.

Cheers LC

The role of the knot polynomial I think is involved with holography. The Lagrangian for spacetime

S = ∫sqrt(g)(R L_m)dtd^3x (1/8π)∮ρdS

Here the curvature ρ is evaluated on the null boundary of the spacetime. This is composed of the extrinsic curvature K_{μν} which is K = dP, for P a displacement on the surface, and in addition the CS term L_{cs}. This is a division of a cochain into a coboundary plus cocycle.

Physically the cocyle can be seen according to Lorentz transformations. Near the horizon as measured afar there is a Lorentz contraction of the radial direction. In the (M^3/KxS^1)xS^1 the contraction eliminates the spatial S^1 and this leaves the knot on the stretched horizon. This is a part of the quantum information on the horizon.

I have yet to assign scores. The problem is that some "trolls" have been assigning ones, and this has the effect of making a 5 in a sense the "new 10." I would probably give your paper an 8 to 10, the 8 reflecting a bit of a problem I might have with something, but with the renormalized score I have been unsure what to do. I decided to give your paper a 7, which reflects a top or near top score with this unfortunate tendency for these "troll scores" that drop everyone's score down.

Cheers LC