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

Hi Torsten,

This is very nice to see you here. Excellent essay! In my 2011 FQXi essay I have used your publication (Torsten Asselmeyer-Maluga, Helge Rosé. On the geometrization of matter by exotic smoothness. arXiv:1006.2230v1) in references. That is very rare that academic entrant, as you, accepts physics as a manifestation of geometry. Even though barely all of physicists accept General Relativity they also accept a sudden jump from the big distance scale (GR) to the small one (QM) and the geometrization disappears. What about the distance scale invariance of the laws of physics? No one matters. It is explicitly showed in the ratings of essays.

In my essay in Table 1 I have defined that the conformally flat spacetime is the Bit and the matter is the It emerging from the spacetime but the reverse way is also possible. The matter and space have the same root (ancestor, predecessor). As you see we generally agree. Differences are in technical details.

In my opinion we need to find the one, universal, distance scale invariant metric, reducing to Einstein GR metric within Solar System distance scale and having ability to generate predictions. The first prediction of my (and also yours) concept is my spin experiment outcome. Then depending on the outcome we shall create the proper metric. The progress I have made since 2011 is that experiment.

Best regards

Hi,

Thanks for the message on my page. I will have time to read some essays this weekend, and yours is at the top of the list.

I wrote further on my blog page on exotic manifolds and its possible role in quantum gravity. You are right, as I remember, that there is a Godel-type result with result with respect to exotic R4s.

Cheers LC

Hi Torsten,

OK, I've had a little time to read now, so I can perhaps try to add my two cents. First of all, I think your realization that diffeomorphism invariance implies that a continuous manifold in GR doesn't contain more information than some triangulation is something that deserves being shouted from the rooftops---I've always thought that continua are something of an ontological burden, and should be avoided if possible. (The argument, or rather side remark, I make on this in my essay is essentially due to Achim Kempf---maybe you're familiar with his approach.)

I also was struck by the relationship you uncover between measurement and undecidability---this is another one of those ideas that keeps cropping up in unexpected places, and something I keep coming back to without, however, coming up with much of anything concrete (I've said a few things about this on the thread of Lawrence Crowell's essay). Perhaps the paper by Paterek et al., where they propose that outcomes of quantum experiments are random iff the proposition they encode is undecidable (from some set of axioms encoded in the preparation procedure), is of interest, but maybe it's a blind lead. Brukner has done some further work in this direction, and together with Zeilinger, has also professed views drawn very much from Weizsäcker.

Regarding Weizsäcker, yes, I think many of his cosmological arguments look somewhat quaint from a modern perspective, especially all the 'large numbers'-stuff, so I wouldn't want to commit myself to a 3-sphere cosmos as well. But I believe the argument for the 3-dimensionality of space being related to the 3-dimensionality of the qubit state space is not without merit; recently, it's been put into a modern information-theoretic form by Müller and Masanes. Of course, I suppose that to you, the advantage of 4-d spacetime is that it gives you a lot of smoothness structures to play with! (Incidentally, with your argumentation regarding 'the spacetime is the Bit', I'm not sure you're that far from the Weizsäckerian picture, in particular when you're talking about Stern-Gerlach measurements.)

Coming from the quantum side of things, I must confess that any attempt to 'geometrize the quantum' instead of quantizing geometry finds me a bit hesitant, but your argument regarding 'wild' embeddings as deformation-quantized versions of tame ones is nevertheless intriguing, I'll have to think about it for a bit. Maybe there's a sort of dual perspective thing here: you can start with the quantum, and get a 3(+1) dimensional spacetime out, or you start with the spacetime, and out pops the quantum (via a suitable embedding). That's probably a bit fanciful, but that way, everybody gets what they want...

Anyway, I've got to go now, it was a joy reading your essay, and I hope you do well in the contest!

Cheers,

Jochen

Copoied from my page:

Hi Torsten,

I remember reading an article back in the 1990s about how the classification of exotic R^4s was not enumerable, which had connections to Godel's theorem.

The exotic R4 structure has its origin in the Casson handles as pointed out by Freeman. A thickened disk D^2 --- > D^2xR^2 can produce various structures, which by the self duality of four dimensions leads to these strange conclusions. In scanning your paper I see you invoke Casson handles. The number of such structures by h-cobordism turns out to be infinite, which as I say above, I remember this to be nonenumerable. This result was proven by one of the big mavens in this area, Atiyah, Freeman, Taubes, ... ?

The one element of this is that the e8 Cartan matrix as the eigenvalued system for an E8 manifold, an exotic R4. It has been a while since I have studied these matters, but as I remember this tells us how to tie 3-manifolds in 7 dimensions in the Hopf fibration S^3 --- > S^7 --- > S^4. The dual to this structure are 4-manifolds. The 7 manifold this knotting is performed is in the heterotic S^7 --- > S^{15} ---- > S^8, and the e8 Cartan matrix gives the eigenvalues for the 7-space.

The interesting thing about the E8 is that the 8-dimensional space is equivalent to the group in a lattice construction; the root-weight space is ~ the space itself. The E8 manifolds of Freeman are I think embedded in the set of possible 8-spaces. This suggests a duality between the smooth manifold in 4-dim and a discrete or noncommutative manifold in a quantum sense.

Physically this seems evident from data obtained so far. Measurements of the dispersion of light from extremely distant sources invalidate a discrete structure to spacetime. This tells us that a measurement of spacetime structure by measurement of photons that traverse a large distance give no signature of grainy structure. Yet a lattice perspective of spacetime with the Grosset polytope and the 120-polytope of quaternions in 4-dim would suggest a noncommutative geometry. However, if the lattice is equivalent to the space, then this smooth structure is dual to a grainy picture of spacetime. This structure should emerge in an extremely high energy experiment that probes small regions, rather than testing across vast distances.

Cheers LC

Mr.聽Asselmeyer-Maluga,

In your article you write "smooth spacetime contains only a discrete amount of information", are you refering for instance to quantum theory? In that case the gravitational wave is to be observed yet. If you mean every information, then please explain further.

Hi Torsten,

Your essay was excellent reading. I'm unfortunately not fluent in topology but have tried to explore similar ideas with a discrete background independent foundation. It's more like å gut feeling than anything else but I suspect that space-time is structure built with the bits, aka at least one level of complexity above raw bits. Is it room for that in your opinion?

I try to explore these Ideas in my completely non mathematical essay. And think that it should be possible to deduce why we have a velocity limit, and from there why there's no space-time beyond the event horizon in black holes (which is not covered in the essay) Thinking about bits as the foundation can give rise to ideas on how entanglement works, alternative interpretation of what happens in the double-slit experiment. Even how physical laws arises.

If you would take time to read it - and shoot it down if you like - I would be very happy because we have some similar ideas. (I would really like to get some feedback on some of the ideas there - trails of before the interesting ideas there. Probably because it's badly written)

Anyways - excellent essay and best regards

Kjetil

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).

[/math]

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