Why spin is quantized one-dimensionally (spon. dim. red. conjecture)

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  • #26
Fra
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I was just making a small remark since we have some partial agreements before, but maybe I just misinterpreted you.

If you have everything, then it is easy to constrain its variety to be left with something more limited.
Indeed, it's easier to remove or trace out information, and to create it. I think this is a common argument and it's why this is a common approach.

If you start with an infinity of possibilities, that you put constraints on, you somehow end up with a gigantic landscape that is problematic. I can't figure out how that's computable or managable to a limited processing agent.

I just think the idea of an infinite sea of degrees of freedom is doubtful, as it fails to match any reasonable expectation I have on an inside view.

This "infinity of degrees of freedom" does not actually exist. It is just a vagueness, a naked potential. So don't get too hung up on the idea of some actual realm of everythingness. That would be unreasonable!
So maybe you even meant the opposite of what I thought?

I like to make a clear distinction between a defined uncertainty and just undecidability, maybe this is the source of our confusion.

If you consider a probability space; ie. the missing information is still constrained to an event space and comes with some equiprobable states or prior probabiltiy distribution. This way we can quantify the missing information, since our uncertainty is constrained by the context (microstructure of probability space and prior). To actually decide, and quantify and measure uncertainty actually requires alot of information!

Sometimes you can't even do that, and the state space itself is uncertain. If this is what you mean then I agree. I just wouldn't call that infinite possibilities. To me the set of possibilities are a physical in the sense that they determine the action.

Edit: This is also in a sense the essense of evolution, and evolutionary learning in the sense that variation must be there but it must be small, or we loose stability. Ie. we need options, possibilities but not too many of them so that we get lost.

/Fredrik
 
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  • #27
Fra
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I didn't even think this would be controversial given gauge symmetry. Adding constraints has the effect of breaking symmetry and even "complex" spin eventually gets reduced to the simplest possible spin geometry.
I see your point, but I'm not representative to mainstream of course. I object to the use to monster symmetries. I instead advocate a evolving vievw on symmetries. And an evolution that does not just mean self-organisation and reduction in fixed state spaces.

Based on beein here for a while I'd be surprised if anyone else on here raised the objections. I was just suprised to see this come from YOU :) since if I don't mix you up with conrad I think you're the one talking about about evolving constraints.

/Fredrik
 
  • #28
apeiron
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If you consider a probability space; ie. the missing information is still constrained to an event space and comes with some equiprobable states or prior probabiltiy distribution. This way we can quantify the missing information, since our uncertainty is constrained by the context (microstructure of probability space and prior). To actually decide, and quantify and measure uncertainty actually requires alot of information!

Sometimes you can't even do that, and the state space itself is uncertain. If this is what you mean then I agree. I just wouldn't call that infinite possibilities. To me the set of possibilities are a physical in the sense that they determine the action.
We are definitely talking about fundamentally different notions of a probability space. You are taking the standard approach where there is a lot of actual stuff - an ensemble of microstates, a maximum entropy, a requisite variety, etc. Then the task becomes to sort it or order it in some way by an act (or acts) of constraint. The observer has to sort his world into what is signal, what is noise (and either way, it pre-exists as information).

But I am talking about a potentiality space which is instead radically indeterminate. It is neither signal nor noise. It is a pre-information state of untransformed potential. And the task of the observer, the task of constraints, is to transform potential into definite signal and noise.

The other difference here is that we both take a constraints based view of QM collapse. But whereas you favour an ensemble of located observers, all making partial measurements, and reality arising out of that distributed partial information, I am arguing that constraint exists globally for a system. There is an averaging across observers in a way that there is no requirement for local observers to store the efforts of their observing, rather the observers are instead the information the system is creating. So a particle or event would be the signal in effect, and all the places where particles or events are not found would be the noise, to extend the analogy.

But that is a sidelight. What interests me here is a constraints-based approach to generating dimensional structure.

Spin is one aspect of this. But even more important I would say are reasons to think that
3D is uniquely selected by a constraints approach. For example, see Wolfram's argument on how all "higher dimension" network topologies reduce to three edges, but no further.

http://www.wolframscience.com/nksonline/page-476
 
  • #29
apeiron
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I see your point, but I'm not representative to mainstream of course. I object to the use to monster symmetries. I instead advocate a evolving vievw on symmetries. And an evolution that does not just mean self-organisation and reduction in fixed state spaces.

Based on beein here for a while I'd be surprised if anyone else on here raised the objections. I was just suprised to see this come from YOU :) since if I don't mix you up with conrad I think you're the one talking about about evolving constraints.
Yes, I would definitely take an evolving (or developing ) view. But it is possible that our universe developed via a succession of phase transitions, and so higher states of symmetry would have been just passing phases on the slippery slope down to our lowest possible state of self-consistent geometry.

Time of course would also have had to develop - beginning as a vaguely defined direction and become now in our universe a very definite one. So if we wanted to locate these earlier stages in which monster symmetries "existed", they wouldn't be in our past in a definite sense, just also a vague sense.

So the view I am taking is perhaps extreme in two ways here. There is the idea of a self-organising state of constraint that brings around a radical reduction in degrees of freedom (from a potential and chaotically directed infinity to an actual 4D, smoothly expanding and cooling, spacetime).

And then the further assertion that this reduction could only arrive at the one irreducible outcome. There is no landscape of possible dimensional arrangements which makes our outcome somehow accidental and anthropic. And likewise, there is no open-endedness in the evolution, so the universe cannot continue to evolve and wander that landscape in some fashion. It is a simple entropic gradient from an infinity of degrees of freedom (the least constrained state of being) to as few degrees of freedom as are physically possible (the most constrained state of being).

Which is why the irreducibility of spin, and of network topologies, become important principles.

The dimensional reduction literature is intriguing because it too explores the same issue.
 
  • #30
marcus
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So the approach is to invent or identify a 1-dim. structure from which spin, spacetime etc. is emergent in an appropriate limit.
Tom, that's a logical reaction, but it is too ambitious for me. To the extent that I am able to think about this, I am simply wondering how

if a piece of 1D information exists at all, how could that piece of 1D information, from a 1D world, "percolate up" into our 3D world?

You have asked a more fundamental and difficult question: how can a piece of 1D information exist in the first place?! What underlying structures could exist in such a barren, cramped environment? Or perhaps it is a chaotic environment and turns out to be 1D not because it is too simple but because patterns of adjacency have been disrupted. I'm getting embarrassed for having asked ridiculous questions without adequate resources to follow through.

Anyway I can't address your question of what structure might be at that level. All I am asking is: suppose we take seriously what some QG people say about micro-space being 1D, then could there be some information in that world that "percolates up" into our world, and talks to us.

As it percolates up in dimensionality, I picture it gradually clothing itself with new degrees of freedom which are in some sense spurious (fake) and maybe they are new degrees of freedom which it gets from us, from the questions we ask.

It puts on these new degrees of freedom merely for appearance sake, so it can exist in our 3D world. They are contingent "costuming" because at heart, basically, the piece of information knows that it is only 1D.

Thanks to everybody (esp. TomS) for giving this wacky notion your attention.

As I must remind myself, it could well be that the QG people are mistaken and spatial dimensionality does not go down to 1D at very small scale.
==================

BTW Lewandowski has a new LQG paper that is very much in a Thiemann direction. Canonical quantum general relativity (CQGR). Not the manifoldless "new look". No spin foams. Embedded networks. An actual Hamiltonian. Lewandowski is defining a different direction for LQG---namely to complete the CQGR program. Real diversification.
 
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  • #31
arivero
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Thanks for both responses. Pro and con, both are helpful. It's just a thought. I would be glad to know either your intuitive hunch or any reasons you see to reject this line of thinking. Maybe there is no connection between the conjectured 1D of microspace and the way that particle spin behaves. Could be no connection either logical or intuitive.
It seems to me that a discussion on spin should rely heavely on discussing the rotation group. Amusingly, this thread is able to discuss spin without mentioning this point. Of course tom classification of approaches is related to it, and the only comment actually using the word "rotate" is interesting because it use it in a very liberal way, sort of dual to usual sense. In 0 and 1 dim, particles do not rotate. But if the spin is intrisecal, apeiron seems to hint, then precisely dimensions 0 and 1 are the ones where spin becomes more relevant. But hten, what is the rotation group that spin is covering? Why should it be SO(3)?
 
  • #32
MTd2
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Yesterday I was thinking about this and I also think that starting from 0D is more aesthetically appealing than 1D. But the reason is different from the stated above. Since we are talking about space dimensions, 1D means 1+1, which is the dimensionality of the world sheet of string theory.

In that case we can already work with discreet discreet spins because as a consequence of being able of defining spin statistics for the fields living on it.

But, with 0D, we cannot define any statistics whatsoever without violating causality. That means, to define a statistic, we have to have a field where particles can at least go forward and backwards in time, so that a phase in the wave function can show up. So spin, in 0D, should just be a 2 form charge.

So, in 0D, spin = graviton? So, if somehow we made several 0D worlines cross and interact, we would have naturally quantized gravity AND quantized spin? Does this look like LQG?
 
  • #33
marcus
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... discussion on spin should rely heavely on discussing the rotation group. Amusingly, this thread is able to discuss spin without ...
In whatever I contributed here, I was intentionally and pointedly NOT mentioning rotation. I want to leave open the possibility that the spin of a point particle is not to be thought of as a rotation.
It could simply be a quantized length, a vector with no definite direction in 3D space. As this vector "comes out" into the macroscopic world it has to decide, when presented with another direction, whether it is aligned with it, or aligned against it. (A little bit like a naive primitive who comes to the city with no political alignment who is confronted with choices on various issues, one after the other, until at last he discovers how he fits into the broader picture. Except it doesn't work like that with particles--asking new questions can destroy earlier answers.)

I suppose that the vector might acquire the clothing of rotation to cover its nakedness, when it comes out in public. Custom dictates that, for decency's sake, a vector must be the vector of something, and this newcomer has nothing to stand for except the rotation around itself. The 1D vector brings no force or motion with it, so we dress it as a conventional rotation even though nothing rotates.

What do you think about "dimensional reduction" at small scale, Arivero? Are you skeptical of it? If you consider the possibility, how do you imagine that space could be 1D at small scale? How do you picture it, if you do?
 
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  • #34
MTd2
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I suppose that the vector might acquire the clothing of rotation to cover its nakedness, when it comes out in public. Custom dictates that, for decency's sake, a vector must be the vector of something, and this newcomer has nothing to stand for except the rotation around itself.
Something to rotate must have a referential. If you are isolating just 1D, without a space time, just space, there is no rotation. Unless you just write a vector and say this is spin. In this case a vector is just a scalar, so, you rather think of 0D. For 1D we already have strings.

Well, but given that we are really interested in angular momentum, choose a 2 form by definition and that scalar, make its determinant.
 
  • #35
marcus
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Thanks MTd2 and everybody else who has responded to this somewhat odd train of thought. This is as far as I want to go, or can go, with such a speculative question.
I want to shift my attention to a new Lewandowski paper that came out yesterday, so will start a thread on that.
 
  • #36
apeiron
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the only comment actually using the word "rotate" is interesting because it use it in a very liberal way, sort of dual to usual sense. In 0 and 1 dim, particles do not rotate. But if the spin is intrisecal, apeiron seems to hint, then precisely dimensions 0 and 1 are the ones where spin becomes more relevant. But hten, what is the rotation group that spin is covering? Why should it be SO(3)?
This wasn't quite what I meant. Again, what Marcus finds troubling and paradoxical (his comments on how what "exists" as 1D might percolate upwards to construct a higher-D effect) is a natural part of a systems approach to causality.

So I am not arguing that there is spin as the lowest, most fundamental, form of existence. Instead, that it represents the limit of a process of dimensional constraint.

When you have asked every other question about a location (removed all its degrees of freedom for spatial motion) you still do not yet know if it spins. At 0D, there is still that open question.

Now if we are asking that question from the perspective of a 3D world, then spin can have three orientations (or the answer could also be that there is no spin).

And then the gauge point, asking from a 3D realm is not enough to prevent higher dimensional varieties of spin. For all we know, the spin may be a rotation through 720 degrees, not 360 - the double cover of SO(3). If I understand gauge symmetry correctly (OK, long shot :tongue2:) then you would have to be making measurements from the perspective of a higher dimensional realm to pin down all the potential facets of a spin.

Yes, I am making spin sound literal - a rotation. But that is just because I do mean to remind that spin judgements do require a context, a realm from which measurements are being made, questions are being asked, as so constraints on localised freedoms are being imposed.

"Spin" itself could be considered as a raw or vague potential. So it is not a particular state of rotation but the general possibility of an irreducible local symmetry - which gains a definite character in a definite measuring context. So a standard gauge view as I understand it.

To answer your question on why SO(3)? A constraints based approach (as in condensed matter physics) says that as dimensionality cools, it becomes dimensionally reduced. So higher, more complex, spin can only be seen from a hotter perspective.

Early in the big bang, SO(3) would be a symmetry still found "everywhere". Now it is only expressed at certain "hot locations" - certain massive particles, certain energetic events.

If the universe did cool to a 1D dust, then even 3D polarised spin could not be observed. There would be no reference frame to ask the proper questions. How we would describe spin in such a reduced world is another question. It would not be an actual rotation clearly (how could that be defined?).

But as I said, my own view is that extended spatial degrees of freedom - the freedoms of linear motion, of positional uncertainty - cannot be reduced to less that three. Which is why we find ourselves in a 3D realm as the result of sliding all the way to the bottom of a process of dimensional reduction.

Now CDT and other attempts to marry GR to QFT do seem to give a picture of actual further dimensional reduction. It is as if they are finding a way to cool reality further. New constraints spontaneously appear.

But no! In fact, I am arguing, these models are reheating spacetime. They are selectively melting the background (the other other degrees of freedom that make up the other spatial directions are being melted and rendered vague) while preserving (keeping cool) other remaining degrees of freedom (the naked spatial action of a vector - and as Marcus was worrying about, the open question of what kind of spin might remain for a 1D vector no longer able to rotate in the framework of some crystalised set of dimensions, but instead "rotate" in some co-ordinateless version of space).

This is why I say the current dimensional reduction is smuggling in the planck scale, rather than generating it as an output of the modelling. You get apparent further dimensional reduction because the planck machinery melts your backdrop. And melting the backdrop makes if vague - returns it to a "higher" state of dimensional indeterminacy.

Perhaps someone will show me that I'm wrong in my view that dimensional reduction approaches achieve their results in this fashion. Marcus did not in the end contradict me the last time I argued the case.

I'm not saying models like CDT are bad or deceptive. But I do believe their intent is misguided.

The view I am arguing towards (supported by arguments I have only hinted at here - such as the lessons of network theory) is that 3+1D is the "coldest possible state of dimensionality" if reality arises as a condensation, a self-organising phase transition, from a potential infinity of degrees of freedom to a lowest entropy balance of degrees of freedom.

If this is correct, then it seems to me that collapsing GR to QFT is doomed to failure (so long as it is framed as the job of collapsing the state of things past 3+1D).

Fundamental theories are seeking a big reason why reality is bounded by limits (such as the planck scale). And just here - taking a constraints-based approach to the self-organisation of dimensionality - is where we can already find some natural arguments as to why a dimensional reduction does not just go all the way and shrivel into a dust of nothingness.

Luckily for us, there was an irreducible limit on constraint. The 0D story of spin is just one aspect of it (and yes, the irreducibility of spin is dependent on the irreducibility of the other dimensions in which spin as a property is embedded).

So get down to 1D dust, and spin as we know it can't be found. But as a question to be asking, a way to be thinking, I believe it is subtle and profound.

Arivero, I've enjoyed very much your two papers on ancient greek metaphysics and you will know from Rhythmos, Diathige, Trope in particular how the question of irreducible properties is a very old and established one. Need we mention platonic solids?

So this is exactly that approach applied in a modern setting - where we now know about higher-D symmetries and gauge spin, where the question becomes what is irreducible about dimensionality, physical degrees of freedom, itself.
 
  • #37
arivero
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Arivero, I've enjoyed very much your two papers on ancient greek metaphysics and you will know from Rhythmos, Diathige, Trope in particular how the question of irreducible properties is a very old and established one. Need we mention platonic solids?

So this is exactly that approach applied in a modern setting - where we now know about higher-D symmetries and gauge spin, where the question becomes what is irreducible about dimensionality, physical degrees of freedom, itself.
Let me to keep the answer in the ancient setting. Platonic solids are a good example, yes, of the peculiarities of low dimensions. We have a infinity of them in 2D, five in 3D, and only three in 4D and beyond. Low dimensional peculiarities abound in mathematics.

In the case of Rithmos, Diathige and Trope, what happens in 1D geometry is that there are only two properties, perhaps still to be named Rithmos and Diathigue. Or perhaps only one; in any case it is clear that Trope, "which makes Z different of N" is not needed. It could still be questioned if we need a way to keep "p" diferent of "q".
 
  • #38
apeiron
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Let me to keep the answer in the ancient setting. Platonic solids are a good example, yes, of the peculiarities of low dimensions. We have a infinity of them in 2D, five in 3D, and only three in 4D and beyond. Low dimensional peculiarities abound in mathematics.
Yes, in 2D we get a pathological landscape :rolleyes:. Again evidence of a failure of constraint.

In 3D, I would argue that the five platonic solids are actually three self-dual solids, so the actual count is "just three". This may be important if the argument is that 3D represents the lowest available minima.

But I am not basing any strong opinions on platonic solids as such. We have to take other issues into account such as spatial curvature. For instance, what does tiling on a flat plane tell us (and the ability to tile pentagons on a curved surface). So there are no simple answers in ancient metaphysics and mathematics, but rather clues to a style of thinking which has been rather lost.

In the case of Rithmos, Diathige and Trope, what happens in 1D geometry is that there are only two properties, perhaps still to be named Rithmos and Diathigue. Or perhaps only one; in any case it is clear that Trope, "which makes Z different of N" is not needed. It could still be questioned if we need a way to keep "p" diferent of "q".
Shape would indeed be gone. Which is not a problem for those who then want to claim it is precisely what would be constructible from 1D fragments. And what I, in turn, am arguing is in fact an irreducible aspect of a reality that is formed by a process of dimensional self-constraint.

Orientation seems to be gone too - that was my argument about CDT, imagining 1D vectors against a now vague backdrop that offers no proper purchase for the making of orientation measurements.

Some kind of relative position still exists as we have a bunch of local 1D vectors sprinkled around in different locations.
 
  • #39
MTd2
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http://arxiv.org/abs/physics/0006065v2

Rhythmos, Diathige, Trope

Alejandro Rivero
(Submitted on 26 Jun 2000 (v1), last revised 29 Nov 2000 (this version, v2))
It is argued that properties of Democritus' atoms parallel those of volume forms in differential geometry. This kind of atoms has not "size" of finite magnitude.
-----
Se arguye que las propiedades de los atomos de Democrito son paralelas a las de sus formas de volumen en geometria diferencial. Este tipo de atomos no tiene tamanno de magnitud finita.
 
  • #40
apeiron
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http://arxiv.org/abs/physics/0006065v2

Rhythmos, Diathige, Trope

Alejandro Rivero
(Submitted on 26 Jun 2000 (v1), last revised 29 Nov 2000 (this version, v2))
It is argued that properties of Democritus' atoms parallel those of volume forms in differential geometry. This kind of atoms has not "size" of finite magnitude.
-----
Se arguye que las propiedades de los atomos de Democrito son paralelas a las de sus formas de volumen en geometria diferencial. Este tipo de atomos no tiene tamanno de magnitud finita.
In english here....

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.15.9983&rep=rep1&type=pdf
 
  • #41
atyy
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