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

[marcus]

Spin behaves as if it is one dimensional, along any axis you select.

This behavior would be just what one expected from a vector which, owing to spontaneous dimensional reduction, lived in a one-dimensional world.

But according to several approaches to QG, very small "things" or degrees of freedom DO in fact live in a world which is spatially one-dimensional.

Steve Carlip recently posted on Arxiv a paper about the curious agreement among several quite different QG approaches that spatial dimensionality (which is not limited to whole number values) gradually goes down from 3D to 1D as one approaches Planck scale. There is also a video of his talk about this last summer. If anyone wants links on this, please ask.

 

[marcus]

Tom Stoer started a thread about that Steve Carlip paper a few days ago. It doesn't mention the quantization of spin, as I recall, but it's a handy source for information about spontaneous dimensional reduction:
https://www.physicsforums.com/showthread.php?t=427649

My first exposure to dimensional reduction with scale was in 2005, with this paper. The exposition is very clear and I still find it an impressive result.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure, Physical Review Letters 95:171301 (2005)
(Submitted on 12 May 2005)
"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be 'self-renormalizing' at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

They plot the gradual decline of spacetime dimensionality with scale, from 4D down to 2D, as computed using their MonteCarlo QG sims. Carlip's 2010 paper gives references to a bunch of other papers that followed on the heels of this one including AsymSafe QG and Loop QG. So I'll give a link to it:

http://arxiv.org/abs/1009.1136
The Small Scale Structure of Spacetime
Steven Carlip
(Submitted on 6 Sep 2010)
"Several lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional. I summarize the evidence for such 'spontaneous dimensional reduction', and suggest an additional argument coming from the strong-coupling limit of the Wheeler-DeWitt equation. If this description proves to be correct, it suggests a fascinating relationship between small-scale quantum spacetime and the behavior of cosmologies near an asymptotically silent singularity."

Basically all I am pointing out here is the obvious fact that spacetime 2D at small scale corresponds to spatial 1D. Which is how the spin vector acts.

 

[MTd2]

At 2D, spin is not quantized in discreet values. So, something crazy happens in the middle of the way.

 

[arivero]

At 2D, spin is not quantized in discreet values. So, something crazy happens in the middle of the way.

SpaceTime dimension 2, or Space dimension 2?

 

[tom.stoer]

I don't know whether this has anything to do with spin.

In LQG you have a colored graph (spin network) on which one could study random walks and the spectral dimension. Looking at the dual "triangulation" of this graph this triangulation can have any "dimension" from 1 to arbitrary high N. But the spin degrees of freedom stay what they are; they are neither used in this triangulation, nor do they change in some way. The dimension is a "topological quantity" depending only on the graph which is independent of the coloring.

I asked some weeks ago regarding spin networks constructed from other groups, e.g. SU(N). It would be interesting to see if their spectral dimension depends on N. But this is a question regarding the dynamics by SU(N).

 

[MTd2]

SpaceTime dimension 2, or Space dimension 2?

Space dimension 2, since marcus was talking about going down to 1.

 

[MTd2]

To tell you people the truth, I don't think spin has anything to do with what Carlip said. His observation is more related to the asymptotic statistical behavior of the dynamics of quantum gravity theories going to 1D in the UV, no matter what are its fundamental definitions.

This is creepy.

 

[tom.stoer]

To tell you people the truth, I don't think spin has anything to do with what Carlip said. His observation is more related to the asymptotic statistical behavior of the dynamics of quantum gravity theories going to 1D in the UV, no matter what are its fundamental definitions.

This is similar to what I said.

 

[atyy]

http://arxiv.org/abs/0911.0401

"We notice first of all that our result agrees with that found for d = 4 in [8], i.e. we find Ds(sigma -> 0) 2 both in three and four dimensions. We should have a word of caution here, in light of the recent observation [2] that, at least in the 4D case, the Planck length is of order the lattice spacing a at the currently possible parameter settings in Monte Carlo simulations, and remains fixed when the “bare Newton coupling” is fixed. This is something that we have not tested in our 3D simulations, but which could well hold true in this case. This observation implies that we are at the moment unable to probe physics well beyond the Planck scale in a reliable way (i.e. without discretization artifacts)"

"In [14] the general sigma -> 0 limit of the spectral dimension for asymptotically safe gravity in d dimensions was derived, resulting in Ds(sigma -> 0) = d/2. Interestingly, for d = 2 this coincides with the AJL results, but for d = 3 it disagrees with the results found here."

"In (n + 1)-dimensional Horava-Lifgarbagez gravity with characteristic exponent z one finds [19] that Ds(sigma -> 0) = 1 + n/z. Hence Ds(sigma -> 0) = 2 in the (2 + 1)-dimensional case with z = 2, which is the (2+1)-dimensional analogue of the (3+1)-dimensional case with z = 3 proposed in [26], n = z being the critical dimension of models characterized by z."

"In the context of spin foam models of three-dimensional quantum gravity, the results of [18] give Ds(sigma -> 0) = 2, but only after a transition at Ds ~ 1.5 for small positive sigma. This behaviour could either be an artifact of the method used to determine the spectral dimension or something characteristic of spin foams. In any case, keeping in mind the existence of a minimal length in spin foam models, the limit sigma -> 0 should probably be interpreted with some care."

"We note that, as far as short scale spectral dimension is concerned, kappa-Minkowski agrees with CDT in d = 3 but disagrees in d = 4, a situation which seems to be opposite to that found about asymptotically safe gravity."

 

[atyy]

http://arxiv.org/abs/0912.0220
"For smaller diffusion times, the spectral dimension approaches zero"

http://arxiv.org/abs/1009.1136
"one can always find a time t small enough that the second term dominates, leading to an effective spectral dimension of two."

 

[marcus]

Thanks to all who responded! It seems my idea was probably mistaken---both TomS and MTd2 say that the 1-D behavior of spin can not have anything to do with the conjectured 1D nature of space at small scale.

I will try to explain why intuitively I thought it might be. First, I don't know whether the CDT etc result is right or not, but suppose it is. Suppose for discussion's sake that spacetime is 2D and space is 1D down around Planck scale. Not just *spectral* dimension, that is just one way to measure, but Haussdorf. The volume goes up linearly with the radius. There is only one direction to point (more exactly two, this way plus and back the other way minus. Or if you prefer, up or down.).

What I'm thinking is, imagine something very small that has a vector, which necessarily lives in a 1D world, and suppose you look at it and start to increase the dimensionality and give it more ways to point, what happens? Would it not depend on what choice you offered it?

The vector does not have to be a spin vector, just any vector attached to a very small object. But the spin of a point particle does provide an example to think about.

Of course, this train of thought can lead nowhere or simply be silly (as MTd2's comments suggested.)

What I am trying to do here is to take seriously the idea that space could be 1D at small scale. What would that mean? How would the richer dimensionality come into play as you gradually expanded scale and new degrees of freedom became excited? What would that look like?

And maybe it is NOT 1D at small scale, but try to think how things would be if it were.

 

[tom.stoer]

Not just *spectral* dimension, that is just one way to measure, but Haussdorf.

I am not so sure if you are always able to define the dimension as you like. In the end all definitions should agree (or should agree within a certain range; it's clear that values for fractional dimensions will never match exactly).

The vector does not have to be a spin vector, just any vector attached to a very small object. But the spin of a point particle does provide an example to think about.

What I am trying to do here is to take seriously the idea that space could be 1D at small scale. What would that mean? How would the richer dimensionality come into play as you gradually expanded scale and new degrees of freedom became excited? What would that look like?

I don't think that you are able to do this.

I think there are several options
1) start with spin networks (LQG)
2) start with a smooth manifold (e.g. AS)
3) start with something else (e.g. CDT)

1) if you start with spin networks then the topology of "space" has to be defined on a spin network, i.e. on a colored graph. But the coloring does in no way affect the the definition of "dimension". The dimension is a "topological quantity" defined entirely by the uncolored graph (its vertices and links). The coloring itself affects the dynamics which causes the graph to change, which eventually even causes the "dimenion of the graph" to change, but it does not affect the definition of the coloring. So in LQG SU(2) spin networks stay SU(2) spin networks, regardless what you are doing or what they are doing ...

2) if you have a smooth manifold then its symmetry structure is fixed. Therefore you can define a change in dimension only if you abandon the manifold. That's to some extend what happens in AS.

3) if you start with something else then it's questionable how to define spin at all. I do not see how you can define spin on a triangulated space. It's symmetries according to Poincare invariance are lost and can be established only in the continuum limit. Therefore you would have to introduce approximate symmetries or some discrete structure which converges to a smooth symmetry structure in a certain limit. I do not know what they are doing in CDT, but I have never seen something out of which spin could emerge.

Last but not least it seems that this dimensional reduction is rather robust w.r.t. changes in the fundamental approach. That's why I asked some time ago regarding SU(N) "spin networks". I could imagine that spin networks with some appropriate dynamics defined on a different symmetry group may as well show a short-distance scaling to two-dim. spacetime.

What I see is that there are two different approaches:
A) start with SU(2) spin as a fundamental concept which cannot be questioned and which will never change, regardless what happens to dimension; this is the LQ approach
B) start with something different, something which neither relies on SU(2) nor manifolds nor anything else; the you would have to explain how all these different concepts can emerge.I doubt that you will find "spin of 2-dim. spacetime" corresponding to SO(1,1) at small distances as there will be no spacetime at all to which you can apply this concept.

 

[Fra]

What I am trying to do here is to take seriously the idea that space could be 1D at small scale. What would that mean? How would the richer dimensionality come into play as you gradually expanded scale and new degrees of freedom became excited? What would that look like?

It seems natural to me if you see it from the information perspective, that has you reduce the complexity of the interacting parts (which it presumably means to look at the "microstructure of space") then from the inference perspective the simplest extension of diversity or history of a distinguishable state would be like an ordered set, where the ordering either is timeordering or simply ordered as by it's information divergence from any point. This seems to naturally form the backbone for a 1D as the first emergent one as the complexity increases from zero.

Maybe another conceptual picture would be that the parts interacting down at this level are so "stupid" that they can only encode 1D interactions. A 2D code of that size may be unstable.

I vote for Tom's third option

"if you start with something else then it's questionable how to define spin at all. "

Maybe just consider a quantum of action, in general state spaces? The action system will then define a "structure" in the abstract state space by ordered transitions. What dimensionaliy this can be seen as a backbone for then depends on the rules of the action network, and ideally these can evolve. And maybe there are some information theoretic abstractions as to what the simplest possible action networks are, as the constrain the total capacity.

The step to 1D strings seems not so far. It's hard to envision how to construct a 2D continuum without first cosntructing a 1D continuum. Somehow the dimensions need ordering, and the limiting procedures for the respective continuum constructions need to be done in order.

/Fredrik

 

[marcus]

What I am trying to do here is to take seriously the idea that space could be 1D at small scale.

...What I am trying to do here is to take seriously the idea that space could be 1D at small scale...

I don't think that you are able to do this.

I think there are several options
1) start with spin networks (LQG)
2) start with a smooth manifold (e.g. AS)
3) start with something else (e.g. CDT)

...

Just to clarify. What I'm trying to do is take seriously the smallscale 1-dimensionality idea. I am not trying to do this in some specific theoretical framework (like LQG, AS, CDT). I want to face this possibility directly, and try thinking about it, regardless of how the theoretical framework might ultimately turn out. A naive mental experiment, if you like.

I do not believe that, in trying to take the possibility seriously, I am necessarily required to opt for one of those three options.

I do not see how you can define spin on a triangulated space. ... I do not know what they are doing in CDT, but I have never seen something out of which spin could emerge.

Last but not least it seems that this dimensional reduction is rather robust w.r.t. changes in the fundamental approach.

I also do not see how to define a particle's spin in a triangulated space. I have not yet seen CDT people even put matter into 4D CDT. But now I am not trying to think in the CDT framework, or any particular QG framework.
I agree with what you say about "rather robust". Rightly or wrongly, dimensional reduction comes out in several fundamentally different approaches.

Therefore I want to consider, independently of any approach, what it would be like if in fact space is 1D at micro level. So there are fewer DoF than we expect. And then suppose there is some physical quantity associated with a spatial vector...living at micro level.

Suppose there is a physical quantity associated with direction, orientation, that lives at that micro level. How do we, in macroscopic 3D space, experience that quantity?

It is an orientation, or direction, which only knows 1D dimensionality. I don't necessarily think of it as the axis of a real spin, just as a vector living in the 1D microspatial environment (which CDT and a variety of other QG tell us to expect, but don't necessarily tell us fully how to accommodate conceptually.)

 

[tom.stoer]

OK, now it's clear; sorry for the confusion

 

[atyy]

Just to clarify. What I'm trying to do is take seriously the smallscale 1-dimensionality idea. I am not trying to do this in some specific theoretical framework (like LQG, AS, CDT). I want to face this possibility directly, and try thinking about it, regardless of how the theoretical framework might ultimately turn out. A naive mental experiment, if you like.

Let's take the one of the CDT proposals measuring dimension seen by a random walker - the thing that I don't understand is how is this background independent? In their mathematics, the random walker does not affect the background fractal(like) geometry, but in real life, the random walker must be a material probe with mass that presumably affects background fractal geometry.

 

[MTd2]

So, how about getting rid of spin? Well, spin is related to space time and with so many things emergent, so should spin.

 

[marcus]

OK, now it's clear; sorry for the confusion

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.

 

[MTd2]

I see I connection now. Hmm. If space time is emergent, meaning, you get a manifold out of something else, spin might come in the process. If the smallest bit of information is a direction, nothing more natural than choosing something like spin.

 

[marcus]

Let's take the one of the CDT proposals measuring dimension seen by a random walker - the thing that I don't understand is how is this background independent? In their mathematics, the random walker does not affect the background fractal(like) geometry, but in real life, the random walker must be a material probe with mass that presumably affects background fractal geometry.

Heh heh, Atyy you are back with your old worry about the "test particle" nullifying background independence. But this time it is the "random walker". I don't think the idea of test particle introduces any problem. And this random walker is not a real thing. It is just a method of calculating what dimensionality would be observed.

The dimensionality of the space around you at some given scale is a quantum observable.

If you imagine yourself as a tiny person looking around you in a quantum geometry world, one of the things you can do is measure the radius and volume of a sphere. The V will grow as some power R^d. d could be 1.5 or 2.6 or whatever. That is the dimensionality.

Or you can measure by sending an even smaller person out for a random walk and seeing if he gets lost or happens by accident to return. The higher d is the more apt to get lost.

The CDT people do not have the real thing, they only have a jagged approximation in the computer. They find it is easiest to measure dimensionality by the random walker. In any case it is just imagination. They actually do it in their simulated approximate 4D universe. It would be impossible for anything to actually take a walk in a 4D block universe----there is no time. So we don't worry about anything deforming the 4D geometry.

They generate a random 4D block universe, pick a point in it, and perform a diffusion from that point (a random walk) for a certain length. They do this over and over again from the same point until they know the probability of returning or getting lost. A diffusion rate. It is not a real thing, only imagined. It is just a mathematical way to explore the geometry of the given 4D block universe.

The mass of the "test particle" does not matter. I think

 

[marcus]

I see I connection now. Hmm. If space time is emergent, meaning, you get a manifold out of something else, spin might come in the process. If the smallest bit of information is a direction, nothing more natural than choosing something like spin.

MTd2 in your two previous two comments you said "crazy" and in another one "creepy".
I think these are encouraging comments and I also think the thought is crazy and creepy, but maybe good to try out anyway.

 

[apeiron]

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.

I would suggest that spin is the result of 0D rather than 1D.

As argued previously, the dimensional reduction story is about the constraint of a physical action (a spacetime event represented by a vector) to the point where a context for the action no longer seems visible. You just have a naked vector "floating" in a vagueness. The model has collapsed the other two dimensions that served to distinguish the vector as a third direction, so all you are left with is the image of raw actions and no background space in which they are embedded.

So the physical interpretation of the 1D vector is just that - an action with a direction and no context.

But if we could further constrain the situation so that there is just a point (existing in time) then we can know we have successfully removed all possibilities of action...apart from spin. Spin becomes the remaining irreducible property of a point. And what is not forbidden becomes something that must exist. A 0D point has that last degree of freedom that cannot be removed.

So I think you are getting at something essential - if we strip away everything that can be stripped away (ie: if we constrain dimensionality) is there something irreducible? For a 0D locale that exists in time (so at least has that dimension), then you can take away everything but its potential to rotate.

And the step before that in dimensional reduction takes away all directions of motion apart from one, so you have just a naked vector with no embedding spatial backdrop to orient its action - define the other directions it is not pointing.

Of course, all this seems a little artificial in that dimensional reduction seems to me - as far as I can follow the work - to fall out of the fact that the Planck scale is wired into the models. Dimensionality becomes vague at the Planck scale in these models because all actions by simple definition do start to have the one scale.

You probably won't agree with my answer as you will want to think about physical action as a matter of construction. It seems natural to start with nothing (no degrees of freedom) and then start gluing them together to construct your desired spacetime.

But the reason I like dimensional reduction approaches is because they are in fact coming from the other direction - a subtraction of degrees of freedom. Or more accurately, a constraint on degrees of freedom. You can start with a presumption that there exists an infinity of degrees of freedom (a naked QM-like potential or indeterminacy). And then - through self-organising constraint - there is a dimensional reduction.

 

[Fra]

You can start with a presumption that there exists an infinity of degrees of freedom

Is this a rational or reasonable starting point in your view?

/Fredrik

 

[tom.stoer]

So the approach is to invent or identify a 1-dim. structure from which spin, spacetime etc. is emergent in an appropriate limit.

 

[apeiron]

Is this a rational or reasonable starting point in your view?

/Fredrik

Yes, why not? Though you may not understand in detail what I am arguing.

I'll make a few simple points, and if you want to pursue the argument further, that's fine.

Is it more reasonable that reality develops from a state of nothingness or a state of everythingness? If you have everything, then it is easy to constrain its variety to be left with something more limited. But from nothing must come nothing. So it is a "reasonable enough" approach just on those grounds.

And it is indeed an approach that has been considered reasonable since ancient greece - Anaximander's cosmology, Aristotle's being and becoming, Plato's plenum.

Now models like CDT already presume a greater state of "somethingness" as they start their dimensional reduction from 4D. But is it unreasonable in modelling to seek greater generality? And the most general possible concept of a number of dimensions, of degrees of freedom, is an infinite number.

And this is in fact really demanded by the notion of constraint itself. A constraint is a limit, and so what it must act on is the unlimited. And so as our raw material here, our initial conditions, we must imagine a state which is completely unlimited. Partially unlimited is just not good enough otherwise we would have to explain why there are limits - and they are not the product of constraints?

So if you are going to take a constraints based approach, then the raw material, the raw potential, or however else you want to characterise the initial conditions, must properly be taken as that which lacks limits. ie: an infinity of degrees of freedom.

Of course, the argument gets more complex after here. 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!

Anyway, don't worry about the background machinery of the approach if it troubles you as even a dimensional reduction from some definite number of dimensions like 4, or 11 if you prefer, would still give you the same view of why spin is an irreducible property even after all other degrees of freedom have been constrained.

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.

 

[Fra]

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 a lot 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

 

[Fra]

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

 

[apeiron]

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 a lot 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

 

[apeiron]

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.

 

[marcus]

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.

 

[arivero]

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)?

 

[MTd2]

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?

 

[marcus]

... 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?

 

[MTd2]

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.

 

[marcus]

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.