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

In summary, there are various approaches to quantum gravity that suggest a gradual decrease in spatial dimensionality from 3D to 1D at the Planck scale. This is supported by evidence from simulations and theoretical calculations. Spin, on the other hand, does not seem to be directly related to this phenomenon.
  • #36
arivero said:
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 :-p) 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.
 
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  • #37
apeiron said:
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
arivero said:
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.

arivero said:
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
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.
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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
MTd2 said:
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
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