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

  1. Sep 11, 2010 #1

    marcus

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    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.
     
    Last edited: Sep 11, 2010
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  3. Sep 11, 2010 #2

    marcus

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    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.
     
    Last edited: Sep 11, 2010
  4. Sep 12, 2010 #3

    MTd2

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    At 2D, spin is not quantized in discreet values. So, something crazy happens in the middle of the way.
     
  5. Sep 12, 2010 #4

    arivero

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    SpaceTime dimension 2, or Space dimension 2?
     
  6. Sep 12, 2010 #5

    tom.stoer

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    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).
     
    Last edited: Sep 12, 2010
  7. Sep 12, 2010 #6

    MTd2

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    Space dimension 2, since marcus was talking about going down to 1.
     
  8. Sep 12, 2010 #7

    MTd2

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    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.
     
  9. Sep 12, 2010 #8

    tom.stoer

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    This is similar to what I said.
     
  10. Sep 12, 2010 #9

    atyy

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    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."
     
    Last edited: Sep 12, 2010
  11. Sep 12, 2010 #10

    atyy

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  12. Sep 12, 2010 #11

    marcus

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    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.
     
    Last edited: Sep 12, 2010
  13. Sep 13, 2010 #12

    tom.stoer

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    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).

    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.
     
  14. Sep 13, 2010 #13

    Fra

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    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
     
  15. Sep 13, 2010 #14

    marcus

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    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 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.)
     
    Last edited: Sep 13, 2010
  16. Sep 13, 2010 #15

    tom.stoer

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    OK, now it's clear; sorry for the confusion
     
  17. Sep 13, 2010 #16

    atyy

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    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.
     
  18. Sep 13, 2010 #17

    MTd2

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    So, how about getting rid of spin? Well, spin is related to space time and with so many things emergent, so should spin.
     
  19. Sep 13, 2010 #18

    marcus

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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.
     
  20. Sep 13, 2010 #19

    MTd2

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    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.
     
  21. Sep 13, 2010 #20

    marcus

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    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 Rd. 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 :smile:
     
    Last edited: Sep 13, 2010
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