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Minimum Length from First Principles

  1. May 17, 2005 #1


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    in case anyone is interested

    Minimum Length from First Principles
    Xavier Calmet, Michael Graesser, Stephen D. H. Hsu
    8 pages, Honorable Mention in the 2005 Gravity Research Foundation Essay Competition

    "We show that no device or gedanken experiment is capable of measuring a distance less than the Planck length. By "measuring a distance less than the Planck length" we mean, technically, resolve the eigenvalues of the position operator to within that accuracy. The only assumptions in our argument are causality, the uncertainty principle from quantum mechanics and a dynamical criteria for gravitational collapse from classical general relativity called the hoop conjecture. The inability of any gedanken experiment to measure a sub-Planckian distance suggests the existence of a minimal length."

    if they happened to be right, this would not mean that space is necessarily discrete, only that you can't MEASURE a length smaller than such and such.
    quantum theories tend to be about measurement, about what one system can know or not know about another system, about observation of one thing by another, they tend to shy away from a notion of some absolute reality (that is "there" regardless of who is measuring or observing what).

    so? so you can't measure any length finer than planck length accuracy? it does not mean that smaller lengths do not exist, only that measurment fails. well if you cant measure it shouldnt we say it doesnt exist :confused:

    anyway CDT and LQG are based on topological continuums, LQG is even based on a (smooth) differentiable manifold of definite preselected dimension (usually 4D). CDT is a continuum limit and the main authors say they have found no indication that spacetime is discrete.

    so maybe spacetime really is a continuum, but some mechanism we do not understand yet puts a limit on the precision of measurement

    have fun
  2. jcsd
  3. May 17, 2005 #2


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    The thing is that when you are dealing with quantum mechanics the reason you can't measure less than a certain length is because lengths smaller than that size are inherently ill defined.
  4. May 17, 2005 #3
    since the metric of space and time emerge from the relationships of events- the smallest space/time would be the simplest signal between two events- this relationship would correspond to an emergent and discrete area/duration- to say that a 'smaller' or 'briefer' area/duration exists would be impossible- as it would require a relationship of less than 2 separate events- a single isolated event does not establish any relationships with other events- and thus cannot provide the fundamental substrate for a distance metric or a duration-

    a smaller/briefer area/duration can only be established if two events interact in a way that provides a smaller/briefer relationship than any other relationship in the observed universe- such relationships might randomly or orderly occur 'below' the Planck scale= but since they appear to have no compatible/relative relationship with the rest of the world such events would have no meaning or 'reality' to us- and no map-able comparability with our world
  5. May 17, 2005 #4


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    I think you hit upon the heart of the issue, marcus. Unobservable regions of the universe are not scientifically useful - be it what lays outside our hubble bubble or inside a planck length.
  6. May 19, 2005 #5
    Marcus rather than go and pose a Question elsewhere, I would like to ask it here, if you feel its Irrelevant please alter accordingly.

    Question, what is the finite dimensional limit that a 3-Dimensional length componant can exist within?..can a 'bit' of 3-D space, be transfered to 2-D space.

    If one has a Triangle of certain size, and separates it into three equal 'bits', lengths, take one of those lengths, fold it (reform) into another triangle you end up with a scaled down representation of the original triangle, which can be further reduced by repeating the action of 'separation and reforming'.

    Now in the action of reduction, or transferance, between 3-D to 2-D 'bits', the reduction of a 3-D 'anything', must be broken up at the finite limit of reduction, and the way to imagine this is to place the first scale reduced Triangle inside the original Triangle, is this first scale representation 'continuous'?...or is it discrete?

    Important to realize that the FIRST scale 'triangle' reduction, HAS to be broken in order to reach the very first scale reduction!

    It may be that to define a minimum length, is to define a scale transformation?

    Smolin has a paper that uses 'Voronoi Diagrams?' I believe?..but to image what a 2-D boundery(lines), emerging from a structure of 3-D look here:
    http://www.cosy.sbg.ac.at/~held/projects/voronoi_2d/04_vd.gif where n=5 (look at the mathworld link below for visual correlation).

    But for a visual description of what is involved for clarity: http://www-2.cs.cmu.edu/~quake/triangle.html

    and geometric terms:http://www-2.cs.cmu.edu/~quake/triangle.defs.html

    There is a question of what can be achieved and what cannot be achieved?

    In 3-D a :http://mathworld.wolfram.com/EquilateralTriangle.html

    is 'perfect'..it is the perfect geomerty for a given portion of Space, which can be embedded into another 'perfect' Sphere Space, see half way down to :


    So Mohammad H. Ansari and Lee Smolin's paper gives a good overview, page 3 figure 2 :http://uk.arxiv.org/PS_cache/hep-th/pdf/0412/0412307.pdf
    Last edited: May 19, 2005
  7. May 19, 2005 #6


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    Spin, I think it is fine to ask this question in this thread. And I can't alter other folk's posts (so it wouldn't matter if I DIDN'T think it was fine

    but my plate is full now with other stuff. so all i can do is hope someone else replies!

    BTW thanks for reminding us that Ansari has co-authored with Smolin.
    I will look Ansari up, I think he must be a graduate student or postdoc at Perimeter. Ansari just co-wrote a CDT paper with Fotini Markopoulou
  8. May 19, 2005 #7


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    a knack for aphorism :smile:
    also you tend to have neat sigs---I like the fish one and especially the Douglas Adams

    I am not sure what I think about these issues. Only posted the article because I thought the topic was of general interest.

    I am intrigued what I see as a theoretical difference between CDT and LQG. The CDT people say that in CDT there is no indication of a minimal length or of a fundamental discreteness (to area, volume, length whatever). Or so the CDT people say. they are frankly indiscrete. But the LQG people are more equivocal about it.

    maybe there is a smallest MEASUREABLE length but so far it has no role in the theoretical picture. I do not feel philosophically equipped to deal with a situation where something has no operational meaning, with no practical proceedure for measuring it, and yet to keep the theory SIMPLE you have to imagine that it exists :yuck:
    because it would be more bother than it's worth to modify the theory so that the unmeasurable things are excised.

    basically they can say whatever they want about unobservable accessories as long as they eventually make good predictions about what we CAN measure, would that be fair?
    Last edited: May 19, 2005
  9. May 19, 2005 #8


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    Actually, I am not confortable about things being inside a Planck length. I'd be happier if at least they were inside a Planck area or inside a Planck volume...
  10. Feb 12, 2007 #9


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    Had the investigation of minimum length continued to asking,
    “What does the Quantum Minimum Length Structure look like?”
    I think that the conclusion would have been,
    The “quantum minimum length structure” is the building block of our universe.

    See my blog
  11. Feb 13, 2007 #10


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    I should include some inspirational references for my model in my blog.

    Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness
    Authors: G. 't Hooft
    http://arxiv.org/PS_cache/quant-ph/pdf/0701/0701097.pdf [Broken]
    Gerard ’t Hooft
    15 Jan 2007

    Last edited by a moderator: Apr 22, 2017 at 3:46 PM
  12. Mar 6, 2007 #11


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    To measure something with an unlimited precision you would need a point-like measuring device, that is to say, an ideal mathematical entity. In the real world the position and momentum of any particle is measured by using another particle. If the particles are not mathematical points (i.e. having finite energies) then it is obvious that the accuracy of any measurement would be restricted. It is pretty much transparent in composite models, but even the SM particles by having form-factors are not point-like objects, so that accuracy is always limited. There is no need in lengthy papers to explain this simple fact. :wink:
    Last edited: Mar 6, 2007
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