Minimum Length from First Principles

In summary, the paper argues that there is a finite dimensional limit to the length component of a three dimension object. Any reduction of the object must be broken up at the finite limit in order to maintain the object's perfection.
  • #1
marcus
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in case anyone is interested

http://arxiv.org/abs/hep-th/0505144
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 can't measure it shouldn't we say it doesn't 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
 
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  • #2
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.
 
  • #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
 
  • #4
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.
 
  • #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 transferred 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 :
http://mathworld.wolfram.com/RegularPolygon.html

n=3

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
 
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  • #6
Spin_Network said:
Marcus rather than go and pose a Question elsewhere, I would like to ask it here, if you feel its Irrelevant please alter accordingly.
...

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
:smile:)

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
 
  • #7
Chronos said:
... be it what lays outside our Hubble bubble or inside a Planck length.

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?
 
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  • #8
Chronos said:
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.
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...
 
  • #9
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
jal
 
  • #10
I should include some inspirational references for my model in my blog.

http://xxx.lanl.gov/abs/gr-qc/9601014
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
ON THE FREE-WILL POSTULATE IN QUANTUM MECHANICS
Gerard ’t Hooft
15 Jan 2007

jal
 
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  • #11
marcus said:
this would not mean that space is necessarily discrete, only that you can't MEASURE a length smaller than such and such

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:
 
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1. What is the concept of Minimum Length from First Principles?

The concept of Minimum Length from First Principles is a fundamental principle in physics that states that there is a minimum distance that can exist between two objects. This distance is known as the Planck length and is approximately 1.6 x 10^-35 meters.

2. Why is the Planck length considered to be the minimum length?

The Planck length is considered to be the minimum length because it is the smallest length that can be measured according to the laws of quantum mechanics. At this length, the concept of space and time becomes meaningless, and the laws of physics as we know them break down.

3. How is the Planck length calculated?

The Planck length is calculated using the Planck constant (h) and the speed of light (c) in the following equation: lP = √(hG/2πc3), where G is the gravitational constant. This results in a value of approximately 1.6 x 10^-35 meters.

4. What implications does the concept of Minimum Length have on our understanding of the universe?

The concept of Minimum Length has significant implications on our understanding of the universe. It suggests that there is a limit to how small distances can be, and there may be other fundamental particles and forces that play a role at this scale. It also challenges our understanding of space and time and may lead to the development of a theory that unifies quantum mechanics and general relativity.

5. How does the concept of Minimum Length relate to other principles in physics?

The concept of Minimum Length is closely related to other principles in physics, such as the Heisenberg uncertainty principle and the concept of quantum foam. It also plays a role in theories such as string theory and loop quantum gravity, which attempt to explain the fundamental nature of the universe at the smallest scales.

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