# The Planck Circle Problem

1. Aug 31, 2006

### ModernTantalus

I was doing a thought experiment and came across something that I'm going to term the Planck Circle Problem for the sake of naming it something. The logic is as follows:

1.) All lengths contain an integer number of Planck Lengths.
2.) The radius and circumference of a circle are both lengths.
3.) Therefore, the radius and circumference of a circle must both contain an integer number of Plank Lengths.
4.) A circle cannot have both a radius and circumference with an integer number of units because pi is a transcendental number.

I asked a mathemetician about this problem and his response was that the concept of a fundamental unit of length is impossible. However, physicist are just as adamant in claiming that a universal unit of length does exist. What am I missing?

2. Aug 31, 2006

### Mike2

There may be curvature in the space which would then allow the radius to be a different fraction of the circumference than pi.

3. Aug 31, 2006

Staff Emeritus
If the circlumference contains only an integer number of lengths then it is not a circle, but a polygon, and the use of pi is inapprpriate.

4. Aug 31, 2006

### Careful

The physicists circle is not the Platonic circle of mathematics. The Planck length is about 10^{-35} meter, so that any circle of atomic size, say radius of 10^{-9} meter, is going to contain roughly 2 \Pi * 10^{26} planck units, this number is so huge that you can safely neglect all digits behind the 27'th digit in \Pi.

Careful

5. Aug 31, 2006

### arivero

To me, the answer is that Planck Length does not measure a length but a force, its definition comes from the fact that Planck *area* is the coupling constant of gravity.

6. Aug 31, 2006

### arivero

As for your mathematician friend, are you sure he spell "unit"? Of course a unit can exist. For instance the meter is a unit of length, and you have distances smaller than one meter. A fundamental unit is just as the meter, only that it is fundamental in the sense it comes given from Nature.

7. Aug 31, 2006

### ModernTantalus

I'm reading up and it seems that you can't measure anything beneath the Planck length, so something with a length of 0.5 Planck lengths would have no meaning. What about something with 3.5 Planck lengths though? Measuring an object at 3.5 Planck lengths wouldn't produce a micro-black hole, so would this measurment be valid, or must all measures of length be in integer numbers?

8. Aug 31, 2006

### topovrs

It is not clear that all lengths must be a multiple of the Planck length, or areas a multiple of the Planck area. While the smallest eigenvalues of the length and area operators may have these values, the spectra do not necessarily have just integer multiples, and they may even become continuous at high multiples.

The minimum values are really only of interest if you want to investigate a universe that is Planck scale - not at all like our real universe. I also know of no way of performing relevant measurements, since that would be like checking to see whether the minimum change in the size of the entire universe is a Planck unit!

9. Sep 1, 2006

### Careful

You are thinking here about spin foam or spin network models based on the Lorentz group instead of SU(2) ? All this does not really matter, it is just some result which cannot be verified against any reasonable test.

10. Sep 1, 2006

### ModernTantalus

Thanks for the responses. Careful is probably closest to the solution that I concluded after some introspection. It occured to me that because of the way that a Planck length is defined that it does not represent the smallest unit of measure, but a minimum uncertainty in line with the Heisenburg Uncertainty Principle. Basically, what can I know about a circle with a diameter (D) = 1 Planck Length? Well, I can't tell that the length is exactly one. Not even thoretically. The best I can do is determine given length to within + or - 1 Planck Length. I can, however, determine that 0<D<=2. This means that my circumference (C) is 0<C<=7. In other words, due to the inherent uncertainty in measurements, I can only estimate to whole number values of the Planck Length.

11. Sep 1, 2006

### etera

If your initial motivation comes from quantum gravity considerations, it is true that the standard black hole argument tells you that the minimal length uncertainty is the Planck length. Indeed, one says that if the probe (photon for example) has a wavelength of the Planck length, then it would create a black hole and you would not be able to perfom any measurement. Of course, this relies on the assumption that the black hole laws (Schwardzschil metric) holds down to the Planck scale, which can not be true once you reached the conclusion that the space(time) must be discrete at the Planck scale... but let's forget that tiny loophole, and simply consider what kind of discrete space(time) structure one can have at the Planck scale.

So your Planck circle "paradox" is very similar to the issue of the Lorentz contraction. Usually, if one wants to keep a Lorentz invariant theory (such as special relativity), one will naturally get the Lorentz contraction of length for boosted observers. Then the Planck length would get squeezed to something smaller.. and thus problem..! Obviously, if we want to keep a discrete structure for lengths, we should also quantize the rapidities (speeds) allowed in the theory. At an even more basic level, if we imagine two points distant from the origin (you, the observer) from a planck length unit lp, but such that the two directions form a 90deg angle, then the distance between them should be lp*sqrt(2). So either we must assume that the two directions can not form a 90deg angle and therefore angles should also get quantized, or that what we call distance is not as simple as this.

A first idea would be to go to a squared lattice. Then we directly realize that the length is not a integer multiple of the Planck length but that the length squared is a integer multiple of the Planck length squared by Pythagoras theorem. By the way, this fits with the LQG expectation that it is the area that is quantized. Of course, the circle remains a problem. But of course, what we call a "circle" is not really circular anymore, it is a polygon. And the 2*Pi law is only approximate. More precisely, if you define the circle as the set of points on the lattice at a certain given distance from the origin, and you define the circumference as the sum of the distance between each closest neighbour, then you will see that 2*Pi is the (never reached) superior limit of the ratio circumference/radius.

Okay, but it does not seem very satisfying to describe the beautiful space(time) as a "stupid" squared lattice... So one way to do better is to go quantum. Indeed, if we look at the spin of a particle, then we say it is quantized. What we mean is that its measured value is discrete but its expectation value can be an arbitrary real number. The same way, we can promote the length to a quantum observable and make it an operator. It could have a discrete spectrum while its expectation value can still be arbitrary. For example, we can now boost a Planck length ruler, a boosted observer (an ensemble of observers) would measure in average the usual length contraction. In practice, they would measure the Planck length again, or sometimes bigger, or actually 0 (the ruler becomes a point..). This is the route followed by non-commutative geometry.

But, at the end of the day, i think the best answer is, as someone said earlier, that when you do a length measurement, you always cut your distance in small straight pieces and always end up measuring a polygon. So 2*Pi is only an idealization anyway.

12. Sep 27, 2006

### jal

Good day!
This is my first post.
Spontaneous Symmetry Breakdown and Perspective of Higgs Mechanism
21 March 2006

between the negative value φ(−)0 and the positive one φ(+) 0
Please explain why you can apply the uncertainty over an area that does not exist?
YOU CANNOT PRETEND THAT THE PLANCK LENGTH IS A POINT. THAT IS A MAJOR ERROR. THE PLANCK LENGTH IS 100% CERTAIN.
The Planck lengths must be conserved. Therefore, you can only use “Uncertainty Principle”, “entangled states”, “Quantum indeterminacy” , where the waves are located and there cannot be any waves less than the Planck length.

2. What is the smallest distance in our universe?
4. What is the smallest sphere, bubble, in our universe?
The answer has been worked out by James G. Gilson at
http://www.maths.qmul.ac.uk/~jgg/gil0.pdf [Broken]

http://www.geocities.com/j_jall/4pi.gif
Don’t forget….. that is only the 2D. You must bring it to a 3d size.
http://www.geocities.com/j_jall/instanton.gif
Lets see ….. would it also mean that the minimum size would also scale to 2 X of what we had…. 2(3(2 pi ))? (Check my math/logic)

Spontaneous Symmetry Breakdown and Perspective of Higgs Mechanism
21 March 2006

between the negative value φ(−)0 and the positive one φ(+) 0
That is how the mexican hat is determined... by applying uncertainty around zero/a point.
Please explain why you can apply the uncertainty over an area that does not exist?
A point…..
If CERN does not find SUSY it does not find a point.
If they do not find a point then our understanding/explanation of our universe must be modified.
jal

Last edited by a moderator: May 2, 2017
13. Sep 29, 2006

### jal

I guess that I'm doing a bad job of posting.
I have opened my journal for viewing.
Maybe .....???
WHAT HAPPENS IF THEY DON’T FIND THE MEXICAN HAT?
(Which was arrived by applying uncertainty around zero/a point.)
You make a PLAN “B”.
There are more amateurs then professionals.
You therefore, organize a party and ask everybody to come with anything that they can find that does not break the Planck scale.
If there is anything that looks promising then it can be pursued further.
The professionals have the abilities and tools to do more than the amateurs.
Most professionals will not be interested or inclined to look for a PLAN “B”.
Some will be interested. See:
http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008.pdf [Broken]
The mathematical basis for deterministic quantum mechanics
By; Gerard ’t Hooft
Doing this endeavor might help the professionals and it will certainly help the amateurs.