# Half a Planck Length

1. Mar 25, 2006

### flotsam

I know the concept of half a Planck length is wrong. But could someone explain exactly why it is wrong. Also could someone tell me what we conclude about the photon that has moved at C in a planck time (for example) Do we conclude that the photon was at A and is now at B after the passing of a planck time unit and that is all? What I mean; is it pointless to say the photon was at some point halfway? as I suspect it is.

I eagerly await your response as I am engaged in an debate over this.

2. Mar 25, 2006

### Haelfix

There is absolutely nothing wrong with saying 'half a planck length', at least in our vanilla theories. The planck length is a quantity that is found by dimensional analysis of the fundamental constants in general relativity, roughly it is an order of magnitude quantity that says roughly at what scale (length or energy depending on how you look at it) gravity effects will become comparable to other forces.

Depending on what theory you are talking about, things may or may not be complicated. For instance as was explained to me by Lubos Motl, in string theory it is roughly the place where if you keep subdividing lower, any and all field will collapse into a mini blackhole state. So you no longer really are talking about smaller *things* but rather probing bigger and bigger blackholes (length of course is now somewhat of a meaningless classical concept as it will of course be warped into something new and distinctly quantum gravity like).

In other theories it may be a fundamental atom of space.

3. Mar 25, 2006

### arivero

It should be added that dimensional analysis of Gravity by itself gives a "quantum" of Area, and Planck length is to be extracted by other tricks, either directly taking a square root, or indirectly using comparision of this area against another (wave)length.

Also this quantum of Area is not very great deal if we consider space-time areas, ie areas where one side is x and the other one is ct. Because then a minimum A=x t c does not imply a minimum in the product x t in the classical non relativistic limit (c goes to infinity) .

4. Mar 25, 2006

### f-h

In the sense in which we expect it to be wrong it will be wrong like speaking of a harmonic oscillator with 1/4 hw Energy. You can never meassure it and the Energy can't have this expectation value either.

5. Mar 26, 2006

### arivero

f-h, I expect, the sense you speak of, to be valid for area but I am not sure for length.

6. Mar 26, 2006

### duke_nemmerle

Like someone else said, you can talk about half a planck length, you can talk about a millionth of a planck length if you like. It just so happens that there are a number of theories which state that a planck area(planck length squared)is the lowest amount of area we can talk about. That is: an atom of space.

7. Mar 27, 2006

### hossi

I too would say that there is a priory nothing wrong with half a Planck length or time. However, from several argument one would expect the Planck scale to act as a minimal scale as concerning the resolution that is achievable. You may find some phenomenological examinations of the features under the term 'Generalized Uncertainty Relation', some (very readable) reviews on the issue

http://arxiv.org/abs/gr-qc/9403008
Quantum gravity and minimum length
Authors: Luis J. Garay

http://arxiv.org/abs/gr-qc/0305019
Selected topics in Planck-scale physics
Authors: Y. Jack Ng

http://arxiv.org/abs/hep-ph/0410122
The Minimal Length and Large Extra Dimensions
Authors: Sabine Hossenfelder

It is pointless to say that the photon is a point. I.e. you can't localize it enough to make the statement at all.
Best,

B.

8. Mar 27, 2006

### louis arthur

i think it is ok for something to move half of a planck length. if it was unachievable and we made a right triangle with planck length on two sides, and let that something travel the hypotenuse, it is a valid distance to travel, otherwise we would have a world that is less dynamic.

hossi, you like to write alot. did i finally sound smart on a thread?

9. Mar 28, 2006

### hossi

Hi louis,

yeah, I write a lot. It scares my advisors.

Sounds smart. Still it doesn't make sense to me to talk about moving of 'something' from one 'point' to another, by fixing the points to a precision better than the Planck length, when you can't localize that 'something' up to a Planck length in the first place.

Best,

B.

10. Apr 16, 2006

### garrett

Hi Sabine,

I'm kind of confused by the justification and accommodation of a minimal length. The usual argument for a minimal length being bad is that it breaks Lorentz invariance, since an observer who sees a ruler fly by sees its length Lorentz contracted to
$$L' = \sqrt{1-(\frac{v}{c})^2} L$$
And if the ruler needs to be kept above the minimal length, a Plank length, then you have to go in and deform the Lorentz transformations to keep it from contracting the ruler -- the DSR approach.

Quantizing gravity is almost certainly going to give a minimal scale, from many points of view (as I saw listed in your paper). But aren't all these arguments equally satisfied by a minimal area? This would seem much more natural from LQG, certainly. And I think the Lorentz contraction argument goes away, since an area $T L$ (represented by a ruler and clock) flying by would be invariant under Lorentz rotation:
$$T' L' = (\frac{1}{\sqrt{1-(\frac{v}{c})^2}} T) (\sqrt{1-(\frac{v}{c})^2} L) = T L$$
And I think this holds under Lorentz transformations of areas with any orientation, no? Unless I'm being stupid, which wouldn't surprise me so much.

So, if all this is true, why do people mess with a minimal length when a minimal area seems to work great?

Thoughts?

-Garrett

11. Apr 16, 2006

### hossi

Hi garrett,

don't know, I am not a LQG person, but I think they have a minimal extension into all directions. I am not really sure what sense to make out of a length times time flying by? I mean, you can always look at some invariants instead, but does that solve the problem?

Anyway, I don't like this approach to the minimal length via DSR specifically, see my latest paper

Interpretation of Quantum Field Theories with a Minimal Length Scale

which is a careful reinvestigation about the interpretation of a minimal length, what this has to do with the recent DSR-approaches, and how it can be incorporated into quantum field theory.

The essence of the statement is that there is no contradiction of the type you mentioned, as long as both observers have not compared the length of their rulers. I.e. the one flies by, so what. The other one doesn't see it unless he measures it. This always requires an interaction. And it is the resolution of this interaction about which both should agree, not the length of the non-interacting ruler.

Meaning, in my interpretation, there is no DSR for the free particle, and this is not in contradiction with usual Lorentz-trafos. Quantum gravity effects become important for the exchange particles in a strongly gravitationally disturbed background, caused by the highly energetic particles. I see it as an alternative DSR approach. Makes less spectacular predictions, but has also less conceptual problems. I also think it goes along better with the emergence of a minimal length in string-theory (but this is just a speculation, not an actual result).

Anyway, this is conceptually different from the usual DSR. In this case, there is a modification of the Lorentz-trafo already for the free particle.

Best,

B.

12. Apr 16, 2006

### garrett

Oh yah, of course -- as long as you talk about a "minimal proper length", there is no Lorentz violation anyway, since that's what Lorentz transformations preserve.

Yikes, nonlinear group representations look scary.

I didn't look at your paper closely enough to see that you were diss'ing DSR -- I suppose your sticking your tongue out was concealed in the typically polite academic verbiage.

13. Apr 16, 2006

### hossi

Hi garret,

well, the paper makes it clear that there are two ways to interpret DSR. Apparently, I favor mine :tongue2: No, seriously, I am very open minded to the DSR approaches, but so far the 'usual' ones fail to come up with a reasonable effective QFT model. As long as I don't see how they solve the soccer-ball or energy-conservation problem (not to mention unitarity and gauge invariance), I will stick to my interpreation as the more physical one. At least, its the more useful one.

I wasn't talking about a proper lenght, but that should be conserved anyway, deformed or not-deformed special relativity. Yeah, indeed I think the group stuff is scary. (It's the more scary as I have to give a seminar about it in 2 weeks.)

Best,

B.

14. Apr 17, 2006

### garrett

Hmm, sounds like the wine and cheese is starting to taste like butter and bread...

But I guess scary can be good, if it's the right kind.

15. Apr 17, 2006

### hossi

My mother tought me never to go out and drink wine without having enough bread.

Seriously, giving a seminar every now and then is annoying, but it forces me to structure my thoughts, and to look into details I would have missed otherwise.

Best,

B.