Is a Planck length really the smallest length?

[Jow]

I am just an interested high school physics student so I don't really know much about the subject, but I am given to understand that a Planck length is the smallest length. It cannot be further subdivided and no length smaller can be measured even with a great advancement in measuring techniques.
My question is this: does this mean that space is really made up of indivisible pockets, or is it simply that measurement looses meaning if you get any smaller?

 

[Meir Achuz]

The Planck length is not the smallest length.
It is just a convenient unit for very short lengths.

 

[Jorriss]

I'm speaking beyond my knowledge here but I suspect there is not a smallest length or else the smallest length in one reference frame would be smaller in a second frame moving with respect to the first.

Is that wrong?

 

[Doc2626]

Jow, I wouldn't say that a Planck length is the shortest length, per se... better said, it's the shortest unit of measurement of length

 

[Jorriss]

Jow, I wouldn't say that a Planck length is the shortest length, per se... better said, it's the shortest unit of measurement of length

Why is that better? One can just define a new unit as half the Planck length.

 

[DiracPool]

Why is that better? One can just define a new unit as half the Planck length.

This is a great thread topic. My understanding is that the Planck length is NOT just an arbitrary unit of measurement, but rather is found and is constrained in the same way all the Planck units are, specifically by analyzing the relationships between the fundamental constants of nature. That would be G, c, h-bar, coulombs constant, and Boltzmann's constant. The Planck length is the square root of (h-bar times G divided by c cubed).

So, again, from my understanding, the value for this length is a theoretical value based on the relationship between these fundamental constants. I don’t think we’ll ever know if there is anything smaller cause we’d have to build a particle accelerator the size of the galaxy to do so. As of now, we can only measure 10^-19m whereas the Planck length is 10^-35m.

This is what puzzles me when researchers say that an electron and a quark have no size, or no measureable size. Does that actually mean no size at all? Or does it mean that if it does have size it is less than 10^-19m? Also, is there an apriori reason to think that the Planck length is the absolute smallest length/size that can exist, or is it just that it represents a fundamental length of some sort. Just because it is a fundamental length doesn’t necessarily mean that it is the shortest possible length, does it? Look at the Planck mass, that is way bigger than the smallest mass we can measure.

 

[Evert]

As far as I get it, Planck length should be measured everywhere in the universe as the same length. (It contains constants so it stays constant right?) So for every observer, nothing happens at scales smaller than Planck length?
So particles appear at 1 spot, disappear for Planck time, and show up 1 Planck length further of that spot? (for light speed particles)

Or is my understanding totally wrong?

 

[Naty1]

For the OP:

No one really knows if space and time are discrete or continuous.

Here are two views from my notes..usuallly from discussions in these forums:

Current research is focused on the nature of spacetime at the Planck scale. Loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity even makes precise predictions about the geometry of spacetime at the Planck scale.

http://arxiv.org/abs/1010.4354

“The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any band limited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the band limit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possesses an ultraviolet cutoff.”

http://arxiv.org/abs/0708.0062
On Information Theory, Spectral Geometry and Quantum Gravity
Achim Kempf, Robert Martin
4 pages
(Submitted on 1 Aug 2007)
We show that there exists a deep link between the two disciplines of information theory and spectral geometry.

Regarding Planck length: it is hypothesized that below Planck length, space and time and other entities we see at maco scales become indistinguishable amid quantum foam...undulations of background vacuum energy. ..

from Wikipedia: Planck length

The physical significance of the Planck length is a topic of research. Because the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is currently no way of probing this length scale directly. Research on the Planck length is therefore mostly theoretical. According to the generalized uncertainty principle, the Planck length is in principle, within a factor of order unity, the shortest measurable length - and no improvements in measurement instruments could change that.
In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; often it is suggested that spacetime might have a discrete or foamy structure at Planck length scale.

 

[Naty1]

[I'm speaking beyond my knowledge here but I suspect there is not a smallest length or else the smallest length in one reference frame would be smaller in a second frame moving with respect to the first.

Is that wrong?/QUOTE]

That's one view from special relativity. But at this scale neither GR nor quantum mechanics can handle the effects of gravity and energy, for example...so we just don't have a consistent theory yet.



The phenomena of a constant Planck scale is known as double special relativity.

http://en.wikipedia.org/wiki/Double_special_relativity

...a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but an observer-independent maximum energy scale and minimum length scale (the Planck energy and Planck length).

 

[Naty1]

Wikipedia on PLANCK LENGTH gives the definition:

L = root [hbar x G]/c³http://en.wikipedia.org/wiki/Planck_length

DIRACPOOL's opening explanations is correct.

when researchers say that an electron and a quark have no size, or no measureable size. Does that actually mean no size at all?

A point particle is an idealization often used in our mathematical models. But no size has yet been detected for elementary particles. However, the concept of a point particle is complicated by the Heisenberg uncertainty principle where an elementary particle, with no internal structure, occupies a nonzero volume.

In Coulomb's law, which describes the electric force between two point charges, the electric field associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero making energy (thus mass) of point charge infinite. That sure doesn't seem right! But the Pauli exclusion principle [of quantum mechanics] prevents such fermions from occupying the same quantum state simultaneously...so maybe they can't be in the same place anyway.

 

[haael]

Planck length might be the smallest length we will ever be able to measure, but that does not mean it is an absolutely the smallest length. It is rather the smallest possible size of a particle as we understand it now.

To see a structure of any object, we must shoot it with some particles of certain energy. De Broglie wavelength of the probe particles must be smaller than the size of the structure. Maximal energy of the probe particles sets the limit of the magnifying abilities of our measurement device. To increase "resolution" of the picture, we have to increase the energy.

But at some point, the probing particles reach so high energy level, that they become black holes. This is the Planck length. We could of course use even more powerful particles, but they would not be smaller. Their Schwarzschild radius will get bigger, so the resolution will not increase.

So, the Planck length is the smallest size of any structure we will be able to see judging from the current state of knowledge. It doesn't mean that smaller structures don't exist - we will just not be able to see them in the conventional sense.

The existence of the Planck length doesn't imply spacetime quantization. While we will never register a particle with size ("diameter") smaller than Planck, it does not mean that the position space is discrete ("difference of positions"). The spacetime may or may not be quantized, but that's a problem independent from the existence of the limit of our magnifying abilities and if it is quantized, the size of the "cell" doesn't have to be one Planck length in particular.

The Planck length sets the maximum energy limit where the quantum mechanics breaks and at the same time the minimum energy limit where general relativity breaks. It doesn't mean that these theories couldn't break at even closer energy limits. We are just sure that the current quantum mechanics is not valid above the Planck length and GR below it.

 

[Duplex]

If the Planck length isn’t the minimum distance, then the Planck time isn’t either, since the units are linked by the speed of light. Right? It worries me a bit.

Between the past and the future is the NOW. Only there we can operate. I have always felt confidence in having at least one Planck time available for a thought, 5.4 x 10 ^-44 s.

How small is the NOW now? Less than Planck time, or disappearing into the quantum foam or - worst case - where the time zone between history and the future is hypothetical zero? Beyond time and space?

My five cents when the New Year evening here approaches the enigmatic Planck time (or less) between 2012 and 2013.

 

[Naty1]

I disagree with these two pieces of Haael's excellent post:

But at some point, the probing particles reach so high energy level, that they become black holes.

No evidence, no theory for such for such a thing. The KE of a particle CANNOT cause it to become a black hole. Why am I so sure: because right now, in some frame, I am seen moving at VERY close to c...yet I am unaware of any physical changes.

The existence of the Planck length doesn't imply spacetime quantization.

It implies exactly that! But it does not prove it as my prior posts suggest...and as you seem to imply as well.

Any formula with h [or h bar] implies quantization.

There is LOTS of theoretical evidence for discreteness at the Planck scale:

The Holographic principle was inspired by the 1970’s Bekenstein bound of black hole horizons and was first proposed by Gerard 't Hooft in the 1980’s ; It was given a precise string-theory interpretation by Leonard Susskind in 1993. Juan Maldacena provided rigorous string theory interpretation.

 

[K^2]

I'm speaking beyond my knowledge here but I suspect there is not a smallest length or else the smallest length in one reference frame would be smaller in a second frame moving with respect to the first.

Is that wrong?

Keeping in mind that Lorentz Contraction is a side-effect of Lorentz Boost as a hyper-rotation in space-time, what you are really getting at is the fact that it's hard to imagine space-time with a finite smallest unit of distance that's also isotropic with respect to (hyper)rotations.

Simplest visualization. Imagine that you do classical physics on a grid. Then if you move along X or Y coordinate, you have smallest step of 1. But if you move diagonally, the smallest step is √2. That's something you are going to notice, unless something very interesting is going on with how space is structured.

So this is a fundamental difficulty of finite distance increment, but it's simply something that would have to be addressed by the underlying theory. Since we don't really have such a theory, discussing whether it's actually a problem is pointless.As far as plank's length, we have no theory to adequately describe physics at a smaller scale. That's all that really means.

 

[haael]

No evidence, no theory for such for such a thing. The KE of a particle CANNOT cause it to become a black hole. Why am I so sure: because right now, in some frame, I am seen moving at VERY close to c...yet I am unaware of any physical changes.

Right. So, a clarification: the rest mass of the target-projectile system must be lesser than the Planck mass. This, together with the condition that the interaction must take place in a volume smaller than the Planck volume, sets the limit of our observations resolution, since increasing energy further would produce black holes.

Any formula with h [or h bar] implies quantization.

By quantization you mean noncommutativity or discretness?

The spacetime does not have to be discrete at the Planck scale. The Planck length might be the minimum possible wavelength, but the position space may be continuous.

If the position space was discrete, then for instance particles close to the Planck scale would not produce interference patterns in the double-slit experiment. Suppose you have two entangled oscillators producing a radiation of the Planck length. To nullify their radiation, you have to position them half a Planck length apart. If only integral Planck length distances are allowed, they would never nullify. When you connect this with Huygens principle, you can see that any field would never nullify itself.

I don't say that the spacetime is not quantized (in fact, no one knows), I just say that the existence of the Planck cut-off for wavelengths does not imply that. You could have the Planck length as minumum length of wave and still have the continuous position space.

 

[Vanadium 50]

There is LOTS of theoretical evidence for discreteness at the Planck scale:

There is no such thing as "theoretical evidence". Evidence is, by definition, experimental.

 

[Jorriss]

Keeping in mind that Lorentz Contraction is a side-effect of Lorentz Boost as a hyper-rotation in space-time, what you are really getting at is the fact that it's hard to imagine space-time with a finite smallest unit of distance that's also isotropic with respect to (hyper)rotations.

Simplest visualization. Imagine that you do classical physics on a grid. Then if you move along X or Y coordinate, you have smallest step of 1. But if you move diagonally, the smallest step is √2. That's something you are going to notice, unless something very interesting is going on with how space is structured.

So this is a fundamental difficulty of finite distance increment, but it's simply something that would have to be addressed by the underlying theory. Since we don't really have such a theory, discussing whether it's actually a problem is pointless.


As far as plank's length, we have no theory to adequately describe physics at a smaller scale. That's all that really means.

This is an area I am very unfamiliar with (I have never formally studied special relativity), thanks for the insight!

 

[Naty1]

Haael:

Any formula with h [or h bar] implies quantization.

By quantization you mean noncommutativity or discretness?

just discreteness...like electron orbitals...


Vanadium:

There is no such thing as "theoretical evidence"...

I take your point, bad wording on my part!

I checked that term in Wikipedia...some funny stuff popped up!

 

[DiracPool]

There is no such thing as "theoretical evidence"...

Sure there is, its just the thought results of your thought experiments, isn't it?