Is distance continuous or "pixel" like?

[iDimension]

Essentially I'm asking if space is divided into stepping stones, pixels if you will. Whereby the absolute minimum distance you can travel is this distance?

Another way of asking about this is, if you took a finite area of space, are there infinite positions within this space?

 

[PeterDonis]

We have no way of knowing the answer to this because we can't make measurements over arbitrarily small length scales. The smallest length scale we can probe is, IIRC, about 1/10 the size of an atomic nucleus, or about $10^{-16}$ meters. Down to that scale there is no sign that space is not continuous. But that scale is still about 19 orders of magnitude larger than the Planck length, which is where our best current theoretical speculations would lead us to expect signs of space being discrete, if in fact it is. (I stress the word "speculations", btw: that's all they are, and for them to be right, our best current actual theories, general relativity and quantum field theory, would have to be wrong, since both of them are built on the assumption that space--more precisely, spacetime--is continuous.)

 

[Hercuflea]

We have no way of knowing the answer to this because we can't make measurements over arbitrarily small length scales. The smallest length scale we can probe is, IIRC, about 1/10 the size of an atomic nucleus, or about $10^{-16}$ meters. Down to that scale there is no sign that space is not continuous. But that scale is still about 19 orders of magnitude larger than the Planck length, which is where our best current theoretical speculations would lead us to expect signs of space being discrete, if in fact it is. (I stress the word "speculations", btw: that's all they are, and for them to be right, our best current actual theories, general relativity and quantum field theory, would have to be wrong, since both of them are built on the assumption that space--more precisely, spacetime--is continuous.)

I'm not good at quantum. Do virtual particles/hawking radiation have anything to do with the Planck length?

 

[PeterDonis]

Do virtual particles/hawking radiation have anything to do with the Planck length?

No.

 

[mfb]

High-energy physics allows a search for effects up to ~4 TeV, or 3*10^-19 meters: still no steps.

 

[EugeneBird]

oIn fact, there are some indirect ideas that lead to certain arguments against space being NOT infinitely divisible:

(1) In classical physics, there is a concept of "equipartition of energy" (Ludwig Boltzmann's idea, I believe). If something is free to move, then it's energy will be divided equally among each of the kinds of movements it can make. So, for example, if subatomic particles were made up of smaller things, and those things were made up of smaller things, and so on, in an infinite regress, then particles would be sucking up a lot of energy to make their smaller components move. Perhaps they could even hold an infinite amount of energy ... presuming that their energy spectrum is made up of small jumps in energy, and that the overall infinite sum of energies was divergent. That seems to be NOT the case, and thus it's one argument for the existence of a 'smallest thing'. (Which is perhaps different from a smallest distance - but related through the concept of Compton wavelengths and virtual particles, I believe.)

(2) In string theory, there is a concept of small things having an equivalent mathematical description of large things ... so that when something gets really small, it behaves indistinguishably from something larger - "T Duality" - this is reminiscent of the idea that when you concentrate enough energy to resolve (look at) an image of something as small as the Planck length, that you accidently create a black hole. If you use even more energy to resolve smaller images, it goes into making the black hole bigger. Technically, you might be able to rearrange the information 'hidden' in the Hawking radiation when that black hole explodes, but no one has an inkling of how. The energy required to resolve such a small picture is unimaginably beyond what a human can make, anyway.

(3) There are theories of non-infinitesimal spacetime: Loop Quantum Gravity, Spin Networks - like String theory, there aren't any real-world testable consequences of them yet (that I have heard).

(4) In the real world, you cannot know if something is truly infinite. There isn't time for you to investigate it! Instead, we can imagine infinity because of mathematical ideas, like "induction" - you presume that there is no "number" that is too big to have the number 1 added to it. But how can you be sure unless you have tried it? And even if you could, is there any "math infinity" that can apply to the real world? To me, the existence of differentials in calculus "dx" and simple-to-understand quandries such as "Zeno's paradox" make me feel like infinity isn't so necessary. I'm not a mathematician or a physicist, but I can't think of anything that requires "full" infinity - just "way more than we could hope to count" seems to be sufficient. Math is a human invention - it might be going too far to presume that infinity exists - it is a rather extravagant concept, don't you think?

It does seem productive to think of infinity, to consider it. I think of Zermelo and Russell, and attempts to codify mathematics in a single set of rules (Principia Mathematica) and how it crumbled with the consequences of allowing "infinite sets". Yet, there are divergent infinite sums that can still provide usable answers via Ramanujan Summation and other techniques ... all over the internet you'll see the mathematical curiosities like "1 + 2 + 3 + 4 + ... = -1/12 + (an "infinite constant" that can be ignored)." The "12" in the preceding equation is what makes 24 out of the 26 dimensions required in a simplified version of heterotic string theory, I have seen in lectures. You have to add up every possible frequency multiple as a separate quantum harmonic oscillator (see: https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯) If you don't, then you can't reproduce the correct particle spins. Again - I only get this from reading a vast amount of lectures - so I could have misunderstood something.

 

[phyzguy]

Essentially I'm asking if space is divided into stepping stones, pixels if you will. Whereby the absolute minimum distance you can travel is this distance?
Another way of asking about this is, if you took a finite area of space, are there infinite positions within this space?

If space(time) is not continuous, it cannot simply be pixelated in the way the screen on your monitor is pixelated, because then the pixels and boundary locations would be observer dependent. If there is a minimum distance or minimum spatial volume, the "pixelization" must occur in a way that is Lorentz invariant and observer independent. It is not at all obvious how to do this. This is one of the motivations behind the concept of non-commutative geometry, which is potentially a description of how this could occur. A good example that you might want to read about is called the "fuzzy sphere", where the surface of a sphere is divided up into N discrete regions in a way that is Lorentz invariant, and as N->∞ it approaches a continuous surface. Another example is quantum mechanical phase space, where there is a minimum phase space "volume" of (2 π ħ)^3. Note that the coordinates in quantum mechanical phase space do not commute, since (x*px - px*x) ≠0. Whether this approach will lead to an accurate description of the universe is an open question.

 

[mfb]

(1) In classical physics, there is a concept of "equipartition of energy" (Ludwig Boltzmann's idea, I believe). If something is free to move, then it's energy will be divided equally among each of the kinds of movements it can make. So, for example, if subatomic particles were made up of smaller things, and those things were made up of smaller things, and so on, in an infinite regress, then particles would be sucking up a lot of energy to make their smaller components move. Perhaps they could even hold an infinite amount of energy ... presuming that their energy spectrum is made up of small jumps in energy, and that the overall infinite sum of energies was divergent. That seems to be NOT the case, and thus it's one argument for the existence of a 'smallest thing'. (Which is perhaps different from a smallest distance - but related through the concept of Compton wavelengths and virtual particles, I believe.)

No, it is just an argument for elementary particles. And it is a very weak one. The energy distribution is only uniform in the classical limit - it is not uniform if we consider quantum mechanics. Large energy steps make excitations extremely unlikely for the temperature ranges we can test. As a simple example: nuclei and even the individual nucleons can be excited because they have a substructure, but those excitations do not happen at room temperature - they do not influence the gas properties. Excitations of particles we today consider as elementary would need even larger energies.

(2) In string theory, there is a concept of small things having an equivalent mathematical description of large things ... so that when something gets really small, it behaves indistinguishably from something larger - "T Duality" - this is reminiscent of the idea that when you concentrate enough energy to resolve (look at) an image of something as small as the Planck length, that you accidently create a black hole. If you use even more energy to resolve smaller images, it goes into making the black hole bigger. Technically, you might be able to rearrange the information 'hidden' in the Hawking radiation when that black hole explodes, but no one has an inkling of how. The energy required to resolve such a small picture is unimaginably beyond what a human can make, anyway.

The discussion is not about the precision of measurements. In addition, you do not have to localize something at the step size to note that those steps exist (if they do).

(3) There are theories of non-infinitesimal spacetime: Loop Quantum Gravity, Spin Networks - like String theory, there aren't any real-world testable consequences of them yet (that I have heard).

There are also theories with continuous spacetime. That is hardly an argument for either side.

(4) In the real world, you cannot know if something is truly infinite.

So what? We can look for ever-decreasing step sizes and fail to see them forever. We might be able to find a theory of everything that works only in continuous spacetime.

 

[iDimension]

Thanks for your replies.

So Planck length is, as far as we can work out, the smallest unit of measure with regards to objects but not space? Space could well be infinitely divisible, unlike matter.

 

[mfb]

Our current laws of physics don't work if distances of the order of the Planck length are relevant. Something new has to happen there, this can involve discrete spacetime but it does not have to.