Are time and space continuous or discrete?

[Haorong Wu]

TL;DR Summary

Are time and space continuous or discrete?

In another forum, some people argue that time and space are discrete, due to Planck time and Planck length.

However, I disagree with this idea. I think, the Planck time and Planck length are just some scales that we can measure, but they do not forbid continuous time and space shorter than them. Besides, I hardly remember any equations in quantum mechanics forbid any continuous values of time or space. Furthermore, I read a idea that if space is discrete, then the continuous symmetries of space would not exist, such as spatial translations.

So I hold the idea that time and space are continuous.

 

[PeroK]

Summary: Are time and space continuous or discrete?

In another forum, some people argue that time and space are discrete, due to Planck time and Planck length.

However, I disagree with this idea. I think, the Planck time and Planck length are just some scales that we can measure, but they do not forbid continuous time and space shorter than them. Besides, I hardly remember any equations in quantum mechanics forbid any continuous values of time or space. Furthermore, I read a idea that if space is discrete, then the continuous symmetries of space would not exist, such as spatial translations.

So I hold the idea that time and space are continuous.

Yes, you are right. It's a common misconception about the Planck units.

The best current theories are based on a spacetime continuum.

 

[A. Neumaier]

For time, this question is thoroughly discussed at https://physics.stackexchange.com/questions/35674/

For space, precisely the same holds:

From a purist point of view, the question is undecidable, but from a practical point of view, space and time must form a continuum, or almost all theory of physics breaks down.

 

[fresh_42]

... but from a practical point of view, space and time must form a continuum, or almost all theory of physics breaks down.

I don't think this is true. They would still be valid models and that is what physics is about. We calculate without hesitation with a continuous length of steel, however, this length is created by a finite number of molecules. The world is a discrete world. I agree that it is undecidable from a purist point of view, but from a practical point of view it might turn out that different models will apply to different scales: a continuous model for all practical purposes and a possibly discrete one for the nature of time itself. I do not claim that this has anything to do with the Planck scale, only that a future model of spacetime could as well be discrete.

 

[A. Neumaier]

I don't think this is true. [...] from a practical point of view it might turn out that different models will apply to different scales: a continuous model for all practical purposes and a possibly discrete one for the nature of time itself.

Did you read the reference I gave? (It is my highest ranked answer on Physics Stack exchange from 636 total answers given by me.) There I argued against your view:

in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous.

Physics would become very awkward if expressed in terms of a discrete time: The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.

Please consider that. Being practical requires being tractable.

 

[fresh_42]

Please consider that. Being practical requires being tractable.

Yes, but this is not what you said in the first place:

Physics would become very awkward if expressed in terms of a discrete time.

This does not contradict what I said: We can calculate with smooth lengths despite the fact that molecules are not. However, this does not change the discrete nature of material lengths. I do not see why this has to be different with time. I do not claim it is, just that there is no reason to prefer one over the other.

 

[A. Neumaier]

This does not contradict what I said: We can calculate with smooth lengths despite the fact that molecules are not. However, this does not change the discrete nature of material lengths.

The material length is fractal and at the submolecular level it is fuzzy, not discrete.

Molecules are not points but have a (somewhat fuzzy) continuous extension in space. All predictive calculations done with molecules, e.g., to determine material properties or chemical reaction processes are based on continuous space and time.

 

[fresh_42]

This is nitpicking. Fact is that material length isn't smooth. Whether an alternative model is called fuzzy (no determined location) or discrete (distance in an atomic lattice, if you like at 0K) is meaningless. It is only an argument that the distinction doesn't affect models on different scales. We have different models for gravity and nobody cares. It is daily business in physics. I only wanted to say, that a discrete time does not automatically lead to:

almost all theory of physics breaks down

Yourself set it in relation in your answer:

Physics would become very awkward if expressed in terms of a discrete time

These are not equivalent statements. Physics would also break down if we had to plan our buildings on a molecular level!

You might say that the first statement was a rhetorical exaggeration, but so was mine as I said the world is discrete. I meant in the sense that we can find differences on almost every scale, but this doesn't mean our continuous models are useless.

 

[A. Neumaier]

Fact is that material length isn't smooth.

But there is a huge difference between not being smooth and being discrete.

fuzzy (no determined location)

''fuzzy'' does not mean ''no determined location'' but ''no location determined to infinite precision''.
The boundary of a cloud does not become less fuzzy by discretizing space since the fuzziness is associated with the fact that there is no natural threshold at which density one should consider the boundary to end.

discrete (distance in an atomic lattice, if you like at 0K) is meaningless.

Even an atomic lattice, with nuclei idealized as fixed discrete points, has in the conducting case an electron flow that is continuous in space and time. Moreover, the atoms oscillate in space and thereby give rise to phonons. Neither the properties of electrons nor those of phonons can be modeled in discrete space or time.

Nothing in solid state physics is truly discrete, in the sense that space or time has gaps.

I meant [discrete] in the sense that we can find differences on almost every scale

Yes, but the OPs question was related to the underlying structure of space and time:

they do not forbid continuous time and space shorter than them.