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I know that it is believed that energy is discrete, in that it travels in quanta. I was wondering if there is any evidence which either proves or disproves something similar with both time and distance?

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Regarding discreteness of time: physics.stackexchange.com/q/35674/11062 –  Waffle's Crazy Peanut May 18 '13 at 18:33
    
LQG and related quantum gravity approaches do somethin like this, but these effects are only expected to kick in at the Planck scale. –  Dilaton May 18 '13 at 18:37
    
Aren't the Planck lengths/time more or less an implementation of discrete space/time? Not exactly, but close enough? –  Manishearth May 18 '13 at 18:39
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@Manishearth, Planck length is just a quantity/scale we came up with by dimensional analysis. We think something must happen at that length scale but to be honest, we don't know what that might be. One wacky possibility would be if spacetime is a field theory, but smoothed over a width of Planck length (like a moving time average) rather than an actual discretization. That would presumably give a length scale without any discreteness. But don't take that idea very seriously; I just made it up to illustrate my point. –  Siva May 18 '13 at 18:51
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@Manishearth: Sure... theories assert very interesting things, but we really don't know if that's correct. For eg: Aristotle asserted that everything in the world could be made up of 5 elements (earth, fire, air, water, aether), and that was the leading theory of his time :-) –  Siva May 18 '13 at 18:57
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As far as we can tell (up to energy scales we've measured so far), spacetime is a nice and smooth manifold. It might happen that the smoothness is approximate and spacetime is discrete at a much more microscopic scale, or it could turn out that spacetime is smooth all the way through. Short answer: We don't know.

About the notion of energy quantization: Energy can only be transferred in quanta, but the size of the quanta are not fixed. You can "tune" the energy of a photon by tuning it's momentum/frequency. Once you fix the momentum, then energy can be quantized only in multiples of that quantum. It's a bit like buying (say) rice... you can buy it only in packets, but you can put how much ever you want into one packet.

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If the space-time were discrete, how could we verify it? –  jinawee May 18 '13 at 20:14
    
If the space-time were continuous "all the way down", how could we verify it? –  Jens May 22 '13 at 13:51
    
I don't think we know of any way to do that, yet. It might be possible with a combination of some very powerful theory which we can verify but whose consistency absolutely demands continuous spacetime. However, based on the physics lessons we've learned in the last few decades, it's much more likely that every theory is "effective" over some energy/distance scales. So you could only say that upto some resolution of $a$, spacetime looks to be continuous. –  Siva May 22 '13 at 16:25
    
How to check if it's not continuous at scale $a$? Run experiments at higher resolution than scale $a$. If spacetime is not continuous, you'll notice predictable weird results. Alternatively, if you can't run experiments at resolution $a$ but much much lesser, then you run high precision experiments to enough decimal places of accuracy and it might just tell you about what happens at high energies. –  Siva May 22 '13 at 16:28
    
The from an experimental point of view not yet ruled out wiggle room for time and/or space being discrete at the smallest scales is quite constrained by now ... –  Dilaton Aug 12 '13 at 21:27
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