The atomic theory as first theorised by Democritus has been successfully applied to matter and to energy (quanta).

Space-time is still generally seen as a continuum. What arguments are there (if any) in support of there being a particulate structure of space-time?

  • $\begingroup$ I may be wrong, but it seems you're confusing two concepts. Space time being continuous does not preclude matter being composed of discreet objects. The discreet objects can move in continuous spaces. $\endgroup$ – BMS Feb 6 '14 at 3:48
  • $\begingroup$ @BMS: well, yes; this is where we are now - roughly speaking. $\endgroup$ – Mozibur Ullah Feb 6 '14 at 4:07
  • $\begingroup$ I understand now. It wasn't clear to me how your first sentence was connected to the question for some reason; I initially thought your first provided evidence for your second. $\endgroup$ – BMS Feb 6 '14 at 5:35
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/817/2451 , physics.stackexchange.com/q/4453/2451 , physics.stackexchange.com/q/9720/2451 and links therein. $\endgroup$ – Qmechanic Feb 6 '14 at 6:39

If you believe that quantum theory (Hilbert space + Hamiltonian) describes our world, then we need to believe that the quantum theory for a space with a finite volume has a finite dimensional Hilbert space. This (plus locality) implies an atomic structure of space:

Space = a collection of many many qubits.
Vacuum = the ground state of the qubits.
Elementary particles = collective excitations of the qubits.

(See http://blog.sciencenet.cn/blog-1116346-736093.html )

  • $\begingroup$ How finite-dimentional Hilbert space implies discreteness of space-time? $\endgroup$ – Anixx Feb 6 '14 at 7:04
  • $\begingroup$ Because continuous space (ie field theory) always has infinite-dimentional Hilbert space even for a finite space volume. $\endgroup$ – Xiao-Gang Wen Feb 6 '14 at 8:11
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    $\begingroup$ I don't see any a priori reason to believe that space has a finite volume. And it is actually false that the hilbert space for a finite volume is finite. Even something as simple as the infinite square well doesn't work if you assume a finite dimensional Hilbert space -- your eigenbasis no longer spans the function space. $\endgroup$ – Jerry Schirmer Apr 7 '14 at 15:45
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    $\begingroup$ @Xiao-GangWen that's just equivalent to blindly asserting the thing you're trying to justify. Saying that spacetime is atomic because there are no real infinities is putting the cart before the horse: there's really no contradiction whatsoever in assuming spacetime is continuous. There's a world of difference between getting infinity as a physical quantity and having some infinite set in the foundations. $\endgroup$ – Robert Mastragostino Apr 7 '14 at 15:48
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    $\begingroup$ Infinite exists in mathematics. But our experience in physics is different. So far, all the infinities in physics are just illusions. In physics, infinity = very very large. For example, in physics (except physics textbook), there is no "infinite square well", just very hard wall which is not quite square. If you know the Planck length, you will know that the number of states in a "infinite square well" is finite. $\endgroup$ – Xiao-Gang Wen Apr 8 '14 at 2:23

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