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Based on my rather circumscribed understanding of modern physics, one of the key insights of quantum mechanics over previous scientific theories is the prediction that there exists an absolute limit to how small things can be in sub-divided the universe (i.e., space-time and energy are quantized). Why does this make more sense than a universe which is continuous and truly unbounded in scale? And how do we know that the universe actually has a minimum dimension beyond which it can no longer be sub-divided?

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The planck length is not necessarily an absolute limit to how small thing can be sub divided. The planck length is theoretical and it is empirically defined by dimensional analysis. At this length scale our knowledge of physics makes no sense.

The planck length $\ell_P$ is defined as: $\ell_\text{P} =\sqrt\frac{\hbar G}{c^3} \approx 1.616\;199 (97) \times 10^{-35} \mbox{ m}$

As you can see it "consists" of the 3 fundamental constants of nature:

$G$, the gravitational constant

$c$, the speed of light

and $\hbar$, Planck's constant

Some theories like String Theory and Loop Quantum Gravity make use of these scales to produce a theory of gravity which is consistent with quantum mechanics.

Since the planck length is just theoretical and many orders of magnitude smaller than the finest equipment we currently have can measure, there is no definite answer to what the significance of this length is.

So to summarise:

The planck length does not necessarily mean that its the absolute limit of how small things can get. It simply means we have no idea what is going on down there. This screams for new physics.

Theories like String Theory and LQG make use of this length scale (and is the main reason it has been so popularised).

It doesn't make more sense than a continuum of space-time. It is simply being used in our currently most popular theories of quantum gravity, hence the publicity.

As I'm not an expert in either String Theory, or LQG, my points could be a bit hand wavy. Experts feel free to correct me.

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  • $\begingroup$ So is there a limit to how small a quanta of momentum or energy can be? $\endgroup$ – Bryson S. Jul 20 '14 at 22:26
  • $\begingroup$ @BrysonS. That doesn't make sense since momentum and energy are not lengths. $\endgroup$ – Constandinos Damalas Jul 20 '14 at 22:30
  • $\begingroup$ I meant in magnitude $\endgroup$ – Bryson S. Jul 20 '14 at 22:31
  • $\begingroup$ Hmmm I'm not sure I understand. Are you asking if the difference between two energy/momentum states can get arbitrarily small? They can get only so small until the uncertainty principle starts kicking in. $\endgroup$ – Constandinos Damalas Jul 21 '14 at 7:50
  • $\begingroup$ If that is true, then there is a limit to the finest resolution of reality that physics can currently understand. But what sets the order of magnitude where the uncertainty principle kicks in? Why is it not an order of magnitude greater or smaller? $\endgroup$ – Bryson S. Jul 21 '14 at 15:54

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