Some Planck units, like time, length, or temperature, describe a physical maximum or minimum, at least approximately: you can't get hotter than the Planck temperature, measure anything smaller than Planck time or length, etc. Others, like the Planck charge, Planck momentum, or Planck energy, seem to have no associated maxima. Which units are of what type, and is there a reason that some are limits while others are in the 'middle' of a spectrum of possibilities? Are there limits to physical units which are distinct from the associated Plank unit?
[Physics] What Planck units are limits
absolute-unitsphysical constantsquantum mechanics
Related Solutions
My understanding of the Planck charge is that it is the unit charge necessary to normalize
- the speed of light (and other "instantaneous" interactions e.g. strong force and gravity), $c=1$
- reduced Planck constant, $\hbar=1$
- Coulomb constant, $\frac{1}{4 \pi \epsilon_0}=1$
So it's a natural definition of charge that has no reference to the elementary charge. So you can represent the value of the elementary charge in terms of this Planck charge and it comes out as
$$ e = \sqrt{\alpha} $$
where $\alpha$ is the fine-structure constnat.
So the significance of the Planck charge is, like the other Planck units, they are defined in such a way that no object or particle (not even any subatomic particles) are needed nor used to define the base units. Only free space. Then you can go about asking what the properties of these elementary sub-atomic particles are, in terms of these units that only normalize properties of free space. For the charge of an electron, it is $-e = -\sqrt{\alpha} \approx -0.0854245$, in terms of Planck units.
I prefer rationalized units over unrationalized units because I think it is useful to equate the concepts of flux density and field strength and that is done if you normalize the electric constant rather than the Coulomb constant. If you do that, the elementary charge comes out as $ e=\sqrt{4 \pi \alpha} \approx 0.302822$. This is done in a few books on QFT or something and makes it more clear that the elementary charge, while not exactly the natural unit of charge, is in the ball park of the natural unit of charge. Well within an order of magnitude.
The Standard Model and General Relativity are both successful in appropriate limits, but they cannot be consistently combined for scales below $\sim\ell_P:=\sqrt{\dfrac{G\hbar}{c^3}}$ for various reasons. (By $\sim$, I mean "give or take a multiplicative constant that's besides the point here and may be hard to compute".) For example, what happens if you try to probe such length scales with a photon? How will its wavelength compare to its Schwarzschild radius?
When you ask about the physical meaning or significance of such short length scales, that's where it gets contentious. I'll try to summarise the range of views on this, but I'll probably fudge or simplify a few details:
- String theory says spacetime is infinitely divisible, but particles have a size $\sim\ell_P$. They therefore have worldsheets instead of worldlines, which smears Feynman diagram vertices as thus. This smearing removes the troublesome infinities from the treatment of gravity.
- Loop quantum gravity, in a sense, says the opposite: particles aren't posited to have size, but spacetime is quantised. Particles live at distinct lattice points. The area and volume of an object have operators in the Hilbert space, and these operators have discrete eigenvalues $\sim\ell_P^{2\,\mathrm{or}\,3}$.
- There have been attempts to combine ST with LQG (motivated by their respective pros and cons and their obtaining similar results from very different precepts, e.g. logarithmic corrections to the Hawking-Bekenstein entropy of black holes), but these are in their infancy. For now, it suffices to say such a union would introduce both deviations from the "point particles in infinitely divisible spacetime" idea that causes SM+GR problems.
- Another proposal is that $[\hat{x}_\mu,\,\hat{x}_\nu]=i\ell_P^2 \theta_{\mu\nu}$ is a non-vanishing antisymmetric tensor. This is far from developing into a full theory of quantum gravity, but it's an idea that's been explored in such attempts. Just as quantum mechanics says $[\hat{x}_j,\,\hat{p}_k]=i\hbar\delta_{jk}$ without the eigenvalues becoming discrete, the above use of noncommutative geometry requires only that $\sigma_{x_\mu}\sigma_{x_\nu}\ge\frac{1}{2}\ell_P^2|\theta_{\mu\nu}|$, not that eigenvalues of $x_\mu$ cannot differ by arbitrarily small fractions of $\ell_P$.
Best Answer
Planck units are constructed in such a way that all fundamental constant are equal to one, so they set a scale where the speed of light, the planck constant and the gravitational constant are relevant in their description, this would imply we would presumably need a quantum theory of gravity to explain phenomena in that setup. Since we no have such theory, many physicist think they mark a boundary to our current understanding of nature. We can not say for sure, for example, if lenght is defined below planck length, since length is a property of space itself, and using Einstein theory, it's closely related to gravity, which we don't know how behaves in quantum regimen. Of course, these ideas are speculative, but are the things we expect to find, we don't know what exactly happen at that scales.