Has Planck length been derived rigorously?

[muzukashi suginaiyo]

ħello. Here is my question: Has the so-called "planck length" (~1.61622837 * 10^(-35) meters) been derived with mathematical rigor within standard quantum field theory? If so, I need help finding this proof. Any literature would be appreciated.

Also: If it has, then doesn't this imply a lowest delta-momentum, since momentum is the complex conjugate of length via Heisenberg Uncertainty principle?:

S(momentum) * S(location/"length") ≥ ħ/2

 

[PeterDonis]

Has the so-called "planck length" (~1.61622837 * 10^(-35) meters) been derived with mathematical rigor within standard quantum field theory?

The Planck length is just a particular combination of the physical constants $G$, $c$, and $h$, that yields a quantity with units of length. It is not "derived" from anything.

doesn't this imply a lowest delta-momentum

Not within standard quantum field theory, no. There are speculative hypotheses about spacetime not being continuous but "quantized" at scales around the Planck length, but we have no way of testing such speculations; our experiments are many orders of magnitude less sensitive than they would need to be.

 

[muzukashi suginaiyo]

Would I be correct, then, if I were to say that the only measurable physical quantity known to be quantized is action itself? - Since Planck's constant is the lowest quanta of action.

This combining of the constants is pretty interesting to me. So if I throw the constants G, c, and h together in some way to produce a quantity, I can come up with a "planck-whatever"?

 

[PeterDonis]

Would I be correct, then, if I were to say that the only measurable physical quantity known to be quantized is action itself?

No, certainly not. Energy and angular momentum are two observables that are quantized in many systems (e.g., electrons in atoms).

So if I throw the constants G, c, and h together in some way to produce a quantity, I can come up with a "planck-whatever"?

Sure. The Planck time is another common one.

 

[jtbell]

So if I throw the constants G, c, and h together in some way to produce a quantity, I can come up with a "planck-whatever"?

Sure. The Planck time is another common one.

For others, see

https://en.wikipedia.org/wiki/Planck_units

 

[Demystifier]

ħello.

 

[Nugatory]

Also: If it has, then doesn't this imply a lowest delta-momentum, since momentum is the complex conjugate of length via Heisenberg Uncertainty principle?:

No, because the Planck length is not the minimum possible length. This misunderstanding is so common that we even have an Insights article about it: https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

 

[Stavros Kiri]

ħello.

ħi! and welcome to PF!

that we evem have

typo (..."even")

 

[Nugatory]

typo (..."even")

Fixed - thanks

 

[edguy99]

ħello. Here is my question: Has the so-called "planck length" (~1.61622837 * 10^(-35) meters) been derived with mathematical rigor within standard quantum field theory? ...

My favorite explanation:

For every length, you can determine two energies, one for a photon (of that wavelength) and one for a black hole (of that Schwarzschild radius). At the Planck length, those two energies will be equal. At longer lengths, the black hole will be more energetic than the photon. At shorter lengths, the photon is more energetic than the black hole.

One can check it out using these three formulas:

E=Mc^2,

E=hc /lambda (photon energy)

R =2G M/c^2 (Schwarzschild radius for black hole)