Has Planck length been derived rigorously?

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Discussion Overview

The discussion centers on the derivation of the Planck length within the framework of standard quantum field theory, exploring its implications and the nature of quantization in physics. Participants also examine the relationships between fundamental physical constants and the concept of quantized quantities.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions whether the Planck length has been derived rigorously in standard quantum field theory and seeks literature on the topic.
  • Another participant argues that the Planck length is a combination of physical constants and is not derived from anything, suggesting that it does not imply a lowest delta-momentum within standard quantum field theory.
  • A participant proposes that the only measurable quantity known to be quantized is action, referencing Planck's constant as the lowest quanta of action.
  • Another participant counters that energy and angular momentum are also quantized in various systems, providing examples such as electrons in atoms.
  • There is a discussion about creating "Planck-whatever" quantities by combining constants G, c, and h, with references to Planck time as another example.
  • One participant clarifies that the Planck length is not the minimum possible length, addressing a common misunderstanding and referencing an Insights article on the topic.
  • A later reply introduces a conceptual explanation involving energies of photons and black holes at the Planck length, suggesting a relationship between these energies.

Areas of Agreement / Disagreement

Participants express differing views on the derivation of the Planck length and its implications, with no consensus reached on whether it has been rigorously derived or whether it implies a minimum length scale. The discussion remains unresolved on these points.

Contextual Notes

Participants note the speculative nature of hypotheses regarding spacetime quantization and the limitations of current experimental sensitivity in testing such ideas.

muzukashi suginaiyo
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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
 
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muzukashi suginaiyo said:
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.

muzukashi suginaiyo said:
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.
 
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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"?
 
muzukashi suginaiyo said:
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).

muzukashi suginaiyo said:
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.
 
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muzukashi suginaiyo said:
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 said:
Sure. The Planck time is another common one.
For others, see

https://en.wikipedia.org/wiki/Planck_units
 
muzukashi suginaiyo said:
ħello.
:biggrin:
 
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muzukashi suginaiyo said:
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/
 
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muzukashi suginaiyo said:
ħello.
ħi! and welcome to PF!

Nugatory said:
that we evem have
typo (..."even")
 
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Stavros Kiri said:
typo (..."even")
Fixed - thanks
 
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muzukashi suginaiyo said:
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)
 
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