Can quantum length be derived not using gravity constant?

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Discussion Overview

The discussion revolves around the concept of quantum length, specifically the Planck length, and whether it can be derived without using the gravitational constant. Participants explore various theoretical frameworks and propose alternative approaches to understanding quantum length in the context of quantum gravity and other models.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants define the quantum of length as the Planck length, expressed as \(\lambda_p = \sqrt{\frac{Gh}{c^3}}\), where G is the gravitational constant, h is Planck's constant, and c is the speed of light.
  • Others question the terminology "quantum of length," suggesting it is meaningful primarily in the context of Loop Quantum Gravity (LQG), where it is derived from spacetime quantization.
  • A participant mentions that the Planck mass and length can be derived from a combination of the fine structure constant, electron mass, and Weinberg angle, referencing Nottale's work.
  • There is a query about the existence of a quantum of length in LQG spacetime, with some suggesting that certain formulations of LQG do include quantized lengths and areas.
  • Concerns are raised about the technical reasons preventing the quantization of volume, with participants expressing that volume seems an intuitive candidate for quantization.
  • Discussion includes the mention of Double Special Relativity (DSR), which posits a kinematical Planck length that may not necessarily rely on gravitational constants.
  • Another participant discusses the relationship between cosmological parameters and fundamental lengths, questioning the nature of mass and its quantization in relation to Planck parameters.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of quantum length, with no consensus reached on whether it can be derived without the gravitational constant or if it is meaningful outside specific theoretical frameworks like LQG.

Contextual Notes

Limitations include the lack of clarity on the definitions of terms like "quantum of length" and the unresolved nature of the relationship between various fundamental constants and their implications for quantum length.

Antonio Lao
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The quantum of length is the Planck length given by

[tex]\lambda_p = \sqrt{\frac{Gh}{c^3}}[/tex]

where G is the gravity constant. h is Planck's constant. c is the speed of light in vacuum.

There might be unknown fundamental energy and force whose ratio can also give the quantum length.

[tex]\lambda_p = \frac{E_0}{F_0}[/tex]
 
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Antonio Lao said:
The quantum of length is the Planck length given by

[tex]\lambda_p = \sqrt{\frac{Gh}{c^3}}[/tex]

where G is the gravity constant. h is Planck's constant. c is the speed of light in vacuum.

There might be unknown fundamental energy and force whose ratio can also give the quantum length.

[tex]\lambda_p = \frac{E_0}{F_0}[/tex]

How can you call this the "quantum of length"? What is the field of which it is the quantum? AFAIK the phrase "quantum of length" is meaningful only in LQG where it is derived from spacetime quantization. And there is is not the Planck length.
 
You can get Planck mass (thus Planck length) as a combination of the fine structure constant, the electron mass, and the sine squared of Weinberg angle (at his unification value, 3/8). Nottale did it.
 
selfAdjoint,

Is there such a thing as quantum of length even in LQG spacetime?
 
Thanks arivero. I will look up Nottale.
 
selfAdjoint,

I'm starting reading Kaku's QFT, chap 19 on quantum gravity. Maybe I can find something from what you mentioned about LQG.
 
Antonio Lao said:
selfAdjoint,

Is there such a thing as quantum of length even in LQG spacetime?

One brand of LQG has quantized lengths and areas (not volumes, though, for technical reasons).
 
Thank you, selfAdjoint. But what's the compelling technical reasons why volume can't be quantized? Common sense is telling me that volume should be the obvious 1st candidate for such quantization.
 
Perhaps DSR, double special relativity, should be mentioned here. It uses a kinematical Planck length, with is assumed to be some multiple of the gravitational one, but it does not need to be.

Also, Marcus recently mentioned a acceleration got from cosmological parameters, which also can be transformed in a fundamental length.
 
  • #10
arivero said:
Also, Marcus recently mentioned a acceleration got from cosmological parameters, which also can be transformed in a fundamental length.
Is the cosmological parameter a unit of force? And this force is proportional to said acceleration which upon using Newton's 2nd law of motion indicates that mass is the constant of proportionality. But if this mass is quantized as the Planck mass, we still some quantum of energy to derive the fundamental length.

Is it not that the combined effect of Planck parameters (length, mass, energy, time) is the domain where all the fundamental forces (EM, weak, strong, gravity) are unified?
 

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