Can quantum length be derived not using gravity constant?

[Antonio Lao]

The quantum of length is the Planck length given by

$$\lambda_p = \sqrt{\frac{Gh}{c^3}}$$

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.

$$\lambda_p = \frac{E_0}{F_0}$$

 

[selfAdjoint]

The quantum of length is the Planck length given by

$$\lambda_p = \sqrt{\frac{Gh}{c^3}}$$

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.

$$\lambda_p = \frac{E_0}{F_0}$$

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.

 

[arivero]

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.

 

[Antonio Lao]

selfAdjoint,

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

 

[Antonio Lao]

Thanks arivero. I will look up Nottale.

 

[Antonio Lao]

selfAdjoint,

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

 

[selfAdjoint]

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).

 

[Antonio Lao]

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.

 

[arivero]

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.

 

[Antonio Lao]

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?