Proof of Planck Length: A Pedological Derivation

[y33t]

Hi all,

Does anybody can provide a source that can prove Planck's length from scratch ? I mean from classical mechanics to quantum mechanics, a pedological proof/derivation ?

Thanks in advance.

 

[Borek]

What is there to prove? Planck length is just a unit that can be obtained using three known physical constants. You can't prove Planck length just like you can't prove 7.

 

[y33t]

What is there to prove? Planck length is just a unit that can be obtained using three known physical constants. You can't prove Planck length just like you can't prove 7.

How did Planck come up with the idea to derive the length from the 3 constants ? There is a background of it's derivation for sure.

 

[Bill_K]

It's not a "proof", but you might be interested in http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm of the Planck units and their interpretation.

 

[y33t]

It's not a "proof", but you might be interested in http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm of the Planck units and their interpretation.

Looks satisfying. Thank you.

 

[JohnLuck]

Actually why is the Planck length thought to be the smallest meaningful measure of length? I would also love to know this.

I know that the Planck constant is the ratio between the energy and the frequency of a photon and I know that the Planck time is the time it takes a photon traveling at the speed of light to move 1 Planck length and is therefore the shortest possible time measurable.

But please, if someone can explain why the Planck length is thought of as the shortest meaningful length, please do so!

 

[Bill_K]

But please, if someone can explain why the Planck length is thought of as the shortest meaningful length, please do so!

Have you taken a look at the reference I cited above in #4?

 

[Fredrik]

Actually why is the Planck length thought to be the smallest meaningful measure of length? I would also love to know this.

Basically it's just an order-of-magnitude estimate. If you have no idea how to calculate the volume of a ball, you can guess that it's the product of any three lengths that characterize a ball, because this gives you a result with the right unit. If you e.g. guess that it's the circumference to the third power, then you're wrong by a factor of factor of 6π², but when you do this sort of thing, you're not likely to be wrong by a factor of 10¹⁰⁰.

So once you have decided that the concept of length will be problematic at some scale, you can estimate that scale by combining relevant constants into a product that has the right unit. The relevant constants here are G,c and $\hbar$. I will try to explain why.

To understand a concept like length, we need to find a theory of physics that defines it as a mathematical term, and then we need to do experiments that show that the theory's predictions are accurate. To understand "length" is to understand the mathematics of the theory that makes the best predictions about results of experiments, and to understand what sort of thing is considered a "length measurement" for the purposes of testing the accuracy of those predictions. The best theory we have about the properties of space and time is general relativity. It describes a spacetime in which there's an invariant speed, c. This is why c is relevant. GR is also a theory of gravity. This is why G is relevant.

GR describes a relationship between the properties of spacetime and the properties of matter. Unfortunately, it describes matter classically, and we know that there are situations where a classical description of matter fails miserably. Because of the relationship between matter and spacetime, we should expect GR to also fail miserably at describing the properties of space and time in those situations. Since GR is the best theory of space and time that we have, this means that all the mathematical definitions we have of "length" are part of theories that are useless in those situtations. And this means that none of those definitions can provide us with any sort of understanding of the real world in those situations. The concept of "length" as we know it (i.e. as it's defined by our best theories) has become irrelevant and useless.

It's possible that a better theory would include a definition of something we might want to call "length" in those situations, but that definition might be very different from the definition of "length" in GR. So maybe in a hundred years, people will be talking about lengths at sub-planck scales, but to them the word will mean something different from what it means to us.

In some of the situations where a classical description of matter fails (specifically those situations where gravity can be neglected), we can use a quantum theory of matter instead. This is why $\hbar$ is relevant.