I Is the Concept of a Continuous Real Line Contradicted by Quantum Physics?

[FallenApple]

Real analysis shows that the real line is complete. All the gaps between the rationals are filled. It is perfectly continuous. Of course, all of this is based off of certain axioms that make it work mathematically.

Are those axioms now sound? According to quantum, the plank length is the smallest unit of space allowable in existence. So if the idea of continuity is flawed in reality, then the current state of much of mathematics is also on shaky grounds.

 

[DennisN]

According to quantum, the plank length is the smallest unit of space allowable in existence.

With kindness I challenge you to find a credible source that says/demonstrates that the Planck length is the smallest unit of space allowable in existence.

Spoiler

There is no experimental evidence of any smallest unit of space.

 

[PeroK]

Real analysis shows that the real line is complete. All the gaps between the rationals are filled. It is perfectly continuous. Of course, all of this is based off of certain axioms that make it work mathematically.

Are those axioms now sound? According to quantum, the plank length is the smallest unit of space allowable in existence. So if the idea of continuity is flawed in reality, then the current state of much of mathematics is also on shaky grounds.

This is about the 4th post in the last week where it's claimed that the Planck length is the "smallest unit of length". This is a misinterpretation. See.

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

Moreover, mathematics is in no way dependent on the physical realities of time and space! How you can apply mathematics may be dependent on that, but not mathematics itself.

 

[FallenApple]

This is about the 4th post in the last week where it's claimed that the Planck length is the "smallest unit of length". This is a misinterpretation. See.

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

Moreover, mathematics is in no way dependent on the physical realities of time and space! How you can apply mathematics may be dependent on that, but not mathematics itself.

ah touche.

I know that math is independent of physical reality. But at the same time, we are using that math to prove things about the real world. I was watching a lecture where the quantization of momentum was derived using the periodic property of the momentum wave function, which is a form of euler's formula. These are continuous trig functions. Then I thought that the plank length might also derived in a similar manner: using continuous sines and cosines. It would be problematic if continuous functions based off of continuous space is used to prove space is quantized; that would be a contradiction.

But after reading the link, I see that space isn't really quantized.

 

[FallenApple]

With kindness I challenge you to find a credible source that says/demonstrates that the Planck length is the smallest unit of space allowable in existence.

Spoiler

There is no experimental evidence of any smallest unit of space.

Understood. I'm new to quantum. I believe my misconception came of popular science books and my knowledge of the descrete energies and momentums and thought plank length may be similar.

 

[DennisN]

Understood. I'm new to quantum. I believe my misconception came of popular science books and my knowledge of the descrete energies and momentums and thought plank length may be similar.

Excellent . You are definitely not alone in thinking what you thought - it is quite easy to make that thought leap from Planck length to thinking of it as a smallest unit of space. Yet, I should add that there could be a smallest unit of space - but to test this is very, very hard, with our technological capabilities of today.

 

[PeroK]

ah touche.

I know that math is independent of physical reality. But at the same time, we are using that math to prove things about the real world. I was watching a lecture where the quantization of momentum was derived using the periodic property of the momentum wave function, which is a form of euler's formula. These are continuous trig functions. Then I thought that the plank length might also derived in a similar manner: using continuous sines and cosines. It would be problematic if continuous functions based off of continuous space is used to prove space is quantized; that would be a contradiction.

But after reading the link, I see that space isn't really quantized.

There is a good, elementary example of the relationship between mathematics and physics in terms of a bouncing ball (there's a homework post about this at the moment). You can model a bouncing ball as an infinite geometric series of increasingly smaller bounces and, if you sum the infinite series you get a finite time at which the ball stops. In reality, of course, you do not have an infinite series of bounces and there comes a point where the mathematical bounces are so small that they are immeasurable, indistinguishable from the internal kinetic behaviour of the ball and, of course, smaller than the Planck length.

The mathematics stands up even though, in reality, you can never take the sum of an infinite series and there can only be in reality a finite number of bounces.