I occasionally come up with what I call “ridiculous thoughts”.
These are ideas that can be disproved in a couple of minutes, a day at most.
This is one such idea. Can you disprove it?

I was playing around with Zeno’s paradox of the arrow.

“Zeno gave us the paradox of the arrow. In this paradox, an arrow must jump from one point in space to another point in space. It can’t jump instantaneously because then its speed would be infinite, so it must stop at each point in space for a time period. The arrow is not moving during that time period so must have a speed of zero. In other words, the arrow cannot move at all.”

and I noticed how the Heisenberg uncertainty principle popped out of it.

“If we know the precise position of a particle then we have no knowledge of its velocity, and if we know the precise velocity of a particle then we have no knowledge of its position”.

You can see how Heisenberg uncertainty principle solves the paradox of the arrow. In the paradox of the arrow, we know the precise location of the particle but, by the Heisenberg uncertainty principle, rather than knowing that its speed is zero at that point, we have no knowledge of its velocity, and so the arrow can continue moving.

Can you feed some numbers into this?

If space is quantised at the length of the Planck length. And if velocity is uncertain within the range imposed by the speed of light. Then what is the mass of the arrow (or other particle in place of arrow)?

Do we get a mass similar to the mass of an arrow, or Planck mass (2.17645×10−8 kg), or the mass of the electron/muon/tau, or the approximate mass of a neutrino, or the mass given by the momentum of light divided by its velocity?

The idea here is that we may find a new TOE in which classical mechanics is equivalent to a quantum mechanics in which the Planck length has been shrunk to zero.

I don’t see how an unknown velocity and an unknown particle size define a mass.

Also I don’t think the Zeno so-called paradox has anything to do with classical mechanics.

The Rev Dodgson said:

I don’t see how an unknown velocity and an unknown particle size define a mass.

Also I don’t think the Zeno so-called paradox has anything to do with classical mechanics.

In quantum mechanics, when the uncertainty in velocity is the speed of light, then the speed of the arrow can be anything permitted by classical mechanics. So rather than be stopped at a point of space as Zeno claimed, it is free to move.

In classical mechanics, the paradox of the arrow also vanishes if the distance between points is allowed to be an infinitesimal dx. In which case the speed dx/dt, the ratio of two infinitesimals, never drops to zero when the particle is at a known point in space or rises to infinity.

The classical mechanics argument fails, of course, if space is quantised, because then dx is not an infinitesimal and the paradox of the arrow reasserts itself.

The Heisenberg uncertainly principle is

dx*(m*dv) >= hbar/2

where m*dv is the uncertainty in the particle momentum.

Substituting c for dv, the Planck mass m_p for m and the Planck length l_p for dx.
Then dx*(m*dv) = l_p*m_p*c = hbar > hbar/2

That’s by definition of the Planck momentum in
https://en.wikipedia.org/wiki/Planck_units#Derived_units

So it works.

In order for the Planck length to shrink to zero without either G or c changing, requires hbar shrink to zero.

That in turn is consistent with classical mechanics. In particular the lack of quantisation of electron orbitals of the hydrogen atom in classical mechanics is consistent with hbar = 0. And, as we know, classical mechanics fails the test of spectroscopy, we need quantum mechanics for that.

But this wouldn’t be new, it would have already been known circa the year 1901.

It’s just startling that Zeno’s paradox of the arrow leads directly to the Heisenberg uncertainty principle when space is quantised.

I can’t claim that Zeno saw the Heisenberg uncertainty principle 2,400 years before Heisenberg. He didn’t.

But I can claim that Zeno saw the possibility of quantised space and saw that classical mechanics would be unable to handle it.

mollwollfumble said:

I can’t claim that Zeno saw the Heisenberg uncertainty principle 2,400 years before Heisenberg. He didn’t.

But I can claim that Zeno saw the possibility of quantised space and saw that classical mechanics would be unable to handle it.

1. You are giving Zeno way too much credit. It’s just a not very interesting attempt to make a paradox out of something non-paradoxical.

2. If you insist on combining the wacky notion of quantised space with classical mechanics, there are plenty of ways it could be done. For instance, the points of matter at each location may continuously flow to adjacent points, until after some finite time point 1 is empty, and all that mass is at point 2 (or has flowed to other adjacent points).

What’s being measured in these so-called paradoxes is the path taken by a moving object, not the moving object itself.

This measured path may be empirically derived but as applied to the moving object, it’s just a modelling tool.