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Is HUP just a way the physicists found to correct the unnatural concept and mathematical formalism of dimensionless-point elementary particles? Making these points more fuzzy and therefore giving them a physical finite size and shape. Now with HUP the elementary point particles have finite size limits (i.e. minimum bounds on their position and momentum are applied). So, is HUP really just an artificial epistemic mechanism to make the elementary particles in the standard particle model theory more physical and less mathematical dimensionless operators.

You can imagine it, as the elementary bare particle being a dimensionless point in the center and HUP is actually defining an outer finite size shell for this particle.

In a nutshell, the bare elementary particle in the Standard Model is this dimensionless-point in space and when applying HUP for this particle we get the "dressed" version of the particle with its virtual particle noise mantle surrounding it.

Is the above physical description correct of what HUP really is?

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  • $\begingroup$ how would you test the difference, you know, this is supposed to be physics not metaphysics... $\endgroup$
    – hyportnex
    Apr 15 at 14:56
  • $\begingroup$ All I am saying is that HUP was the necessary physics formalism correction the bare mathematical model of particles was needing to become physical and finite. $\endgroup$
    – Markoul11
    Apr 16 at 9:38
  • $\begingroup$ Nevertheless, let me then rephrase my initial question, the other way around to: Q: Is HUP the necessary physics formalism correction to the bare mathematical model of elementary particles to make them physical and finite? I mean you cannot have a physical theory based purely on singularities? $\endgroup$
    – Markoul11
    Apr 16 at 10:02
  • $\begingroup$ For example It is known that the reduced Compton radius of the electron equals its HU limit in position and momentum. $\endgroup$
    – Markoul11
    Apr 16 at 10:06

3 Answers 3

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No, Heisenberg's uncertainty principle is actually a physical statement.

But there are uncertainty principles which are even more fundamental than Heisenberg's:

  • the uncertainty principle between time $t$ and frequency $f$ $$\Delta t\ \Delta f \ge \frac{1}{4\pi} \tag{1a}$$
  • the uncertainty principle between position $x$ and reciprocal wavelength $1/\lambda$ $$\Delta x\ \Delta \frac{1}{\lambda} \ge \frac{1}{4\pi} \tag{1b}$$

These Fourier uncertainty principles follow directly from the theory of Fourier transforms. (I recommend the video "Uncertainty Principles and the Fourier transform" for an intuitive explanation about this topic.) So far it is just pure mathematics, no physics at all.

You get something physical by adding the experimentally found relations about the quantum mechanical wave/particle duality (with a physical constant $h=6.6\cdot 10^{-34}$ Js):

  • the Planck-Einstein relation $$E=hf \tag{2a}$$ saying that a particle with energy $E$ behaves like a wave with frequency $f$
  • the de Broglie relation $$p=\frac{h}\lambda \tag{2b}$$ saying that a particle with momentum $p$ behaves like a wave with wavelength $\lambda$

Then you can put (1a,b) and (2a,b) together and derive Heisenberg's uncertainty principles:

  • $$\Delta t\ \Delta E \ge \frac{\hbar}{2} \tag{3a}$$
  • $$\Delta x\ \Delta p \ge \frac{\hbar}{2} \tag{3b}$$

Thus the physical Heisenberg uncertainty principle comes from the physical wave-particle duality and the mathematical Fourier uncertainty principle.

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  • $\begingroup$ "Thus the physical Heisenberg uncertainty principle comes from the physical wave-particle duality and the mathematical Fourier uncertainty principle." Playing devil advocate here: HUP also applies to Spins (Sz, Sx, Sy), but there is no Fourier Transformation between them. $\endgroup$
    – Paradoxy
    Apr 15 at 17:02
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No. The most classical example of HUP is the relationship between time and frequency/Energy. Consider a radar that wants to detect position and velocity of an object in the sky.

If we produce a very long electromagnetic wave with almost definite frequency, we can measure the velocity of that object with great accuracy. It is enough to measure the frequency of reflected light and put it inside Doppler equation. However, we face a problem. Since the wave is long and spread in time, we can't be sure when it is exactly reflected from that object, and to make the matter worse, it might have interacted with other stuff in the air (and not just what we want to observe). So inevitability, some uncertainty in the location of that object arises.

On the other hand, if we use a very short pulse by mixing a lot of different frequencies, we can measure the position of that object accurately. However, since the pulse itself is made of infinitely many frequencies, measuring the frequency of reflected pulse doesn't give any useful information that we could use to put it in the Doppler equation, so the velocity of that object becomes uncertain, to some extent.

Now all of this was a simple classical uncertainty. The quantum version arises from the fact that operators (matrices) don't commute with each other. Physically it may have different interpretation (not the one you have given though), but the most straight-one is perhaps statistical one. If you have some observable (position, momentum, etc) and many particles, you would observe a variance (or uncertainty) around the averages (expected values). For example if you find out that on average the velocity of particles should be zero, what you will observe is that yes indeed in average it is zero, but each individual particle may have a non zero velocity in different directions, which its average is zero. There are other (and perhaps more common) interpretation out there as well, but the point is, it is definitely not made-up as you might suggest.

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No. The Heisenberg uncertainty principle is an expression of the incompatibility of quantum operators, such as $\hat{x}$ and $\hat{p}$ of ordinary quantum mechanics or $\hat{\phi}$ and $\hat{\pi}$ of quantum field theory. For example, not all of the quantum objects are point particles with no geometric extension; atoms, for example, have size and structure and the Heisenberg uncertainty principle is applied to them as well.

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