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The Heisenberg uncertainty principle was found to apply to all particles in physics. Was it an assumption that it should also apply to the vacuum? One of the consequence of this extrapolation is that we get all kinds of infinities - infinite vacuum energy density which we do not see, problems merging gravity with quantum becomes and much more.

I am a physicist myself but nobody ever explained to me the justification except to say that a harmonic oscillator cannot have a zero ground state which is fine because there is a particle involved. But how do you mathematically prove that this rule extrapolates to the vacuum. We also get all kinds of infinities that we renormalize -- clearly we have an issue with our assumptions.

So, why this almost religious belief? Can someone please explain.

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    $\begingroup$ What does the HUP have to do with renormalisation? Also the HUP is technically a purely mathematical statement about non-commuting observables, I don't understand what "extrapolating to the vacuum means". $\endgroup$ – Nihar Karve Jan 11 at 4:15
  • $\begingroup$ The more general uncertainty principle, beyond quantum $\endgroup$ – mmesser314 Jan 11 at 5:38
  • $\begingroup$ Unfortunately there are many things in this question that seem perplexing to say the least. You should explain what you mean. $\endgroup$ – joseph h Jan 11 at 5:45
  • $\begingroup$ From the HUP, we derive that the vacuum energy at every point is infinite. "The quantization of a simple harmonic oscillator requires the lowest possible energy, or zero-point energy of such an oscillator to be E = 1/2 h v. Summing over all possible oscillators at all points in space gives an infinite quantity." Observation does not match up with theory in this case. So how does HUP still apply to the vacuum model given we always remove the infinite energy background in practice? $\endgroup$ – Chahé Adourian Jan 12 at 19:07
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Let us try making your statements and questions more precise. First point: the uncertainty principle is a feature of the quantum mechanics framework, it does not apply to a particular particle but to conjugated observables, such as, but not only, position and momentum.

Second point, because of the first point, you cannot "apply" the uncertainty principle to vacuum. What people usually do is study simple models to try and learn generic features, specifically here I am referring to the harmonic oscillator where, as you may know, the lowest energy eigenstate is not 0 but the zero-point energy, characteristic to the oscillator. This is of course a consequence of non-commuting operators and is related but not directly to the vacuum.

Third one has to move on to quantum field theory (QFT) to be able to speak about vacuum in the sense you mention. It is within this framework which is appropriate for higher energies, where the ground state of the system of fields is generally called the vacuum and one can construct a Fock space, which is the proper structure to handle multi-particle systems and therefore also describes a state without any particles. Within QFT one can compute the analogue of the zero-point energy but it diverges given that fields actually have infinite degrees of freedom (although finite, per space-time point), this is usually addressed by the normal ordering prescription, which takes care of removing this constant energy background.

On top of that there is a related issue called vacuum polarization, which has to do with higher order (in usual Feynman perturbation theory) contributions to this energy. This higher order contributions can be pictured as bubble diagrams made from the field content of the theory. They generally diverge and must be controlled by some renormalization procedure.

Having clarified everything that seemed to be mixed in the question, let me finish by trying to answer your last question.

The uncertainty principle is as it is stated a "principle" meaning is something that was proposed and which makes part of current theories of physics. It has survived already for roughly hundred years for a reason, it works. It helps us understand reality and is consistent with experiments (very different to religion). Having said that, it does not have/need a mathematical proof. However, this does not mean that a deeper explanation coming from further understanding in physics in the future, won't be able to get it as a consequence of even simpler principles.

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