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Often when physics students are introduced to the HUP for position and momentum, the interpretation is that you aren't able to measure position and momentum for a particle to arbitrary precision at the same time. However, a better interpretation that works conceptually even without talking about measurements, is to accept that particles are waves and waves just do not have properties like position or momentum intrinsically. They are kind of emergent properties if you shape the waves the right way, and measure the right way.

(A) The HUP for energy and momentum also has a sort of "school-ish" interpretation. For instance in the book by Aitchison and Hey chapter 1.3. Here they say that two exact measurements of energy of a system will fluctuate more ($\Delta E$) the shorter the time the measuring device is interacting with the system ($\Delta t$), and that the two measurements will only yield the same results in the limit where the measuring device is interacting forever ($\Delta t \longrightarrow \infty$). They don't really explain why, beyond invoking the HUP equation.

(B) In QFT, virtual particles can be off-shell and have any energy they want, and this is explained via the HUP. Virtual particles can "borrow energy from the vacuum," provided they give it back shortly after. This is the interpretation I see again and again in the QFT litterature.

The first explanation (A) I can kinda follow, but ultimately it's not very precise. Also, it's very measurement dependent, and doesn't really give insight into what's going on in the particle waves/fields. Both waves and fields have intrinsic energies, right?

The second explanation (B) seems almost ridiculous to me. Particles can borrow energy? Does that concentrate energy in the vicinity of the particle? Can it borrow enough to create a black hole? How does it know that it has to give it back in a time interval according to the HUP equation? What am I missing?

Depenau
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    (B) is a ridiculous overinterpretation of the physical reality of virtual particles. Virtual particles are elements of Feynman diagrams, which is a technique used to calculate the value of a sum. They are a mathematical tool. If you want to know what virtual particles are, you have to read the first few chapters of a QFT book. The statement in this question seems to come from the erroneous assertion "virtual particles must be ordinary particles somehow right? So maybe they borrow energy from the vacuum but they're ordinary particles?" Or-they just aren't the same thing as ordinary particles. – AXensen Feb 14 '24 at 18:55
  • @AXensen They are a mathematical tool No, they are not only that. Virtual particles may be boosted apart by external fields, like black hole gravity, for example and become real particles. That's how Hawking radiation works of BH. So please, do not underestimate power of quantum foam. – Agnius Vasiliauskas Feb 14 '24 at 19:08
  • @AXensen thank you for the reply. I've read plenty of QFT books by now, and this ridiculous overinterpretation is the one I've seen most often. And I agree with you. To my mind, virtual particles cannot be real, so invoking the uncertainty principle to explain them is absurd... but then what does $\Delta E\Delta t \geq \hbar/2$ mean? – Depenau Feb 14 '24 at 19:31
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    @AgniusVasiliauskas thank you for the source - unfortunately the last two sections which touch on this are very "citation needed" heavy. Do you have any other sources I could take a look at? – Depenau Feb 14 '24 at 19:51
  • @Depenau Google gazillion of references about "Unruh effect" and "Hawking radiation". – Agnius Vasiliauskas Feb 14 '24 at 20:02
  • @Depenau Besides, energy-time principle directly works in broadening laser pulse spectrum frequency ranges when one compresses laser pulse into ultra-short durations. So it's very practical principle. – Agnius Vasiliauskas Feb 14 '24 at 20:09
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    @AgniusVasiliauskas Regarding Hawking radiation, a recent experiment published in "Nature" demonstrates antimatter falls downwards, the same as regular matter. https://www.nature.com/articles/s41586-023-06527-1 This means there is equal probability of the regular particle or the anti matter particle of the pair falling into the black hole so that on average the net loss or gain of the black hole is zero, if we accept the separated virtual particles argument. – KDP Feb 14 '24 at 20:31
  • @KDP You mean that they have proved (according to you) that Hawking radiation does not exist ? I'm not quite sure about this, they do not speak about Hawking radiation directly at all. That's one. I'm not quite sure if falling real electron/positron can be directly compared to the pair of virtual electron and positron which BH materializes due to gravity. Even if fall rates are the same, half of particles does not return to quantum foam (is swallowed by BH), hence vacuum must borrow energy back from black hole, hence it shrinks. Its easy as that. – Agnius Vasiliauskas Feb 14 '24 at 20:47
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    I'm not saying that Hawking radiation does not exist. I 'm just saying the populist press explanation for the cause of Hawking radiation is flawed. – KDP Feb 14 '24 at 20:52
  • @KDP The "press explanation" never relied on antimatter falling away from mass, so the fact that antimatter falls down is just irrelevant (PS I'm an author of that paper). This incorrect press explanation was about a particle and an antiparticle being created on opposite sides of the event horizon. it never predicted that more particles than antiparticles were emitted - a black hole emitting equal numbers of electrons and positrons is still losing energy/mass. The problem with this explanation is that it doesn't explain how all of Hawking radiation for reasonable black hole sizes is photons. – AXensen Feb 14 '24 at 21:15
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    The time-energy HUP is very different than the position-momentum one. The latter is a theorem under the current version of quantum mechanics. Proving rigorously the former is .. well a field of study. There are "proofs" but then the interpretation is different than with the standard HUP – lcv Feb 15 '24 at 12:21

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Can [virtual particles] borrow enough [energy] to create a black hole?

Why not ?

Spawning a micro-black hole of Planck mass $m_P$ from quantum foam has time window uncertainty of (according to uncertainty principle): $$ \tag 1 \Delta t\gtrsim \frac {\hbar}{2~m_P~c^2} = \mathcal{O}(10^{-44}~s) $$

While Hawking radiation tells us that micro black hole of mass $m_P$ will have lifetime of : $$ \tag 2 \tau = {\frac {5120\pi G^{2}m_P^{3}}{\hbar c^{4}}}= \mathcal{O}(10^{-40}~s)$$

Which is within error of 4 orders of magnitude. Given that Hawking evaporation time formula (2) is dedicated for evaporation of astronomical-sizes of black holes and hence may not be very suitable for calculating exact evaporation times for micro-black holes, I find this error range acceptable for believing that virtual particles can borrow large enough energy "from nothing" and then forming micro-black holes upon annihilation, provided that these little-monsters will return energy back to quantum foam in very short time frames.

As about how to understand Energy-time principle at the intuitive level, I will compare it to the well-known saying "easy come,- easy go". For example people who wins millions dollars with lottery tickets,- usually don't know how to manage (or don't want to) their prize in a financially stable way. They buy a lot of crap, until it's over :-) Same here, particles sometimes "wins" huge energies, but alas looses it in a moment.

As for practical considerations of Energy-time uncertainty, it plays very important role in ultra-short laser pulses. When one compresses laser pulse to ultra-narrow time window,- pulse frequency range band widens A LOT. For example $1~fs$ laser pulse has $440~THz$ bandwidth - so full visible EM wave frequency range, until the point it can be called a "white pulse".

Experimental proofs of Hawking radiation

Agnius Vasiliauskas
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    Perhaps you should just add that there is currently no scientific experimental support for the idea of Hawking radiation. Chances are that it does not exist. – flippiefanus Feb 15 '24 at 03:47
  • @flippiefanus Not exactly true, read my post edit, i.e. at least Sonic Black Holes are known to produce Hawking radiation analog. Albeit, it's not a gravitationally related effect which directly proves Hawking radiation, but this renders your saying that "no scientific experimental support for the idea of Hawking radiation" as obsolete, for sure. – Agnius Vasiliauskas Feb 15 '24 at 15:19
  • Sonic black holes are not black holes. The radiation seen from them can be explained by a classical process. So it is not same. – flippiefanus Feb 16 '24 at 03:43
  • They are models of black holes, so it can be used to validate BH mechanics. Nobody says that it's the same, just that we HAVE some form of Hawking radiation laboratory, instead of none like you say. – Agnius Vasiliauskas Feb 16 '24 at 05:43