The Heisenberg Uncertainty Principle (HUP) and "borrowing energy"

Question

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?

Answer 1

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

• Fermi space telescope monitoring for gamma ray bursts emanating from evaporating primordial black holes, have found none as of 2008.

• However, Sonic black holes have been reported to emit Hawking radiation or by this article.

• Sister effect, the Unruh effect when quantum foam induces thermal bath in accelerated systems, may also was observed by CERN laboratory first time. By the principle of equivalence, Unruh effect is not much different from Hawking radiation.