Average momentum of particle/anti-particle pairs

Question

I'm far from an expert in physics, I've gathered bits from wiki, books and educational videos. So I apologize in advance if my question is not well framed or plain stupid.

It's about particle / anti-particle pairs. I understand they can be created in (at least) the following ways:

1. from collisions or other energetic event, in which case their momentum must respect energy conservation, and they are real in the sense that they can interact in measurable ways with other particles.

2. as virtual particles in some computations. In this case, it is not clear to me if they are just mathematical tools or can have some measurable effect. From what I read, it's also not clear if there is a consensus on this.

3. as materialization of quantum fluctuations in a vacuum. I understand they usually annihilate rapidly, except if one of them gets sucked inside a black hole. It is not entirely clear to me if there is a consensus of the "reality" of these particles, although they are a good explanation for the Casimir effect and Hawkings radiation.

My question is about this last case of particle pairs created from empty space. Logically they should have a random momentum direction, because there is no reason for a particular direction to be favored. So if one could measure a sample of these particles' momentum vector, its average over many measurement should be zero. Even if it's not feasible right now, let's pretend this is a thought experiment. Let's repeat the experiment from another inertial frame, moving at a fixed speed w.r.t. the first experiment. Because all non-accelerating frames of reference are equivalent, we should get the same result, i.e. null average momentum vector. But if we consider each individual particle, it seems impossible that the statistical expectation of it's momentum vector is null in both reference frames, since they are moving w.r.t. one another.

Could someone help me resolve this paradox, and in doing so, maybe point out where I err in my reasoning or hypothesis?

Answer 1

Your question hits on a lot of misconceptions, mostly derived from pop sci (YouTube) descriptions of virtual particles.

There are at least two classes of virtual particles. Your favorite content creator says that the vacuum is boiling with virtual particle/antiparticle pairs that have positive energy, $E$, that they "borrow" from the Heisenberg Uncertainty Principle for a time $t\approx E/\hbar$, while conserving 3-momentum.

These virtual particles are from Old Fashioned Perturbation Theory (which was the way to do things before 1948 [note: it is 2024]).

It leads to a lot of noob confusion: ppl proposing harnessing the energy. It also implies an absolute rest frame, since the average momentum of space bubbles is zero in that frame.

Diagrams of these are time ordered, so in tree-level $ep\rightarrow ep$ scattering, there are separate diagrams for electron (proton) emits a virtual photon, which is then absorbed by the proton (electron).

This always begs the question: how can exchanging a virtual particle cause repulsion, and then we have to suffer this:

which gives the idea that a convoluted path leads to repulsion. Really, no physical intuition regarding kinematics is gleaned from this, imho....and the diagrams are in momentum space anyway.

This problem also arrises in modern virtual particles, the ones in Feynman diagrams; however, Feynman diagrams are fully relativistic and Lorentz invariant. In tree level $t$-channel charged scattering:

there is but one diagram. It represents virtual photon exchange in which energy and momentum are conserved at all vertices. The vertices are not time-ordered, and the photons four momentum has:

$$ m^2 = E^2 - p^2 < 0 $$

In fact, there is a frame (the Breit Frame) in which $E=0$ and only 3 momentum is transferred.

One doesn't need a boomerang in a canoe to describe repulsion: the photon momentum has the energy and momentum required to make the external lines (particles that are observed in the lab) have the correct kinematics. No matter what.

In this view, the vacuum is static. It is not boiling. Rather, it's a vacuum at $t=0$: $|0\rangle$, and at a later time it's observed to be a vacuum: $|0\rangle$.

What happened in-between? It went through both slits--meaning it took all possible field configurations that conserve energy and momentum density over the volume observed, and those intermediate states are the virtual particles.

In either-case, you have to imagine an intermediate vacuum that has zero average momentum in your frame (since there is no absolute rest frame), so a moving observer must see different intermediate states.

Noobs (laymen, really) have a habit of taking virtual particles way too seriously and going down rabbit holes that just take them further away from understanding basic relativity principles (e.g.: you can't move with respect to space), and basic quantum principles, too (e.g.: stationary states).

Regarding Blackholes and radiation. A basic blackhole has one parameter: $M$, from which we can derive a length scale ($R_S$) and a temperature, $T$. Ofc, the blackbody radiation associated with $T$, should peak around wavelength $\lambda \approx R_S$, so talking about tiny virtual particles near the horizon may not be the best approach.

Answer 2

In special relativity, we want to keep track of the four-momentum since it transforms as a Lorentz covariant object. Taking that into account you can find a Lorentz invariant quantity, namely $p_\mu p^\mu$. On the other hand, the three momentum is not Lorentz covariant, when you move between frames it will have different values.