Is annihilation a smooth process that requires time?

Question

There are a lot of questions on this site about annihilation and quantum processes (and whether they are instantaneous), but none of them specifically give a satisfactory answer to my question. None of these specifically take annihilation to be instantaneous or not, and none of them specifically address how to model the process smoothly.

Like this one:

We then talk about a tunneling event or an emission/absorption event, and we can attribute to them a characteristic time.

Are quantum events such as decay, emission, absorption and tunneling truly instantaneous or is there some small time period?

I have read this question:

But there is no clearly identifiable "particles" in the middle when the scattering reaction takes place, just a messy and evolving quantum state. We can track neither experimentally nor theoretically the identity of a particle in the input and point to any specific point in time when it "disappears". QFT allows us to compute the probabilities for certain inputs to lead to certain outputs, but it is not tractable to produce some sort of "live view" of the scattering process - it's like a black box, particles go in, particles come out.

At what point of approach in annihilation process the charges of antiparticles disappear? How is relevant to the electron-proton interaction?

Now I am not asking about the intermediate states or what is inbetween. I am simply asking, if nothing is inbetween, just a "black box", and electron-positron pair coming in, and two photons (or a photon near a nucleus, but let's not complicate it with this) going out, then is this process truly instantaneous?

The reason I am asking, is because as far as I understand, quantum mechanical processes always take time and are never instantaneous. The deeper reasoning behind this should have to do with the speed of light not being infinite, but my question is not about this. My favorite example of describing whether quantum processes are instantenous is my previous question about the electron's change of energy levels (in atoms).

In every reasonable interpretation of this question, the answer is no. But there are historical and sociological reasons why a lot of people say the answer is yes. Consider an electron in a hydrogen atom which falls from the $2p$ state to the $1s$ state. The quantum state of the electron over time will be (assuming one can just trace out the environment without issue) $$|\psi(t) \rangle = c_1(t) |2p \rangle + c_2(t) | 1s \rangle.$$ Over time, $c_1(t)$ smoothly decreases from one to zero, while $c_2(t)$ smoothly increases from zero to one. So everything happens continuously, and there are no jumps. (Meanwhile, the expected number of photons in the electromagnetic field also smoothly increases from zero to one, via continuous superpositions of zero-photon and one-photon states.) The wavefunction $\psi(\mathbf{r}, t)$ evolves perfectly continuously in time during this process, and there is no point when one can say a jump has "instantly" occurred.

Do electrons really perform instantaneous quantum leaps?

Now this in knzhou's perfect answer is really naively understandable, the quantum state smoothly changes with the expectation values of the first state and the second state changing smoothly. This is understandable, because we are talking about a probability distribution of the electron around the nucleus.

However, when I try to use this for the process of annihilation, it does not seem to work. The good thing about the electron jump example is that the probability distribution describes the electron's position around the nucleus with co-existing multiple states already at certain probabilities. From there, applying the smooth change is understandable.

Though, this is where it becomes hard to apply for the annihilation, because the particle's type itself is not like that. The particle's type itself is in a definite state (as far as I understand), and does not evolve (without interaction), does not change, and an electron is always an electron. I could not say that the electron's type is described by the wavefunction in a way where the particle is at certain probabilities co-existing as an electron and as a photon at the same time.

But with annihilation, something like that should happen. The quantum state before annihilation describes the electron-positron pair's state as a definite, the type of particles are just that (before annihilation, the electron is 100% electron, and the positron is 100% positron, and after annihilation, the photons are a 100% photons). After the annihilation, though, the particle's type changes instantaneously, to two photons. There is no smooth change, no probabilities.

In other words, can we somehow smoothly model the annihilation process with the similar method (electronic transition) in knzhou's answer, or is it just an abrupt (instantaneous) change?

Question:

1. Is annihilation a smooth process that requires time?

Answer 1

I would say your second quote answers your question just fine. No, of course the process of annihilation is not instantaneous. You start with some state of the electron field and photon field that you can interpret as "1 electron, 1 positron, 0 photons" and it evolves smoothly into some state (or superposition of states) of those fields that you can interpret as "no electrons or positrons, 2 (or more) photons." If you probe the state during the annihilation, you will not see either the input state or the output state, but a messy and random field configuration, just as your second quote says. Though it's not tractable to compute this smoothly evolving intermediate state, its existence answers your question.

You may be running into issues because the idea of "an electron" or "a photon" is in itself problematic. In QFT, the fields are reality, and we name/interpret certain configurations of those fields as containing "particles" for our convenience. But the particles don't have physical reality outside of describing the fields, and if the fields evolve in a way that seems discontinuous in your particle model, then that's just because the particle model has broken down and not because the fields are actually doing anything discontinuous. (I.e. the field state has strayed sufficiently far away from any state we can easily describe by listing particles.)

Also, you can definitely construct (mathematically, if not practically—but of course you could physically realize it if you really wanted to) a quantum state that is a superposition between "there is an electron here" and "there is a photon here". This state will quickly evolve to a superposition between "there is an electron here" and "there is a photon way over there," so it's not particularly useful for calculations, but it exists. We do see something like this in neutrinos. The Sun creates electron neutrinos via nuclear reactions, but excitations of the electron neutrino field oscillate smoothly into excitations of the tau and muon neutrino fields (and back) as they travel. The first neutrino detectors on Earth thus only detected a fraction of the expected number of solar neutrinos, since they were only looking for electron neutrinos, spawning the (now-resolved) solar neutrino problem. Or contemplate the superposition $K_L=\frac1{\sqrt2}(K^0+\bar K\vphantom K^0)$ of two kaon types that lies in the resolution of the tau-theta puzzle (this is analogous to a superposition of an electron and a positron).

Answer 2

can we somehow smoothly model the annihilation process?

The answer is no and yes.

No, because there is no established theory about it. As long as electric and magnetic fields are represented as continuous and steady functions and as long as photons with their electric and magnetic field are treated as structureless, there will be no description of the transition of two particles to pure energy.

Yes, because there is a structural description of me. In it, a Theses about the inner structure of physical fields with only two particles is presented that can represent both the annihilation process and the termination of this process when the electron approaches the proton. The charm of the theory is therefore not so much in annihilation processes, but much more in the description of why the proton-electron interaction leads to photon emission, but this process does not end in annihilation.