[ziofill]

Physicist here. You are right that the mental picture we get when we use the term “particle” is a little ball or something like that. It is unfortunately a confusing name… You need to begin with a field, like the electromagnetic field for instance. When you look at its properties like energy, polarization and so on, in order to write down a state of the field you need to specify all of them in a way or another. In quantum mechanics you can associate a vector space to each property, and then (here is the important bit) you need to pick a basis for your vector space in order to write down its vectors. Obviously there is an infinite number of possible choices, and we usually end up choosing what makes things simple, so in the case of energy we pick the basis of eigenvectors of the Hamiltonian, because to evolve them in time you just need to multiply them by a complex number and that’s it. Well, those basis vectors are the “particles“ because when taken individually they share some properties with macroscopic particles, but the analogy really only goes so far. And the thing is that usually the state of the field is not in a single one of these basis vectors unless the conditions are very special, so even saying that the field is “made of particles” is misleading because it’s like saying that the wind is made of air going vertically, horizontally and across, which sure it’s “correct” because you can combine those directions and get any other direction but it’s also not really that…

[evanb]

Also a physicist.

It's not quite right to call basis vectors particles, even in a free field theory. The particles correspond better to the creation / ladder operators that take you from one Hamiltonian eigenstate to another.

In a perturbative interacting case, people still think of particles as these same ladder operators, but they don't connect eigenstates so simply (the interactions generically mix all the states with the same quantum numbers).

In a strongly interacting case the story is even more subtle, because composite operators may be closer to the ladder operators between the asymptotic states, even though they're built of other... particles? Language isn't great in this instance.

[ziofill]

Yeah it’s hard to use everyday language to exactly describe these things… And even more difficult to decide where to draw the line where some hand waving is acceptable.

[tel]

As an amateur, I think I follow most of this, at least at some level, but I don't follow why you'd unify the basis elements and particles. Thinking of a quantum harmonic oscillator, the eigenstates have some kind of localization that feels particle-like, but the oscillating pattern of a coherent solution seems "more particle-like" and arises out of the interference between those eigenstates. In particle-speak, I might try on a sentence like "this classical particle is generated by the interaction between... other... particles" but I'm clearly at a loss there.

On basis of that, I'd be more likely to say "QM needs to describe everything as a wave, and sometimes certain kinds of localized 'wave-packets' move around coherently, and that's what we'd call 'particles'". That also seems to gel with less coherent states where it feels like there's not really a particle to be found.

So, I'm curious why you'd prefer to relate the eigenstates themselves as particles. Again in the oscillator case, the eigenstates themselves seem less coherent and seem to behave less classically than I'd hope.

My best guess is that the property those states have that is not as well replicated by the "particle as a coherent wave packet phenomenon" is that they have well-defined energy quanta. But that's just a bit of a stab in the dark here. It perhaps makes more sense from the perspective of "particles are the things that we're able to measure in detectors" POV, though.

[ziofill]

You are exactly right at the end when you say that particles are what makes a detector go click. Let me try to clarify further. The "particleness" is not about localization, but about energy quantization. In fact, a single photon can be very non-localized, because spatial position is a different Hilbert space and it can be entirely independent of the energy. So the packets you are referring to are particles only if in their energy Hilbert space they correspond to a well-defined photon numbers, and that's why the analogy with classical particles only goes so far. To add to the confusion, one can speak of packets localized in phase space (e.g. coherent states, like the light produced by a laser) and packets localized in physical space like a short pulse, and these refer to different Hilbert spaces.