What does it mean to create particles by applying creation operator

Question

This question will be a way to ask if quantum mechanics is just a mathematical framework to predict physical outcomes without talking about the 'actual', maybe even 'deterministic' phenomena, such that the equations never make the wrong predictions. The main reason is that its language is quite mathematical, and just mathsy in the sense I can't relate it to things that I call physical processes. Recently I was reading about creation operators and vacuum states in QFT, and the book says the following (I can't find a way to use mathjax here, sorry. If you can let me know of a link for it)

There exists a vacuum state and creation operator indexed by momentum. Applying this operator to the vacuum state gives a 1 particle state with the index momentum.

Now, what I don't understand is, what did really happen? In real life there shouldn't be a operator (at least I think) and somehow we produce a particle out of air. I mean, say I produced a 1 particle state. I don't think I used a mathematical entity , operator, to do it but I guess the statement is that maths of this process looks like operation on vectors with operators and such but we never really talk about the physical process itself. In this sense, quantum mechanics looks like a framework rather than theory.

Is this interpretation correct or is reality made out of vectors and such?

Answer 1

Every theory of physics is a framework; it's just that quantum mechanics is specially removed from day to day experience.

Operators in quantum mechanics don't represent actual experimental actions on the system. They are a way of encoding the several observables that we can measure and their probability of measurement. For example, the creation operator is a way of encoding how many particles (or quanta of some system) are present in that system. Nothing "happens" when you apply the creation operator.

As for your last question, I regret to inform you that physics has no answer for you. It cannot tell you about reality at all. All it can do is employ models to test our experience against. As to the reality that hides behind it, if you can talk about it meaningfully at all, it can't tell you if it's vectors and such, it can only tell you if vectors and such are successful at modelling it.

Answer 2

This question will be a way to ask if quantum mechanics is just a mathematical framework to predict physical outcomes without talking about the 'actual', maybe even 'deterministic' phenomena, such that the equations never make the wrong predictions.

Every physical theory is “just” a mathematical framework to predict physical outcomes. Every framework only talks about the actual phenomena in the sense of what we can “actually” predict and observe. Quantum mechanics just feels unusual because there are things which we might have liked to predict and thought should have been there, but the theory makes no mention of such things. You should see look at Bell’s theorem, discussions of (local) hidden variables, and interpretations of QM to learn more about this.

such that the equations never make the wrong predictions

Well, if the equations made the wrong predictions, then we would assume we had the wrong equations and go looking for the other (right) ones. We are assuming that slices of reality are fundamentally rational and can be predicted with logical axioms and mathematical models - there is no reason reality had to be that way but so far this assumption has been very useful!

I think maybe what you are trying to express here is the feeling that the theory is so general that it could be used to model anything, and is thus not much of a model at all. The point to remember though is that the theory is constrained; namely, the assumption (in ordinary QM) that physical states are vectors (actually rays) and observables correspond to Hermitian operators does limit the types of things we are allowed to expect (which is a good thing)!

I am not as familiar with QFT, but similar considerations apply. The creation of a particle is modeled by a creation/annihilation operator, but we also assume this is a linear operator acting on a rigged Hilbert space. This gives some structure to “reality.”

Now, what I don't understand is, what did really happen?

Again, you should look at interpretations of QM for an attempt to answer this question. At present, there is no consensus among physicists as to what “really happened.”

Finally, as a general point, no language or mathematical model of reality is equivalent to reality itself, but we still expect some (approximate) homomorphism between the theory and reality. In other words, the theory tells us somewhat what reality must be like —- even if it is not reality itself. Similarly, a good map tells us what the actual territory is like, but the map isn’t reality.