What is the predictive power of each object in QFT, and how are they related? [closed]

Question

My background is not in physics or mathematical physics, so this question is mostly out of ignorance, and probably easily known to experts.

Basic Setup

1. You begin with a spacetime $M$. (Minkowski in the textbooks I read.)
2. Inspired by relativity, you interpret $E^2=p^2+m^2$ as an equation of operators (called first quantization) and get the Klein-Gordon Lagrangian $L$ for scalar fields, and Dirac Lagrangian for fermions (which I interpret loosely as a linear variant of Klein-Gordon). In particular, for each choice of particle you need to choose a fiber bundle $E\rightarrow M$, and you interpret the Lagrangian as a function of the first jet bundle $J^1(E)$ of this fiber bundle.
3. Using a standard mechanism called "canonical quantization" you let the fields themselves $\phi$ be operator-valued fields $\hat \phi$, and you re-interpret the Lagrangian (whatever it was) as a function of the operator-valued field and its partial derivatives. (Is there a way to speak of this in terms of jet bundles? I'm not sure.) This is where Fock spaces come up: the operators are operators on Fock-space, some construction where each element of fock-space describes the wavefunction of a potentially multi-particle state of the system throughout all spacetime $M$.

When you are finished, the operator-valued field $\hat \phi$ that minimizes the action (integral of the Lagrangian over spacetime $\int_ML[\hat\phi]$) is the quantum field that you worked so hard to get.

Of course it gets more complicated, you have bosons, and Gauge theory, Yang-Mills Lagrangians, interacting theories, and so forth. But let's keep it simple.

Quantum Field Centric Narrative

Wait, so what does $\hat \phi(x)$ mean? It's somehow a wave function of all of spacetime, but also depends on $x$, which is a particular point in spacetime?

Let's see how people interpret it. Well, they say that because it solves the (say) Klein-Gordon Equation, it must satisfy: $\hat \phi(x) = \int \frac{d^4 p}{(2\pi)^4} \delta(p^2 - m^2)\theta(p^0) [\hat a_pe^{-ip\cdot x} + \hat a^{\dagger}_pe^{ip\cdot x}],$ (or $\int\frac{\hat a_p e^{-ip\cdot x} + \hat a^{\dagger}_pe^{ip\cdot x}}{(2\pi)^{\frac32}\sqrt{2E_p}}d^3p$, if integrating over only 3 momentum) where $\hat a_p$ and $\hat a^{\dagger}_p$ are the annihilation and creation operators, which it takes some time to define properly, but are perfectly respectable operators on Fock space. (A slightly more complicated thing happens with Dirac, and you end up with antiparticle creation/annihilation. Okay...)

Physicists then say things like "$\hat\phi$ creates and annihilates particles". This is how they think of it? Creates them where? Annihilates where? Seems mostly meaningless, this is just an expansion, more like Fourier is for functions than anything particularly deep.

The $\hat\phi(x)$'s are not Hermitian in general, so they don't represent observables... The closest I can get to understand what they represent (based on how they are used) is through propagators. In that case, let's delve into that.

Propagator Centric Approach

Define the propagator to be the Green function of the Euler-Lagrange equation interpreted as an operator. As I understand it this is equivalent to the Feynman propagator, which physicists write as $\langle 0∣T\{\hat\phi(x)\hat\phi(y)\}|0\rangle$, where $T$ is a reordering so that the time of $x$ is before that of $y$. (This assumes $M$ is Minkowski, not sure what to do otherwise.) From its equivalence with the Green function, you can compute it explicitly.

Here $|0\rangle$ is the vacuum state in Fock space. They interpret this heuristically as the density of the probability amplitude that a particle created in spacetime point $x$ end up in spacetime point $y$. (Why would they give it this interpretation other than it works? This doesn't seem to come from the definitions.)

Uhm, okay. This seems like something that has predictive power. But then, with the view that is the Green function of the Euler-Lagrange operator, it seems like we are never actually using $\hat\phi$, so that really makes the propagator into the main event.

Also: if the propagator takes in only two spacetime points, and talks about a single particle. What happened to Fock space? As far as I can see, whenever anything is asked about multiple particles, they end up using multiple propagators, one for each particle. Is that principled? Even in cases where they take into account interaction Lagrangians, they end up using sequential propagators, one for each type of particle coming from their free theory. Is this approximating anything more natural?

Scattering Centric Approach

I am not sure you can really say that there is a scattering centric approach, but you van think of it as a variant of the Propagator centric approach, you can take the limit as time goes to minus infinity and plus infinity and try to compute probability amplitude of initial and final states. Ultimately they end up using the propagator interpretation, as far as I understand.

Representation Theory Centric Approach

Each particle type corresponds to an irreducible projective unitary representation $U$ of the Poincare group.

The interpretation that one gives is then that if $\psi$ is in Fock space, then it becomes $U(a,A)\psi$ for an observer transformed by the element $(a,A)$ in the Poincare group.

I don't see how this jibes with any other interpretation... I suppose you can deduce a Hamiltonian by figuring out what $H$ what satisfy that $U((-t,0,0,0), 1)=e^{-itH}$, and from that figure out a Lagrangian... But $\hat \phi$, the least action quantum field, makes no appearance, nor does the propagator.

Questions

1. It looks to me that the interpretation that the Feynman propagator gives the probability amplitude that a particle created at spacetime point $x$ ends up in spacetime point $y$ is inherently discrete: it assumes that interactions are particle by particle, and sequential, and that there are no multiple particle interaction. Is that understanding correct? Is that an assumption? Is it an approximation of an interpretation that is more natural and allows more complicated things to occur?
2. Why is this interpretation of the propagator at all reasonable? It does not seem to come from the basic setup at all. Was it just a fluke that this turns out to give good results? Is there a way to relate this back to more classical quantum mechanical interpretations such as observables?
3. What meaning, if at all, does the operator-valued quantum field $\hat \phi$ itself at a specific point in spacetime $x$ have OTHER than through the propagator? Does it contain more information that has predictive power or does all of its predictive power go through it being a step to define the propagator?
4. I really like the representation theory approach, it's very elegant. Is there a way to show that that interpretation implies the interpretation of the propagator as giving a probability amplitude density of a particle going from one place in spacetime to another? That would be very interesting.
5. This one is a little less important, but I'm a little confused about why you'd use the vacuum state at all. Fock space is a space of wave functions of all spacetime. Wouldn't the vacuum state mean that nothing whatsoever exists nor has existed throughout all space? I'm assuming this is an oversimplification, but I just wanted verification that I'm not crazy, and that this is indeed at least moderately wild to say that having vacuum in a specific area of spacetime is approximately equivalent to having vacuum everywhere always.