Some conceptual difficulties concerning quantum field theory (QFT)

Question

Currently I am doing an introductory course in quantum field theory. Where most of the calculations so far a straightforward, the main difficulties I have are conceptual. Let me walk you trough.

In the beginning of the class we started with quantizing the solution of the (classical) Klein-Gordon equation. The commutation relations between $a(\mathbf{k})$ and $a^\dagger(\mathbf{k})$ allowed us to interpret them as creation/annihilation operators, in analogy with the quantum harmonic oscillator. Only now, our quantum field works on Fock space.

After the canonical quantization method we moved to the path integral method. As far as I understand it, this method is concerned with the direct calculation of scattering amplitudes. The generalization from path integrals in QM to path integrals in QFT was straightforward.

Alright, so now some of my difficulties.

1. The canonical quantization method gives us a quantum field $$\phi(\mathbf{x}, t) = \int \tilde{dk} \left[a(\mathbf{k})\exp(ikx) + a^\dagger(\mathbf{k})\exp(-ikx)\right]$$ which at every space point assigns a linear superposition of creation/annihilation operators evolving in time $t$.
   So what exactly does this field do to our Fock space? More precisely, if you start at a time $t$ and let the field evolve, how does the Fock space change? Is there some preferred Fock state this field wil evolve to?

2. The path integral methods gives us a way of calculating the "overlap between field configurations", in analogy with quantum mechanics. But I find it hard to understand this. I feel like what we should be talking about is the overlap between to states in our Fock space. More precisely, assuming at time $t$ we start with some state $\vert{N}\rangle$, under the time evolution driven by the Hamiltonian the system evolves to a state $\vert{M}\rangle$, we are interested in calculating the overlap between these two states in Fock space. PI-method then tells us we have to sum over all intermediate (field)configurations, with the appropriate weighing factor. Is it the case that these initial and final configurations are to be thought of as the initial and final states in Fock space?

3. I feel like by the expression in 1) we exactly knów how our field evolves. Exactly in the sense that, I have lost sight of where the 'quantum uncertainty' is in this expression. With respect to 2) I feel like that starting in some Fock-state at time $t$, having our field evolve from this point on, and knowing the answer to my question 1), we know exactly what every other state at time $t'$ will be. What his happening with 'the probability to find the system in this state at a later time etc' as we know from quantum mechanics.

Thanks in advance!

Answer 1

$3$. Given a system in state $i$ at time $0$, there is a nonzero probability to find the system in another state $f$ at a time $t$ - just as in quantum mechanics. This probability might be nonzero even the time $t$ is the same: $t=0$. In the Heisenberg picture the states remain fixed with time, and this probability is just $|\langle f|i\rangle|^2$. Notice that the expession in 1) is not evolution of states; it is evolution of operators of particle creation. The states remain fixed.

For instance, if a state consists of just one particle at point x at time $0$, then there is a nonzero probability to find a particle at a different point y at the same time $0$ (because $\langle 0|\phi(y,0)\phi^\dagger(x,0)|0\rangle\ne 0$). (EDIT: But the latter probability has no sense, see this answer.)

$2$. A bit offtopic. If we were doing the same in Euclidean field theory, then we would fix the initial and the final functions $\phi_i(x,0)$ and $\phi_f(x,t)$ (not field operators, not states), fix the functions $\phi(x,t)$ at the boundary of some ball and integrate over all functions inside the resulting cylinder (whatever this integration means). But the question obviously concerns Minkowski field theory: no idea what to do here.

Beware: having absolutely no understanding of the free quantum field, writing just to get a chance to be corrected by those who have.

Answer 2

I am not really sure if I have understood your questions correctly, but here is my attempt.

(1 and 3)

(a) When one says we promote the field to an operator, what exactly does the operator act on, and what does it give? Consider the action of $\phi$ on the vacuum, $\phi(x)|0\rangle$. You get what looks like a superposition of one particle states with definite momenta, with coeffecients that go as $e^{-ipx}$. So this 'looks' like a 'position eigenstate'. The interpretation is now clear-the field $\phi(x)$ 'creates' a particle at $x$.

(b) I'm not sure what you mean by 'what does the fock space evolve to'. All these states(the vacuum, one particle states, etc) are basically members of different Hilbert spaces which we sort of bring under one roof(the technical definition involves direct sums and tensor products) and call it the Fock space. All states are in the same space now, the 'space' is not evolving, just as a rotation (implemented by a rotation operator) doesn't 'evolve' the $(x,y)$ plane-it just takes one element of it to another. Now, in (a), I was vague on purpose about the field $\phi$ being in Schrodinger representation or Heisenberg representation. And that is because either way, the interpretation is fairly obvious-$\phi(\vec{x})$ creates a particle at $x$, and this 'action' is considered at some reference time $t=0$. If I allow time evolution of the field now, the interpretation simply becomes "$\phi(\vec{x},t)\equiv\phi(x)$ creates a particle at position $x$ at time $t$".

(c) It should be clear that the uncertainty principle is not violated anywhere- at each $t$, the position eigenstate is made up of a superposition of one particle states $|p\rangle$. You still can't have a spatially localised one particle momentum state, and vice versa.

(d) It is interesting to note that the fields in the free free Klein Gordon Lagrangian, in momentum space, behave as if each degree of freedom(i.e. the fourier conjugate of $\phi(x)$, $\tilde{\phi}(p,t)$ at each $p$) evolves independently in $t$, for each $p$. Each mode behaves like an independent oscillator, oscillating undisturbed in time without interacting with other oscillators at different $p$.

(2)

Don't get too worked up about the idea of 'Fock spaces'. It is just the space containing all possible states of your theory. So when you say you are measuring an overlap between states, it is obvious that the states belong to the Fock space, by definition. In regular one particle quantum mechanics, this fock space is just some familiar Hilbert space. Here, it is just a larger set.