Some questions about correlation functions and amplitudes in quantum field theory

Question

I have been trying to learn some quantum field theory recently and I have a few questions which should be easy to answer for experts. I understand the basics of quantum mechanics / statistical mechanics, but I am having some trouble scaling this understanding up to qft. Before diving in, I want to emphasize that this is not a question about the foundations of qft. Right now, I am not so interested in thinking about how to define measures on spaces of fields or any of the more mathematically elaborate definitions of qft (although cobordisms do show up naturally). I just want to understand what is going on from the physics perspective.

The Setup: Choose some space time manifold $M$ and a vector space $V$. The space of fields we are going to study is ${\rm maps}(M,V)$. Choose an action $S : {\rm maps}(M,V) \to \mathbb{R}$ which is presumably given by the integral of some Lagrangian. Then we associate the amplitude $$e^{ \frac{i S(\phi)}{\hbar}}$$ to the field $\phi \in {\rm maps}(M,V)$, where $\hbar$ is some value used to make the exponent unitless.

Example 1: Consider the case $V = \mathbb{R}^3$ and $M = \mathbb{R}$. Then the fields $\gamma \in {\rm maps}(\mathbb{R},\mathbb{R}^3)$ are particle trajectories. If we choose the Lagrangian correctly, we can recover quantum mechanics. Indeed, we can recover the matrix entries of the unitary evolution operator from the following path integral: $$ \langle b,T | a,0 \rangle = \int_{\gamma(0)= a,\gamma(T)=b} e^{\frac{i S(\gamma)}{\hbar}} D\gamma$$

Question 1: How should I interpret the correlation function $$ C(t_1,t_2) = \int_{\gamma(0)= a,\gamma(T)=b} \gamma(t_1) \gamma(t_2) e^{\frac{i S(\gamma)}{\hbar}} D\gamma $$

Example 2: Now consider the case where $M = \mathbb{R}^4$ is Minkowski space and $V = \mathbb{C}$. I am not sure how to interpret the fields $\phi \in {\rm maps}(\mathbb{R}^4,\mathbb{C})$. It feels funny to call them wave functions in the quantum mechanics sense.

Question 2: Choose a 4-dimensional tube $T \subseteq M$ and let ${\rm maps}_\partial(T,\mathbb{C})$ be the fields with some fixed boundary condition. How should I interpret the complex number $$ \int_{\phi \in {\rm maps}_\partial(T,\mathbb{C})} e^{\frac{i S(\phi)}{\hbar}} D\phi$$

Question 3: How should I interpret the correlation function $$ \int_{\phi \in {\rm maps}_\partial(T,\mathbb{C})} \phi(m_1) \phi(m_2) e^{\frac{i S(\phi)}{\hbar}} D\phi$$ I have read many times that this correlation function should be the amplitude of a particle propagating from $m_1$ to $m_2$ but I don't really understand this very well. Maybe it is supposed to be related to the fock space sitting inside ${\rm maps}({\rm maps}(M,V),\mathbb{C})$, but I am not sure.

It is possible that I am completely on the wrong track. If this is the case, I would very much appreciate some pointers!

Answer 1

General principle: classical fields of the classical field theory become operator valued fields of the quantum field theory and insertion of classical fields in the path integral compute the matrix element of the time ordered product of the corresponding operators in the quantum theory. Time ordering means that operators are inserted at increasing times when going from the right to the left.

Question1: in the quantum theory, $\gamma (t)$ becomes an operator (corresponding to the position of the particle at the time t). Insertion of $\gamma (t_1) \gamma (t_2)$ compute the matrix elements of the time ordered product of these operators.

Question2: In a quantum field theory defined on spacetime of dimension n, there is a Hilbert space of states associated to every closed manifold Y of dimension (n-1), and every manifold X of dimension n and of boundary Y defines a state in this Hilbert space. The formula given in question 2) is the path-integral realization of this fact: the Hilbert space of states on the boundary is made of fields on the boundary and to define a state from the interior, it is enough to give the pairing between this state and any boundary configuration: this pairing is the complex number obtained by doing path integral over the interior with the fixed boundary configuration.

Question3: in the quantum theory, $\phi(m)$ becomes an operator and insertions in the path integral compute matrix elementts of these operators. The interpretation in terms of particles propagating from $m_1$ to $m_2$ is not correct because there is no well-defined position operator for a particle in quantum field theory. The physical interpretation of these correlations functions in terms of particles is done through the LSZ reduction formula https://en.wikipedia.org/wiki/LSZ_reduction_formula , which is the way to extract scattering amplitudes from correlation functions.