I am still learning QFT, on my own. I am using A. Zee's nice book called quantum field theory in a nutshell. It is funny too. When I got to Wick's theorem, I couldn't help but notice an analogy between a formula I came across while working on the Atiyah problem on configurations of points and Wick's theorem. This led me to want to ask related questions here. The questions are vague. This is due in part to my beginner's understanding of QFT (I am still at the scalar field level for now, and did not yet get to spin 1/2 fields for example).

Suppose you are given $2m$ elements $\psi_i \in \mathbb{C}^2$, for $i = 1, \ldots, 2m$. There is a quantity one may define, say $$ S(\psi_1, \ldots, \psi_{2m}), $$ as a sum of products of $\omega(\psi_i, \psi_j)$, where $\omega$ is a non-degenerate complex symplectic form on $\mathbb{C}^2$, in such a way that each $\psi_i$ is used exactly once (for $1 \leq i \leq 2m$) and satisfying some other rules. For example, if $m = 2$, we have a quantity $$ S(\psi_1, \ldots, \psi_4) = \omega(\psi_1, \psi_2) \omega(\psi_3, \psi_4) + \omega(\psi_1, \psi_3) \omega(\psi_2, \psi_4) + \cdots$$ I did not describe what these other rules are, but hopefully this won't matter much, at the level of details of this post!

My questions are:

1. Is there a QFT (existence in a mathematical sense only, meaning that I am fine if there are no physical applications of this QFT) for which either the propagator, or the 2-point Greens' function is given by: $$ S(\psi_1, \psi_2) = \omega(\psi_1, \psi_2) ?$$
2. If so, and please provide some detail, what would Wick's theorem tell us in this case? I am hoping it would give us an equality between $S(\psi_1, \ldots, \psi_{2m})$ and something else. What is this something else please?

I am still learning QFT, on my own. I am using A. Zee's nice book called quantum field theory in a nutshell. When I got to Wick's theorem, I couldn't help but notice an analogy between a formula I came across while working on the Atiyah problem on configurations of points and Wick's theorem. This led me to want to ask related questions here. The questions are vague. This is due in part to my beginner's understanding of QFT (I am still at the scalar field level for now, and did not yet get to spin 1/2 fields for example).

Suppose you are given $2m$ elements $\psi_i \in \mathbb{C}^2$, for $i = 1, \ldots, 2m$. There is a quantity one may define, say $$ S(\psi_1, \ldots, \psi_{2m}), $$ as a sum of products of $\omega(\psi_i, \psi_j)$, where $\omega$ is a non-degenerate complex symplectic form on $\mathbb{C}^2$, in such a way that each $\psi_i$ is used exactly once (for $1 \leq i \leq 2m$) and satisfying some other rules. For example, if $m = 2$, we have a quantity $$ S(\psi_1, \ldots, \psi_4) = \omega(\psi_1, \psi_2) \omega(\psi_3, \psi_4) + \omega(\psi_1, \psi_3) \omega(\psi_2, \psi_4) + \cdots$$ I did not describe what these other rules are, but hopefully this won't matter much, at the level of details of this post!

My questions are:

1. Is there a QFT (existence in a mathematical sense only, meaning that I am fine if there are no physical applications of this QFT) for which either the propagator, or the 2-point Greens' function is given by: $$ S(\psi_1, \psi_2) = \omega(\psi_1, \psi_2) ?$$
2. If so, and please provide some detail, what would Wick's theorem tell us in this case? I am hoping it would give us an equality between $S(\psi_1, \ldots, \psi_{2m})$ and something else. What is this something else please?