Is there, mathematically speaking, a QFT with the following properties?

Question

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?

Answer 1

First of all, strictly speaking you are talking about a finite-dimensional (in fact, 2-dimensional) caricature of QFT. More precisely, the only way in which your model can be thought of as a QFT is if the underlying space-time consists only of two discrete points. For a (Minkowski) space-time continuum, the arguments of the two-point (Wightman) function $\omega$ are not elements of $\mathbb{C}^2$, but rather (temperate) test functions in space-time taking values in (say) a finite-dimensional complex vector space $V$ whose dual is where the fields take values in. In other words, the two-point function $\omega$ is a (two-point temperate) distribution on space-time.

Algebraically, the "complex symplectic" character of $\omega$ is more conveniently expressed by a property of positive definiteness using an involution operator on $V$ which behaves analogously to complex conjugation, just like a positive definite sesquilinear form ( = Kähler form) associated to a complex symplectic form. This property will be crucial later on to recover the Hilbert space where the field operators act, and (unlike the complex symplectic property) is spin-independent.

With those provisos out of the way, the answer to question 1.) would be yes - under certain circumstances. There is a class of QFT's which is completely determined by its two-point function: the so-called generalized free field theories. All you need from $\omega$ to get one of these is (besides the aforementioned temperedness and positive definiteness properties) is the so-called spectral condition (which restricts the support of the Fourier transform of $\omega$ - by the way, the main reason why we need temperedness is to be able to define that Fourier transform) and microcausality (which says that $\omega$ is symmetric or antisymmetric in its arguments whenever they are causally disjoint, depending on whether the field has integer of half-integer spin respectively). Neither temperedness nor the spectral condition make any sense if $\omega$ has finite-dimensional arguments, by the way. These four properties comprise a minimal subset of the so-called Wightman axioms for $\omega$.

Generalized free fields are thus called because they have no interaction, that is, their S matrix is the identity operator. This is expressed by the fact that all its truncated $n$-point functions vanish except for $n=2$, which is equivalent to expressing the $2n$-point functions in terms of $\omega$ pretty much as you wrote in the OP. The difference from the usual free fields is that the field operator does not necessarily satisfy an equation of motion coming from a Lagrangian variational principle - as such, there may be no action functional associated to it. The translation invariance of the relativistic vacuum state (entailed by the spectral condition, which on its turn defines such a state) implies that the one-point function is constant, and by redefining the field operator one can set this constant to zero - this, on its turn, forces the $(2n-1)$-point functions to vanish as well (not just the truncated ones). The ensuing $n$-point functions satisfy (appropriate versions of) the same Wightman axioms. Finally, the field operator(-valued distribution), the Hilbert space where it acts and the relativistic vacuum state vector can all be recovered from the $n$-point functions through the so-called Wightman reconstruction theorem.

One can think of a generalized free field's $n$-point functions as a quantum version of the moments of a Gaussian probability measure for a random variable, which is characterized by its mean ( = one-point function), which can be set to zero by appropriately shifting the corresponding random variable, and its covariance ( = 2-point function).

As for question 2.), the particular form of the $n$-point functions for a generalized free field and the Wightman axioms together ensure that Wick's theorem holds with essentially the same form as for free fields, keeping in mind the changes for half-integer spin due to the spin-statistics connection, which is encoded in the appropriate version of the microcausality property.