Is there any relationship between cross-terms in QFT and double slit experiment? Suppose I have a fermion-fermion interaction with two channels $t$ and $u$, the matrix element is $\tilde M$ = $\tilde M_t + \tilde M_u$. Then when we square the matrix element, we have $$\sum_s\tilde |M|^2 = \sum_s |\tilde M_t|^2 + |\tilde M_u|^2 +\tilde M_t^*\tilde M_u+ \tilde M_t\tilde M_u^*.$$
I wonder if there is any relationship between the cross-terms $\tilde M_t^*\tilde M_u+ \tilde M_t\tilde M_u^*$ and interference in the double-slit experiment. If so, what does constructive and destructive interference mean here? Since the matrix element describes the amplitude, rather than probability, is there a connection between matrix elements and a wave function?
 A: They are similar. In both cases, we are interested in the components of the time-evolved wavefunction. In the double-slit case, we are interested in the position basis components. In scatrering, we're interested in the Fock basis components.
Now, if, as a result of our computation techniues, we compute this vector component as a sum :
$$\langle i|\psi\rangle=\langle i| \psi_1\rangle +\langle i|\psi_2\rangle=a_1+a_2$$
Then inevitably:
$$|\langle i|\psi\rangle|^2=|a_1|^2+|a_2|^2+a_1a_2^*+a_1^*a_2$$
This is just pure mathematics.
Now, why is it that our computation techniques result in expressing our desired vector component as a sum?
This has different answers for both cases, because the computation techniques are different.
In the double slit case, we are exploiting the linear nature of the Schrodinger equation. We're evolving the wavefunctions from Slit1 and Slit2 individually and adding the results, which is why this computation technique results in a sum as the final answer.  This technique is reminicient of the path integral technique, which is interpreted as a sum over possibilities.
In the Scattering case, we are using perturbation theory techniques like the Dyson series and the Wick's theorem to split-up the time-evolution operator:
$$\langle i| e^{-iHt}|\psi \rangle= \langle i| (\sum S_n) |\psi \rangle = \sum a_n$$
I've written this naive time evolution for simplicity. In general, we should be using the perturbation series arising from the LSZ formula.  But in either case, we end up expressing our desired vector component as a sum, which is why there are cross terms between the sum components in the final probability.
Terms of this sum are tensor products of Green's functions, which is why they can be drawn pictorially.
Should we be interpreting these diagrams as a sum over physical possibilities?
I would say perturbation theory is only a computation technique. Only the wavefunction is measurable.
Nevertheless, there is an interpretation of the individual lines of Feynman diagrams as a sum over possibilities. This is because it happens that you can calculate the factor contributed by each line as a path integral of the relativistic particle action.
Still, AFAIK it takes perturbation theory to derive the full diagram description : like how many lines should be attached to a vertex. I would say they are a purely computational technique without any "particles splitting into virtual particles" description.
