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When I was in school, I couldn't accept the concept of binary pairwise additive interactions, aka "Force" in classical physics. For me, the emergence of magnetism was a clue that for a 3-body problem we need the inclusion exclusion principle.

I self learned quantum mechanics and I realized that QM is nothing but a model to hide the concept of interaction. I mean most of the theory is still about the field-particle approximation, where there is a giant field that the effect of the particle on it is always negligible and only the particle is being affected. What QM says about the same size interactions is the Hohenberg-Kohn theorems that says there is a wavefunction but hasn't been found yet.

The violation of bell inequality is another hint that binary causality can't explain quantum interactions like entanglement, there are also other ways of interactions in QMs like the Pauli exclusion principle.

With all these, why can't we explain interactions with inclusion exclusion principle? if we assume that the Hamiltonian of an N particle system obeys the inclusion exclusion principle that is: $$H=|H_1 \cup H_2 \cup H_3 \cup . . .|=|H_1| + |H_2| + . . . - |H_1 \cap H_2| - . . . + |H_1 \cap H_2 \cap H_3|- . . .+ . . .$$

What is a counter example that these types of Hamiltonians can't explain the physical world?

Does the Bell theorem also violate an inclusive exclusive version of classical physics?

By the way I'm not a native English speaker nor a physicist so please edit my question to be more accurate.

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Yes you can, and this technique actually is used in literature to some extent. Physicists call it "Perturbative many-body expansion".

For example, even for the well-known electrostatic interaction usually understood as pairwise additive, it is no longer true if there are interfaces. Some recent study by K. Freed (U. Chicago) in 2014 about this problem and his approach is exactly using the many-body expansion: https://aip.scitation.org/doi/full/10.1063/1.4890077?casa_token=5-3VxYBZGRIAAAAA%3AVMxN5EEVMndz7gOH_FyEfxzsskGU-ckJ2PqP8ngOzk22GqLeyA4IU8UmljvG1Ddk5A8wM82rrFtiTw

Some related discussions:What is the origin of the many-body expansion?

Anyway, one may understand the pairwise additive form as truncating the many-body expansion, based on the fact that in many case it converges raplidly, the approximation can be good enough (and of course, one also need to assume the self-interaction is constant so neglected also).

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