The answer from physical theory is a resounding yes, but look at some first experiments along these lines:
Bosons obey boson statistics – which means they are not huge players in Quantum interactions. You can jam as many as you like into one state. In other words you can pile trillions of photons up in one place, they will all ignore each other.
Fermions are nice quantum particles. They don’t pile up on the nucleus and instead support the existence of matter with the pauli exclusion principle. All quantum level determining experiments are done with charged fermions. But are there uncharged fermions? (Yes – Neutrinos)
Experiments that might show QM effects on non charged particles
Photon experiments. Experiments with light are pretty boring. Photons are bosons, or put another way, they simply do not interact with one another. The existence of the photon is always determined by an interaction with a charged particle. So no way to do a purely photonic QM experiment, I would think.
Neutrons: Uncharged and fermonic so it seems – but in reality Neutrons are composite particles made of charged quarks. There are no uncharged quarks. So any experiment on QM that uses any charged fermion can’t be included.
Neutrinos: Well here we have an uncharged fermion, so that would seem to rule that there are quantum effects on non – charged particles. But of course neutrino experiments are very primitive and only concern neutrino – charged particle interactions. Its wildly impossible right now to do an experiment where neutrinos are say dropped into some potential well and we detect the pauli exclusion principle on them.
Gravitons and other bosons fall into the uncharged category for the most part, W bosons sticking in this regard. But I would bet that the QM nature of W+ interaction has not been experimentally studied.
I don’t know why the physics community has not spent more time on this. QM effects and charge seem to be locked together. A hypothetical all Boson universe would not need to use QM.
Do Bosons Feel Quantum Mechanics?
More on this hypothetical bosonic universe. If we construct one where all fermions are missing, but the laws of physics are the same, would we need QM at all?