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The interesting thing is that, while the microscopic physics is entirely determined by microscopic maxwell + the matter distribution, the macroscopic physics doesn't seem to strongly depend on anything … We don't know a priori all of the small-scale physics because the details of how gravity has to interact with all of the standard model fields isn't determined in advance. …

Generally speaking, from a 'physics' point of view, the reason particles are labeled according to representations is not too different than how, in normal quantum mechanics, states are labeled by eigenvalues … In other words, the $o_{ij}$ define states, from the physics point of view. Really, this defines a basis where our operators are diagonal. We can--and do! …

You will probably have to delve into the physics literature (particularly, review articles). …

Unitarity ensures conservation of probability. For example, the Hamiltonian is Hermitian, so the time evolution operator $U(t,t_0) = e^{iH(t-t_0)}$ is unitarity. The unitarity of $U$ corresponds to …