We study generalized transition probabilities as defined by Colombeau-Gsponer and introduce H. Bohr’s almost periodic functions into this context. We revisit examples given in dimension 2 and give exact values of transition amplitudes and transition probabilities in this case. We also introduce generalized transition probabilities for Hilbert modules, i.e., for moderate nets \(T = (T_{\varepsilon })\) with \(T_{\varepsilon } : \mathbb {H}\longrightarrow \mathbb {H}\) , a symmetric linear operator defined on the Hilbert space \(\mathbb {H}\) . In particular, the case of self-adjoint Hilbert-Schmidt operators is considered. In the finite dimensional case, we show that transition probabilities exists under certain conditions and give an explicit formula for their value. This is another step alluding to the possibility of the existence of transition probabilities in the Fock space, in the sense referred to by Colombeau-Gsponer, if the spectrum of the operators involved are pure infinities or infinitesimals. Papers coauthored by J. Aragona and J. F. Colombeau already indicated to this possibility.
