  Unstable particles, together with their stable decay products, constitute
probability collectives which are defined as Hilbert spaces with dimension
higher than one, nondecomposable in a particle basis. Their structure is
considered in the framework of Birkhoff-von Neumann's Hilbert subspace
lattices. Bases with particle states are related to bases with a diagonal
scalar product by a Hilbert-bein involving the characteristic decay parameters
(in some analogy to the $n$-bein structures of metrical manifolds). Probability
predictions as expectation values, involving unstable particles, have to take
into account all members of the higher dimensional collective. E.g., the
unitarity structure of the $S$-matrix for an unstable particle collective can
be established by a transformation with its Hilbert-bein.
