A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → 0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.