A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representation r : LjVªRn , where L is the collection of closed subspaces of an n-dimensional real, complex, or quaternionic Hilbert space H, or equivalently, the projection lattice of this Hilbert space, where V is the set of all states v : Lªw0, 1x. The value that v gV takes in agL is given by the scalar product of the representative points r a. and r v... The representation r akb.of the join of two orthogonal elements a, bgL is equal to r a.qr b.. The convex closure of the representation of S, the set of atoms of L, is equal to the representation of V.