Projective Hilbert space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes [ v ] {\displaystyle [v]} of non-zero vectors v ∈ H {\displaystyle v\in H} , for the equivalence relation ∼ {\displaystyle \sim } on H {\displaystyle H} given by w ∼ v {\displaystyle w\sim v} if and only if v = λ w {\displaystyle v=\lambda w} for some non-zero complex number λ {\displaystyle \lambda } . This is the usual construction of projectivization, applied to a complex Hilbert space.