Editing Projective Hilbert space (section)

In [[mathematics]] and the foundations of [[quantum mechanics]], the '''projective Hilbert space''' or '''ray space''' <math>\mathbf{P}(H)</math> of a [[complex number|complex]] [[Hilbert space]] <math>H</math> is the set of [[equivalence class]]es <math>[v]</math> of non-zero vectors <math>v \in H</math>, for the [[equivalence relation]] <math>\sim</math> on <math>H</math> given by

:<math>w \sim v</math> if and only if <math>v = \lambda w</math> for some non-zero complex number <math>\lambda</math>.

This is the usual construction of [[projectivization]], applied to a complex Hilbert space.{{sfn | Miranda | 1995 | p=94}} In quantum mechanics, the equivalence classes <math>[v]</math> are also referred to as '''rays''' or '''projective rays'''. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed.