Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space ${\displaystyle \mathbf {P} (H)}$ of a complex Hilbert space ${\displaystyle H}$ is the set of equivalence classes ${\displaystyle [v]}$ of non-zero vectors ${\displaystyle v\in H}$, for the equivalence relation ${\displaystyle \sim }$ on ${\displaystyle H}$ given by

${\displaystyle w\sim v}$ if and only if ${\displaystyle v=\lambda w}$ for some non-zero complex number ${\displaystyle \lambda }$.

This is the usual construction of projectivization, applied to a complex Hilbert space.^[1] In quantum mechanics, the equivalence classes ${\displaystyle [v]}$ 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.