Normalization, ray equivalence and the phase orbits of the unit sphere

[(a) normalization is well defined and lies in the ray] For $\psi \mathrm{\ne }0$ the vector $\nu (\psi )=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\psi$ is a nonzero complex multiple of $\psi$, hence lies in $[\psi ]$; its norm is $\mathrm{\parallel }\nu (\psi )\mathrm{\parallel }=\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }=\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\mathrm{\parallel }\psi \mathrm{\parallel }=1$ by [F2] and [F3], so $\nu (\psi )\in S(H)$; and $\nu (\nu (\psi ))=\nu (\psi )$ because $\nu (\psi )$ is a unit vector. For $z\mathrm{\ne }0$ the norm is $\mathrm{\parallel }z\psi \mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }$ by [F2], hence $\nu (z\psi )=\mathrm{\mid }z{\mathrm{\mid }}^{-1}\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}z\psi =(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\psi =(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })\nu (\psi )$, using $z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }=\mathrm{\mid }z{\mathrm{\mid }}^{-1}z$ for the unit complex number $z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$. Finally $\nu$ is surjective because every $u\in S(H)$ satisfies $\nu (u)=\mathrm{\parallel }u{\mathrm{\parallel }}^{-1}u=u$.

[(c) the unit representatives of a ray form one phase orbit] Let $u\in S(H)$. If $v\in [u]\cap S(H)$ then $v=zu$ for some $z\mathrm{\ne }0$, and $1=\mathrm{\parallel }v\mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\parallel }u\mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }$ by [F2] and [F3]; hence $z\in U(1)$ and $v=zu\in U(1)u$. Conversely if $\zeta \in U(1)$ then $\zeta u$ lies in $[u]$ by definition and in $S(H)$ by [F4]. Thus $[u]\cap S(H)=U(1)u$.

[(b) ray equivalence is phase equivalence of normalizations] If $[\psi ]=[\phi ]$, then by definition $\phi =z\psi$ with $z\mathrm{\ne }0$, so by step 1.1 $\nu (\phi )=\nu (z\psi )=(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })\nu (\psi )$ with $z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }\in U(1)$; uniqueness of the phase factor holds because $\nu (\psi )\mathrm{\ne }0$. Conversely, if $\nu (\phi )=\zeta \nu (\psi )$ for some $\zeta \in U(1)$, then $\phi =\mathrm{\parallel }\phi \mathrm{\parallel }\nu (\phi )=\mathrm{\parallel }\phi \mathrm{\parallel }\zeta \mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\psi$ is a nonzero complex multiple of $\psi$, so $[\phi ]=[\psi ]$.

[(c) the induced map on phase orbits is a bijection] By step 2.1 the fibre of $\psi \mapsto [\psi ]$ over a ray $R$ is $\{u\in S(H):[u]=R\}=R\cap S(H)$, which is a full phase orbit; and every phase orbit $U(1)u$ is such a fibre because $U(1)u\subseteq [u]$. The map therefore descends to a well-defined injective map from the set of phase orbits onto $R(H)$, and it is surjective because every ray contains the unit vector $\nu (\psi )$ for any of its representatives $\psi$ by step 1.1.

[(d) phase uniqueness for unit representatives] Let $\psi ,\phi \in S(H)$ with $[\psi ]=[\phi ]$. Then $\phi =z\psi$ for some $z\mathrm{\ne }0$, so $1=\mathrm{\parallel }\phi \mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }=\mathrm{\mid }z\mathrm{\mid }$ by [F2] and [F3]; writing $\zeta :=z\in U(1)$ gives $\phi =\zeta \psi$, and if ${\zeta }^{\mathrm{\prime }}\in U(1)$ also satisfies $\phi ={\zeta }^{\mathrm{\prime }}\psi$, then $(\zeta -{\zeta }^{\mathrm{\prime }})\psi =0$ with $\psi \mathrm{\ne }0$, which forces $\zeta ={\zeta }^{\mathrm{\prime }}$ by [F2] and [F3]. Moreover $⟨\phi ,\psi ⟩=⟨\zeta \psi ,\psi ⟩=\zeta ⟨\psi ,\psi ⟩=\zeta \mathrm{\parallel }\psi {\mathrm{\parallel }}^2=\zeta$ by linearity in the first argument, so $\mathrm{\mid }⟨\phi ,\psi ⟩\mathrm{\mid }=1$. The converse implication is the definition of ray equivalence.