State vectors, unit vectors, rays, normalization and phases of a Hilbert space

The Hilbert space and its norm. Throughout this item $H\mathrm{\ne }\{0\}$ is a nonzero complex Hilbert space, with pairing $⟨\cdot ,\cdot ⟩$ linear in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length, Hilbert space), and with the induced norm $\mathrm{\parallel }\psi \mathrm{\parallel }:=\sqrt{⟨\psi ,\psi ⟩}$ (The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The induced length is a norm). A vector is an element of $H$, and a unit vector is a vector $\psi$ with $\mathrm{\parallel }\psi \mathrm{\parallel }=1$. The unit sphere of $H$ is $S(H):=\{\psi \in H:\mathrm{\parallel }\psi \mathrm{\parallel }=1\},$ and the set of nonzero vectors is $H^{\times }:=H\setminus \{0\}$. The hypothesis $H\mathrm{\ne }\{0\}$ says exactly that $H^{\times }$ is nonempty; then $S(H)$ is nonempty as well, since for every nonzero $\psi$ the vector $\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel }$ has norm one, because $\mathrm{\parallel }\psi \mathrm{\parallel }>0$ and $\mathrm{\parallel }\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel }\mathrm{\parallel }=\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }=1$.

Rays. For $\psi \in H^{\times }$ the ray of $\psi$ is $[\psi ]:=\{z\psi :z\in \mathbb{C},z\mathrm{\ne }0\}\subseteq H^{\times }\mathrm{.}$ A ray of $H$ is a subset of the form $[\psi ]$ for some nonzero $\psi$, and an element of a ray is called a representative of it. The set of rays $R(H):=\{[\psi ]:\psi \in H^{\times }\}$ is the ray space of $H$. Two nonzero vectors $\psi ,\phi$ are ray-equivalent when they lie in a common ray, that is, when $\phi =z\psi$ for some nonzero complex number $z$. Ray equivalence is an equivalence relation on $H^{\times }$: it is reflexive with $z=1$, symmetric with $z^{-1}$, and transitive by multiplying scalars. Its classes are exactly the rays, so $R(H)$ is the quotient set of $H^{\times }$ by this relation, and every nonzero vector lies in exactly one ray.

Normalization. The normalization map is $\nu :H^{\times }⟶S(H),\nu (\psi ):=\frac{\psi }{\mathrm{\parallel }\psi \mathrm{\parallel }},$ where the right-hand side is the scalar multiple $\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}\psi$; the displayed definition is meaningful because $\mathrm{\parallel }\psi \mathrm{\parallel }>0$ for $\psi \mathrm{\ne }0$. The vector $\nu (\psi )$ is the normalization of $\psi$.

Phases. The phase group is $U(1):=\{\zeta \in \mathbb{C}:\mathrm{\mid }\zeta \mathrm{\mid }=1\}$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive), and a phase factor is an element of $U(1)$. Two vectors $\psi ,\phi$ are phase-equivalent when $\phi =\zeta \psi$ for some phase factor $\zeta$. The phase action of $U(1)$ on the unit sphere is the map $U(1)\times S(H)⟶S(H),(\zeta ,\psi )⟼\zeta \psi ,$ which is well defined because $\mathrm{\parallel }\zeta \psi \mathrm{\parallel }=\mathrm{\mid }\zeta \mathrm{\mid }\mathrm{\parallel }\psi \mathrm{\parallel }=\mathrm{\parallel }\psi \mathrm{\parallel }$ for $\zeta \in U(1)$ and $\psi \in S(H)$. Ray equivalence allows multiplication by an arbitrary nonzero complex scalar, while phase equivalence allows multiplication by a scalar of modulus one only; within a given ray the two notions coincide exactly on representatives of equal norm, and the precise statements are proved in the normalization lemma of this page.

Scope. This definition fixes the mathematical objects of the state space of a Hilbert space and their elementary vocabulary: nonzero vectors, unit vectors, rays, the ray space, the normalization map and the phase action. It introduces no topology or measure on $R(H)$, no dynamics and no physical interpretation, and it uses no choice principle. The identification of rays with preparations and of the unit sphere with their normalized representatives is the model assumption stated in the framework's state postulate, not part of this mathematical definition; the orbit structure of the phase action is analysed in the normalization lemma of this page.