Editing Projective Hilbert space (section)

==Overview==
{{see also|Wigner's theorem #Rays and ray space}}
The physical significance of the projective Hilbert space is that in [[Quantum mechanics|quantum theory]], the [[wave function]]s <math>\psi</math> and <math>\lambda \psi</math> represent the same ''physical state'', for any <math>\lambda \ne 0</math>. The Born rule demands that if the system is physical and measurable, its wave function has unit [[normed vector space|norm]], <math>\langle\psi|\psi\rangle = 1</math>, in which case it is called a [[normalized wave function]]. The unit norm constraint does not completely determine <math>\psi</math> within the ray, since <math>\psi</math> could be multiplied by any <math>\lambda</math> with [[absolute value]] 1 (the [[circle group]] <math>U(1)</math> action) and retain its normalization. Such a <math>\lambda</math> can be written as <math>\lambda = e^{i\phi}</math> with <math>\phi</math> called the global [[Phase factor|phase]].

Rays that differ by such a <math>\lambda</math> correspond to the same state (cf. [[Quantum state#Mathematical generalizations|quantum state (algebraic definition)]], given a [[C*-algebra]] of observables and a representation on <math>H</math>). No measurement can recover the phase of a ray; it is not observable. One says that <math>U(1)</math> is a [[gauge group]] of the first kind.

If <math>H</math> is an [[irreducible representation]] of the algebra of observables then the rays induce [[Quantum state#Formalism in quantum physics|pure states]]. Convex linear combinations of rays naturally give rise to density matrix which (still in case of an irreducible representation) correspond to mixed states.

In the case <math>H</math> is finite-dimensional, i.e., <math>H=H_n</math>, the Hilbert space reduces to a finite-dimensional [[inner product space]] and the set of projective rays may be treated as a [[complex projective space]]; it is a [[homogeneous space]] for a [[unitary group]] <math>\mathrm{U}(n)</math>. That is,

:<math>\mathbf{P}(H_{n})=\mathbb{C}\mathbf{P}^{n-1}</math>,

which carries a [[Kähler metric]], called the [[Fubini–Study metric]], derived from the Hilbert space's norm.{{sfn | Kong | Liu |2021| p=9}}{{sfn | Cirelli | Lanzavecchia | Mania | 1983}}

As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one [[qubit]]) is the [[complex projective line]] <math>\mathbb{C}\mathbf{P}^{1}</math>. This is known as the [[Bloch sphere]] or, equivalently, the [[Riemann sphere]]. See [[Hopf fibration]] for details of the projectivization construction in this case.