Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P ( H ) {displaystyle P(H)} of a complex Hilbert space H {displaystyle H} is the set of equivalence classes of vectors v {displaystyle v} in H {displaystyle H} , with v ≠ 0 {displaystyle v eq 0} , for the relation ∼ {displaystyle sim } given by In mathematics and the foundations of quantum mechanics, the projective Hilbert space P ( H ) {displaystyle P(H)} of a complex Hilbert space H {displaystyle H} is the set of equivalence classes of vectors v {displaystyle v} in H {displaystyle H} , with v ≠ 0 {displaystyle v eq 0} , for the relation ∼ {displaystyle sim } given by The equivalence classes for the relation ∼ {displaystyle sim } are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions ψ {displaystyle psi } and λ ψ {displaystyle lambda psi } represent the same physical state, for any λ ≠ 0 {displaystyle lambda eq 0} . It is conventional to choose a ψ {displaystyle psi } from the ray so that it has unit norm, ⟨ ψ | ψ ⟩ = 1 {displaystyle langle psi |psi angle =1} , in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine ψ {displaystyle psi } within the ray, since ψ {displaystyle psi } could be multiplied by any λ {displaystyle lambda } with absolute value 1 (the U(1) action) and retain its normalization. Such a λ {displaystyle lambda } can be written as λ = e i ϕ {displaystyle lambda =e^{iphi }} with ϕ {displaystyle phi } called the global phase. Rays that differ by such a λ {displaystyle lambda } correspond to the same state (cf. quantum state (algebraic definition), given a C*-algebra of observables and a representation on H {displaystyle H} ). No measurement can recover the phase of a ray, it is not observable. One says that U ( 1 ) {displaystyle U(1)} is a gauge group of the first kind. If H {displaystyle H} is an irreducible representation of the algebra of observables then the rays induce 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.