Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space "P"("H") of a complex Hilbert space "H" is the set of equivalence classes of vectors "v" in "H", with "v" ≠ 0, for the relation given by

:"v" ~ "w" when "v" = λ"w"

with λ a scalar, that is, a complex number (which must therefore be non-zero). Here the equivalence classes for ~ are also called projective rays.

This is the usual construction of projective space, applied to a Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions ψ and λψ represent the same "physical state", for any λ ≠ 0. There is not a unique normalized wavefunction in a given ray, since we can multiply by λ with absolute value 1. This freedom means that projective representations enter quantum theory.

The same construction can be applied also to real Hilbert spaces.

In the case "H" is finite-dimensional, that is, $H=H\_n$, the set of projective rays may be treated just as any other projective space; it is a homogeneous space for a unitary group or orthogonal group, in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes

:$P(H\_\{n\})=mathbb\{C\}P^\{n-1\}$

so that, for example, the two-dimensional projective Hilbert space (the space describing one qubit) is the complex projective line $mathbb\{C\}P^\{1\}$. This is known as the Bloch sphere.

Complex projective Hilbert space may be given a natural metric, the Fubini-Study metric. The product of two projective Hilbert spaces is given by the Segre mapping.