Projective Hilbert space - Definition and Overview

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

v ~ w when v = λw

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.

The same construction can be applied also to real Hilbert spaces. In the case H is finite-dimensional 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. See Bloch sphere for the unitary case.