Projective Hilbert space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space of a complex Hilbert space is the set of equivalence classes of non-zero vectors, for the equivalence relation on given by
This is the usual construction of projectivization, applied to a complex Hilbert space. In quantum mechanics, the equivalence classes are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed.
Overview
The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any. The Born rule demands that if the system is physical and measurable, its wave function has unit norm,, in which case it is called a normalized wave function. The unit norm constraint does not completely determine within the ray, since could be multiplied by any with absolute value 1 and retain its normalization. Such a can be written as with called the global phase.Rays that differ by such a correspond to the same state. No measurement can recover the phase of a ray; it is not observable. One says that is a gauge group of the first kind.
If 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 correspond to mixed states.
In the case is finite-dimensional, i.e.,, 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. That is,
which carries a Kähler metric, called the Fubini–Study metric, derived from the Hilbert space's norm.
As such, the projectivization of, e.g., two-dimensional complex Hilbert space is the complex projective line. This is known as the Bloch sphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case.