Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space of a complex Hilbert space is the set of equivalence classes of non-zero vectors in , for the relation on given by if and only if for some non-zero complex number . The equivalence classes of for the relation 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 and represent the same physical state, for any . It is conventional to choose a from the ray so that it has unit norm, , in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine within the ray, since could be multiplied by any with absolute value 1 (the U(1) action) 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 (cf. quantum state (algebraic definition), given a C*-algebra of observables and a representation on ). 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 (still in case of an irreducible representation) correspond to mixed states. The same construction can be applied also to real Hilbert spaces. In the case is finite-dimensional, that is, , 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 so that, for example, the projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line . This is known as the Bloch sphere.