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 vectors in, with, for the relation given by
The equivalence classes for the relation are also called rays or projective rays.
This is the usual construction of projectivization, applied to a complex Hilbert space.

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. 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 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.
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 is the complex projective line. This is known as the Bloch sphere. See Hopf fibration for details of the projectivization construction in this case.
Complex projective Hilbert space may be given a natural metric, the Fubini–Study metric, derived from the Hilbert space's norm.

Product

The Cartesian products of projective Hilbert spaces is not a projective space. Segre mapping is an embedding of the Cartesian product of two projective spaces into their tensor product. In quantum theory, it describes how to make states of the composite system from states of its constituents. It is only an embedding not a surjection; most of the tensor product space does not lie in its range and represents entangled states.