Projective Hilbert space

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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 = 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. 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_)=\mathbbP^]

so that, for example, the two-dimensional projective Hilbert space (the space describing one qubit) is the complex projective line [\mathbbP^]. This is known as the Bloch sphere, which treats the subject in greater detail.

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

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