Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space [math]\displaystyle{ P(H) }[/math] of a complex Hilbert space [math]\displaystyle{ H }[/math] is the set of equivalence classes of non-zero vectors [math]\displaystyle{ v }[/math] in [math]\displaystyle{ H }[/math], for the relation [math]\displaystyle{ \sim }[/math] on [math]\displaystyle{ H }[/math] given by

[math]\displaystyle{ w \sim v }[/math] if and only if [math]\displaystyle{ v = \lambda w }[/math] for some non-zero complex number [math]\displaystyle{ \lambda }[/math].

The equivalence classes of [math]\displaystyle{ v }[/math] for the relation [math]\displaystyle{ \sim }[/math] 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 [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \lambda \psi }[/math] represent the same physical state, for any [math]\displaystyle{ \lambda \ne 0 }[/math]. It is conventional to choose a [math]\displaystyle{ \psi }[/math] from the ray so that it has unit norm, [math]\displaystyle{ \langle\psi|\psi\rangle = 1 }[/math], in which case it is called a normalized wavefunction. The unit norm constraint does not completely determine [math]\displaystyle{ \psi }[/math] within the ray, since [math]\displaystyle{ \psi }[/math] could be multiplied by any [math]\displaystyle{ \lambda }[/math] with absolute value 1 (the U(1) action) and retain its normalization. Such a [math]\displaystyle{ \lambda }[/math] can be written as [math]\displaystyle{ \lambda = e^{i\phi} }[/math] with [math]\displaystyle{ \phi }[/math] called the global phase.

Rays that differ by such a [math]\displaystyle{ \lambda }[/math] correspond to the same state (cf. quantum state (algebraic definition), given a C*-algebra of observables and a representation on [math]\displaystyle{ H }[/math]). No measurement can recover the phase of a ray; it is not observable. One says that [math]\displaystyle{ U(1) }[/math] is a gauge group of the first kind.

If [math]\displaystyle{ H }[/math] 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 [math]\displaystyle{ H }[/math] is finite-dimensional, that is, [math]\displaystyle{ H=H_n }[/math], the set of projective rays may be treated just as any other projective space; it is a homogeneous space for a unitary group [math]\displaystyle{ \mathrm{U}(n) }[/math] or orthogonal group [math]\displaystyle{ \mathrm{O}(n) }[/math], in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes

[math]\displaystyle{ P(H_{n})=\mathbb{C}P^{n-1} }[/math]

so that, for example, the projectivization of two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line [math]\displaystyle{ \mathbb{C}P^{1} }[/math]. 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 product of projective Hilbert spaces is not a projective space. The Segre mapping is an embedding of the Cartesian product of two projective spaces into the projective space associated to the tensor product of the two Hilbert spaces, given by [math]\displaystyle{ P(H) \times P(H') \to P(H \otimes H'), ([x], [y]) \mapsto [x \otimes y] }[/math]. 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.

See also

• Projective space, for the concept in general
• Complex projective space
• Projective representation

References

Ashtekar, Abhay; Schilling, Troy A. (1997). "Geometrical Formulation of Quantum Mechanics". arXiv:gr-qc/9706069.