Editing Projective Hilbert space

In [[mathematics]] and the foundations of [[quantum mechanics]], the '''projective Hilbert space''' or '''ray space''' <math>\mathbf{P}(H)</math> of a [[complex number|complex]] [[Hilbert space]] <math>H</math> is the set of [[equivalence class]]es <math>[v]</math> of non-zero vectors <math>v \in H</math>, for the [[equivalence relation]] <math>\sim</math> on <math>H</math> given by

:<math>w \sim v</math> if and only if <math>v = \lambda w</math> for some non-zero complex number <math>\lambda</math>.

This is the usual construction of [[projectivization]], applied to a complex Hilbert space.{{sfn | Miranda | 1995 | p=94}} In quantum mechanics, the equivalence classes <math>[v]</math> 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==
{{see also|Wigner's theorem #Rays and ray space}}
The physical significance of the projective Hilbert space is that in [[Quantum mechanics|quantum theory]], the [[wave function]]s <math>\psi</math> and <math>\lambda \psi</math> represent the same ''physical state'', for any <math>\lambda \ne 0</math>. The Born rule demands that if the system is physical and measurable, its wave function has unit [[normed vector space|norm]], <math>\langle\psi|\psi\rangle = 1</math>, in which case it is called a [[normalized wave function]]. The unit norm constraint does not completely determine <math>\psi</math> within the ray, since <math>\psi</math> could be multiplied by any <math>\lambda</math> with [[absolute value]] 1 (the [[circle group]] <math>U(1)</math> action) and retain its normalization. Such a <math>\lambda</math> can be written as <math>\lambda = e^{i\phi}</math> with <math>\phi</math> called the global [[Phase factor|phase]].

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

If <math>H</math> is an [[irreducible representation]] of the algebra of observables then the rays induce [[Quantum state#Formalism in quantum physics|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.

In the case <math>H</math> is finite-dimensional, i.e., <math>H=H_n</math>, 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]] <math>\mathrm{U}(n)</math>. That is,

:<math>\mathbf{P}(H_{n})=\mathbb{C}\mathbf{P}^{n-1}</math>,

which carries a [[Kähler metric]], called the [[Fubini–Study metric]], derived from the Hilbert space's norm.{{sfn | Kong | Liu |2021| p=9}}{{sfn | Cirelli | Lanzavecchia | Mania | 1983}}

As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one [[qubit]]) is the [[complex projective line]] <math>\mathbb{C}\mathbf{P}^{1}</math>. This is known as the [[Bloch sphere]] or, equivalently, the [[Riemann sphere]]. See [[Hopf fibration]] for details of the projectivization construction in this case.

==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>\mathbf{P}(H) \times \mathbf{P}(H') \to \mathbf{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 of a function|image]] and represents ''[[quantum entanglement|entangled states]]''.

==See also==
* [[Complex projective space]]
* [[Projective representation]]
* [[Projective space]], for the concept in general

==Notes==
{{reflist}}

==References==
* {{cite book | last1=Ashtekar | first1=Abhay | last2=Schilling | first2=Troy A. | title=On Einstein's Path | chapter=Geometrical Formulation of Quantum Mechanics | publisher=Springer New York | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7137-6 | doi=10.1007/978-1-4612-1422-9_3 | arxiv=gr-qc/9706069 }}
* {{cite journal | last1=Cirelli | first1=R | last2=Lanzavecchia | first2=P | last3=Mania | first3=A | title=Normal pure states of the von Neumann algebra of bounded operators as Kahler manifold | journal=Journal of Physics A: Mathematical and General | publisher=IOP Publishing | volume=16 | issue=16 | year=1983 | issn=0305-4470 | doi=10.1088/0305-4470/16/16/020 | pages=3829–3835| bibcode=1983JPhA...16.3829C }}
* {{cite journal | last1=Kong | first1=Otto C. W. | last2=Liu | first2=Wei-Yin | title=Noncommutative Coordinate Picture of the Quantum Phase Space  | journal = Chinese Journal of Physics | publisher=Elsevier BV | arxiv=1903.11962 | year=2021 | volume=71 | page=418 | doi=10.1016/j.cjph.2021.03.014 | bibcode=2021ChJPh..71..418K | s2cid=85543324 }}
* {{cite book | last=Miranda | first=Rick | title=Algebraic Curves and Riemann Surfaces | publisher=American Mathematical Soc. | publication-place=Providence (R.I.) | date=1995 | isbn=0-8218-0268-2}}

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[[Category:Hilbert spaces]]