Projective Hilbert space: Difference between revisions
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In [[mathematics]] and the foundations of [[quantum mechanics]], the '''projective Hilbert space''' <math>\mathbf{P}(H)</math> of a complex [[Hilbert space]] <math>H</math> is the set of [[equivalence class]]es of non-zero vectors <math>v</math> in <math>H</math>, for the relation <math>\sim</math> on <math>H</math> given by |
In [[mathematics]] and the foundations of [[quantum mechanics]], the '''projective Hilbert space''' <math>\mathbf{P}(H)</math> of a [[complex number|complex]] [[Hilbert space]] <math>H</math> is the set of [[equivalence class]]es of non-zero vectors <math>v</math> in <math>H</math>, for the relation <math>\sim</math> on <math>H</math> given by |
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:<math>w \sim v</math> if and only if <math>v = \lambda w</math> for some non-zero complex number <math>\lambda</math>. |
:<math>w \sim v</math> if and only if <math>v = \lambda w</math> for some non-zero complex number <math>\lambda</math>. |
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The equivalence classes of <math>v</math> for the relation <math>\sim</math> are also called '''rays''' or '''projective rays'''. |
The equivalence classes of <math>v</math> for the relation <math>\sim</math> are also called '''rays''' or '''projective rays'''. |
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This is the usual construction of [[projectivization]], applied to a |
This is the usual construction of [[projectivization]], applied to a complex Hilbert space. |
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==Overview== |
==Overview== |
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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. |
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. |
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⚫ | 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, |
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The same construction can be applied also to real Hilbert spaces. |
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⚫ | In the case <math>H</math> is finite-dimensional, |
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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}} |
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⚫ | |||
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. |
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Complex projective Hilbert space may be given a natural metric, the [[Fubini–Study metric]], derived from the Hilbert space's norm. |
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==Product== |
==Product== |
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* [[Projective space]], for the concept in general |
* [[Projective space]], for the concept in general |
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* [[Wigner's theorem]], for a more detailed description of rays and ray space in quantum mechanics |
* [[Wigner's theorem]], for a more detailed description of rays and ray space in quantum mechanics |
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==Notes== |
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{{reflist}} |
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==References== |
==References== |
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* {{cite journal | last=Ashtekar | first=Abhay | last2=Schilling | first2=Troy A. | title=Geometrical Formulation of Quantum Mechanics | publisher=arXiv | doi=10.48550/ARXIV.GR-QC/9706069 | year=1997 |url=https://arxiv.org/abs/gr-qc/9706069}} |
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* {{cite journal | last=Cirelli | first=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}} |
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{{cite arXiv|eprint=gr-qc/9706069|author1link = Abhay Ashtekar|last1=Ashtekar|first1=Abhay|title=Geometrical Formulation of Quantum Mechanics|last2=Schilling|first2=Troy A.|year=1997}} |
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* {{cite journal | last=Kong | first=Otto C. W. | last2=Liu | first2=Wei-Yin | title=Noncommutative Coordinate Picture of the Quantum Phase Space | publisher=arXiv | doi=10.48550/ARXIV.1903.11962 | year=2021 |url=https://arxiv.org/abs/1903.11962}} |
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{{DEFAULTSORT:Projective Hilbert Space}} |
{{DEFAULTSORT:Projective Hilbert Space}} |
Revision as of 07:03, 25 November 2023
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.
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 (the circle group 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.
In the case is finite-dimensional, i.e., , 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 . That is,
- ,
which carries a Kähler metric, called the Fubini–Study metric, derived from the Hilbert space's norm.[1][2]
As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line . 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 . 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
- Complex projective space
- Projective representation
- Projective space, for the concept in general
- Wigner's theorem, for a more detailed description of rays and ray space in quantum mechanics
Notes
References
- Ashtekar, Abhay; Schilling, Troy A. (1997). "Geometrical Formulation of Quantum Mechanics". arXiv. doi:10.48550/ARXIV.GR-QC/9706069.
{{cite journal}}
: Cite journal requires|journal=
(help) - Cirelli, R; Lanzavecchia, P; Mania, A (1983). "Normal pure states of the von Neumann algebra of bounded operators as Kahler manifold". Journal of Physics A: Mathematical and General. 16 (16). IOP Publishing: 3829–3835. doi:10.1088/0305-4470/16/16/020. ISSN 0305-4470.
- Kong, Otto C. W.; Liu, Wei-Yin (2021). "Noncommutative Coordinate Picture of the Quantum Phase Space". arXiv. doi:10.48550/ARXIV.1903.11962.
{{cite journal}}
: Cite journal requires|journal=
(help)