nLab
Gelfand-Naimark-Segal construction

Context

Functional analysis

Overview diagrams

Basic concepts

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Theorems

Topics in Functional Analysis

Algebraic Quantum Field Theory

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Concepts

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Theorems

States and observables

Operator algebra

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Local QFT

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Perturbative QFT

Contents

Idea

The Gelfand-Naimark-Segal construction (β€œGNS construction”) represents a state on a star-algebra over the complex numbers, which a priori is defined purely algebraically as a non-degenerate positive linear function

ρ:π’œβŸΆβ„‚, \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,,

by a vector Οˆβˆˆβ„‹\psi \in \mathcal{H} in a complex Hilbert space β„‹\mathcal{H} as the β€œexpectation value”

ρ(A) =⟨ψ|A|ψ⟩ β‰”βŸ¨Οˆ,Ο€(A)ψ⟩ \begin{aligned} \rho(A) & = \langle \psi \vert \, A \, \vert \psi \rangle \\ & \coloneqq \langle \psi, \pi(A) \psi \rangle \end{aligned}

with respect to some star-representation

Ο€:π’œβŸΆEnd(β„‹) \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

of π’œ\mathcal{A} on (a dense subspace of) β„‹\mathcal{H}; where βŸ¨βˆ’,βˆ’βŸ©\langle -,-\rangle denotes the inner product on the Hilbert space.

Originally this was considered for C*-algebras and C*-representations (Gelfand-Naimark 43, Segal 47), see for instance (SchmΓΌdgen 90), but the construction applies to general unital star algebras π’œ\mathcal{A} (Khavkine-Moretti 15) as well as to other coefficient rings, such as to formal power series algebras over β„‚[[ℏ]]\mathbb{C}[ [\hbar] ] (Bordemann-Waldmann 96).

The GNS-construction plays a central role in algebraic quantum field theory (Haag 96, Moretti 13, Khavkine-Moretti 15), where π’œ\mathcal{A} plays the role of an algebra of observables and ρ:π’œβ†’β„‚\rho \colon \mathcal{A} \to \mathbb{C} the role of an actual state of a physical system (whence the terminology) jointly constituting the β€œHeisenberg picture”-perspective of quantum physics; so that the GNS-construction serves to re-construct a corresponding Hilbert space of states as in the SchrΓΆdinger picture of quantum physics. In this context the version for C*-algebras corresponds to non-perturbative quantum field theory, while the generalization to formal power series algebras corresponds to perturbative quantum field theory.

Details

under construction

For C *C^\ast-algebras

Theorem

Given

  1. a C*-algebra, π’œ\mathcal{A};

  2. a state, ρ:π’œβ†’β„‚\rho \;\colon\; \mathcal{A} \to \mathbb{C}

there exists

  1. a C*-representation

    Ο€:π’œβŸΆEnd(β„‹) \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

    of π’œ\mathcal{A} on some Hilbert space β„‹\mathcal{H}

  2. a cyclic vector Οˆβˆˆβ„‹\psi \in \mathcal{H}

such that ρ\rho is the state corresponding to ψ\psi, in that

ρ(A) =⟨ψ|A|ψ⟩ β‰”βŸ¨Οˆ,Ο€(A)ψ⟩ \begin{aligned} \rho(A) & = \langle \psi \vert\, A \, \vert \psi \rangle \\ & \coloneqq \langle \psi , \pi(A) \psi \rangle \end{aligned}

for all Aβˆˆπ’œA \in \mathcal{A}.

For C *C^\ast-categories

The GNS construction for C *C^\ast-algebras is a special case of a more general construction of Ghez, Lima and Roberts applied to C*-categories (horizontal categorification of C *C^\ast-algebras).

Theorem

Let π’ž\mathcal{C} be a C *C^\ast-category. Fix an object A∈Obπ’žA \in \operatorname{Ob}\mathcal{C} and let Οƒ\sigma be a state on the C *C^\ast-algebra π’ž(A,A)\mathcal{C}(A,A). Then there exists a **-representation

ρ Οƒ:π’žβ†’Hilb \rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}

together with a cyclic vector ξ∈ρ Οƒ(A)\xi \in \rho_\sigma(A) such that for all xβˆˆπ’ž(A,A)x \in \mathcal{C}(A,A),

Οƒ(x)=⟨ξ,ρ Οƒ(x)ξ⟩. \sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.

A C*-algebra π’œ\mathcal{A} is a C *C^\ast-category with a single object β€’\bullet, where we make the identification A=π’œ(β€’,β€’)A = \mathcal{A}(\bullet,\bullet). In this case the theorem reduces to the classical GNS construction.

References

The original construction for C*-algebras and C*-representations is due to

  • Israel Gelfand, Mark Naimark, On the imbedding of normed rings into the ring of operators on a Hilbert space. Matematicheskii Sbornik. 12 (2): 197–217 (1943)

  • Irving Segal, Irreducible representations of operator algebras (pdf). Bull. Am. Math. Soc. 53: 73–88, (1947)

see for instance

  • K. SchmΓΌdgen, Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. BirkhΓ€user, Basel (1990)

The application to algebraic quantum field theory is discussed in

  • Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics. Springer (1996).

  • Valter Moretti, Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation, Springer Verlag, Berlin (2013)

See also

For general unital star-algebras:

  • Igor Khavkine, Valter Moretti, Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction, Chapter 5 in Romeo Brunetti et al. (eds.) Advances in Algebraic Quantum Field Theory, , Springer, 2015

For formal power series algebras over β„‚[[ℏ]]\mathbb{C}[ [ \hbar ] ]:

Discussion in terms of universal properties in (higher) category theory is in

Last revised on December 11, 2017 at 12:24:20. See the history of this page for a list of all contributions to it.