nLab
Gelfand-Naimark-Segal construction

Context

Functional analysis

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system

free field quantization

gauge theories

interacting field quantization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The Gelfand-Naimark-Segal construction (“GNS 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:𝒜, \rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,,

by a vector ψ\psi \in \mathcal{H} in a complex Hilbert space \mathcal{H} as the “expectation value

ρ(A) =ψ|A|ψ ψ,π(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

π:𝒜End() \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

of 𝒜\mathcal{A} on (a dense subspace of) \mathcal{H}; where ,\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 (Schmüdgen 90), but the construction applies to general unital star algebras 𝒜\mathcal{A} (Khavkine-Moretti 15) as well as to other coefficient rings, such as to formal power series algebras over [[]]\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 𝒜\mathcal{A} plays the role of an algebra of observables and ρ:𝒜\rho \colon \mathcal{A} \to \mathbb{C} the role of an actual state of a physical system (whence the terminology) jointly constituting the “Heisenberg picture”-perspective of quantum physics; so that the GNS-construction serves to re-construct a corresponding Hilbert space of states as in the Schrödinger 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, 𝒜\mathcal{A};

  2. a state, ρ:𝒜\rho \;\colon\; \mathcal{A} \to \mathbb{C}

there exists

  1. a C*-representation

    π:𝒜End() \pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})

    of 𝒜\mathcal{A} on some Hilbert space \mathcal{H}

  2. a cyclic vector ψ\psi \in \mathcal{H}

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

ρ(A) =ψ|A|ψ ψ,π(A)ψ \begin{aligned} \rho(A) & = \langle \psi \vert\, A \, \vert \psi \rangle \\ & \coloneqq \langle \psi , \pi(A) \psi \rangle \end{aligned}

for all A𝒜A \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 𝒞\mathcal{C} be a C *C^\ast-category. Fix an object AOb𝒞A \in \operatorname{Ob}\mathcal{C} and let σ\sigma be a state on the C *C^\ast-algebra 𝒞(A,A)\mathcal{C}(A,A). Then there exists a **-representation

ρ σ:𝒞Hilb \rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}

together with a cyclic vector ξρ σ(A)\xi \in \rho_\sigma(A) such that for all x𝒞(A,A)x \in \mathcal{C}(A,A),

σ(x)=ξ,ρ σ(x)ξ. \sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.

A C*-algebra 𝒜\mathcal{A} is a C *C^\ast-category with a single object \bullet, where we make the identification A=𝒜(,)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. Schmüdgen, Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. Birkhäuser, 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 [[]]\mathbb{C}[ [ \hbar ] ]:

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

Revised on December 11, 2017 12:24:20 by Urs Schreiber (178.6.238.60)