nLab Gelfand-Naimark-Segal construction

Contents

Context

Functional analysis

Algebraic Quantum Field Theory

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

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 18, 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 ρ\psi_\rho \in \mathcal{H}

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

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

for all A𝒜A \in \mathcal{A}.

Namely, consider on the underlying complex vector space of 𝒜\mathcal{A} the sesquilinear form (inner product)

A,B ρρ(A *B). \langle A,B \rangle_\rho \;\coloneqq\; \rho \big( A^\ast B \big) \,.

Since ρ\rho is a “positive functional”, hence taking non-negative values, this is, in general, positive semi-definite. It becomes positive definite on the the quotient vector space

(1)𝒜/N \mathcal{H} \;\coloneqq\; \mathcal{A}/N

by the subspace of 0-norm elements

(2)N{A𝒜|ρ(A *A)=0}. N \;\coloneqq\; \big\{ A \in \mathcal{A} \,\vert\, \rho(A^\ast A) = 0 \big\} \,.

In fact, NN is a left ideal in 𝒜\mathcal{A} so that the left multiplication action of 𝒜\mathcal{A} on itself descends to an action on the quotient Hilbert space (1)

𝒜 π (A,[ψ]) [Aψ]. \array{ \mathcal{A} \otimes \mathcal{H} & \overset{ \;\;\;\;\; \pi \;\;\;\;\; }{ \longrightarrow } & \mathcal{H} \\ (A, [\psi]) &\mapsto& [A \cdot \psi] \,. }

Therefore, the cyclic vector in question can be taken to be that represented by the unit element 1𝒜1 \in \mathcal{A}:

ψ ρ[1]. \psi_\rho \;\coloneqq\; [1] \,.

Hence on this Hilbert space \mathcal{H}, the original operator-algebraic state ρ\rho is now represented by the tautological density matrix

|ψ ρψ ρ=|[1][1]. \left\vert \psi_\rho \right\rangle \left\langle \psi_\rho \right\rangle \;=\; \left\vert [1] \right\rangle \left\langle [1] \right\rangle \,.

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.

Functorial Aspects

See Functorial Aspects of the GNS Representation.

quantum probability theoryobservables and states

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)

    reprinted in:

    Robert Doran (ed.), C *C^\ast-Algebras: 1943–1993, Contemporary Mathematics 167, AMS 1994 (doi:10.1090/conm/167)

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

Textbook accounts:

in the context of algebraic quantum field theory:

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

  • Valter Moretti, Spectral Theory and Quantum Mechanics :Mathematical Structure of Quantum Theories, Symmetries and introduction to the Algebraic Formulation, 2nd ed. Springer Verlag, Berlin (2018)

Review with an eye towards quantum probability and entropy:

  • A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Section II of: Algebraic approach to entanglement and entropy, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)

See also

For general unital star-algebras:

in relation with the classical moment problem and the notion of POVM

  • Nicolò Drago, Valter Moretti, The notion of observable and the moment problem for *\ast-algebras and their GNS representations, Lett. Math. Phys. 2020 (arXiv.org:1903.07496)

For formal power series algebras over [[]]\mathbb{C}[ [ \hbar ] ]:

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

Last revised on July 9, 2023 at 08:11:20. See the history of this page for a list of all contributions to it.