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 Hermitian inner product on the Hilbert space.

Originally and typically by default this is considered for C*-algebras and C*-representations (Gelfand & Naimark 1943, Segal 1947), see for instance (Schmüdgen 1090), but the construction applies to general unital star algebras 𝒜\mathcal{A} (Khavkine& Moretti 2015) as well as to other coefficient rings, such as to formal power series algebras over [[]]\mathbb{C}[ [\hbar] ] (Bordemann & Waldmann 1996).

The GNS-construction plays a central role in algebraic quantum field theory (cf. Haag 1996, Moretti 2017, Khavkine & Moretti 2015), 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} in the sense of

    1. a linear map on the underlying vector space

      (1)ρ:𝒜, \rho \,\colon\, \mathcal{A} \to \mathbb{C} \,,
    2. which takes star-involution to complex conjugation

      (2)A𝒜ρ(A *)=ρ(A)¯, A \in \mathcal{A} \;\;\; \vdash \;\;\; \rho(A^\ast) = \overline{\rho(A)} \,,
    3. and satisfies “positivity in the sense that

      (3)A𝒜ρ(A *A) 0, A \in \mathcal{A} \;\;\; \vdash \;\;\; \rho(A^\ast A) \in \mathbb{R}_{\geq 0} \subset \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 pure 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}.

Proof

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

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

By the “positivity”-condition (3) on ρ\rho, the pairing ,\langle-,-\rangle is positive semi-definite, and therefore it satisfies the Cauchy-Schwarz inequality:

(5)|ρ(A *B)| 2ρ(A *A)ρ(B *B). {\big\vert \rho\big( A^\ast B \big) \big\vert}^2 \;\leq\; \rho\big(A^\ast A\big) \, \rho(B^\ast B) \,.

However, (4) will in general not be definite, in that there may be a non-trivial linear subspace of 0-norm elements:

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

and hence we have to quotient out by this null-space in order to produce the desired Hilbert space.

Observe that this null space (6) is not just a linear subspace but a left ideal for the algebra structure, as follows by the Cauchy-Schwarz inequality (5):

(7)N𝒩 A𝒜}{|ρ((AN) *AN)| 2 =|ρ((N *A *A)N)| 2 ρ((N *A *A)(N *A *A) *)ρ(N *N).=0 \left. \begin{array}{l} N \in \mathcal{N} \\ A \in \mathcal{A} \end{array} \right\} \;\;\; \vdash \;\;\; \left\{ \begin{array}{l} \big\vert \rho\big( (A N)^\ast A N \big) \big\vert^2 \\ \;=\; \big\vert \rho\big( (N^\ast A^\ast A) N \big) \big\vert^2 \\ \;\leq\; \rho\big( (N^\ast A^\ast A) (N^\ast A^\ast A)^\ast \big) \, \underset{ = 0 }{ \underbrace{ \rho\big( N^\ast N \big) \,. } } \end{array} \right.

Similarly, the Cauchy-Schwarz inequality (5) implies that any inner product with a null vector vanishes:

(8)N𝒩 A𝒜}ρ(A *N)=0. \left. \begin{array}{l} N \in \mathcal{N} \\ A \in \mathcal{A} \end{array} \right\} \;\; \Rightarrow \;\; \rho\big( A^\ast N \big) \;=\; 0 \,.

Therefore ,\langle-,-\rangle first of all descends to the quotient vector space

(9)𝒜/𝒩, \mathscr{H} \;\coloneqq\; \mathcal{A}/\mathcal{N} \,,

because

A 1,A 2+N 2 ρ(A 1 *(A 2+N 2)) =ρ(A 1 *A 2)+ρ(A 1 *N 2) =ρ(A 1 *A 2), \begin{array}{l} \big\langle A_1 ,\, A_2 + N_2 \big\rangle \\ \;\equiv\; \rho\big( A_1^\ast (A_2 + N_2) \big) \\ \;=\; \rho\big( A_1^\ast A_2 \big) + \rho\big( A_1^\ast N_2 \big) \\ \;=\; \rho\big( A_1^\ast A_2 \big) \,, \end{array}

where in the last step we used (8), and from this:

A 1+N 1,A 2 =A 2,A 1+N 1¯ =A 2,A 1¯=A 1,A 2, \begin{array}{l} \big\langle A_1 + N_1 ,\, A_2 \big\rangle \\ \;=\; \overline{ \big\langle A_2 ,\, A_1 + N_1 \big\rangle } \\ \;=\; \overline{ \big\langle A_2 ,\, A_1 \big\rangle } \;=\; \big\langle A_1 ,\, A_2 \big\rangle \,, \end{array}

where in the first and last step we used (2).

Now descended to the quotient, the pairing ,\langle-,-\rangle becomes positive definite by construction, so that it defines a Hermitian – by (2)inner product on \mathscr{H}, making it the desired Hilbert space.

Finally, again by the left-ideal property (7), the left multiplication action of 𝒜\mathcal{A} on itself also descends to an action on the quotient Hilbert space (9):

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

and so the cyclic vector in question is that represented by the unit element 1𝒜1 \in \mathcal{A}:

ψ ρ[1]. \psi_\rho \;\coloneqq\; [1] \;\in\; \mathscr{H} \,.

Hence on this Hilbert space \mathscr{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 \;=\; {\big\vert [1] \big\rangle} {\big\langle [1] \big\vert} \,.

Notice in summary that this GNS construction constitutes a kind of operator-state correspondence modulo null operators:

𝒜/𝒩 [A] Aψ ρ \begin{array}{rcl} \mathcal{A}/\mathcal{N} &\xleftrightarrow{\phantom{---}}& \mathscr{H} \\ [A] &\mapsto& A \cdot \psi_\rho \end{array}

For C *C^\ast-categories

The GNS construction for C *C^\ast-algebras is a special case of a more general construction of Ghez, Lina & Roberts 1985, Prop. 1.9 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:

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

In the generality of C * C^\ast -categories (and with an eye towards W * W^\ast -categories):

For general unital star-algebras:

and 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 January 14, 2024 at 09:21:04. See the history of this page for a list of all contributions to it.