nLab Gelfand-Naimark-Segal construction

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

$$\rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,,$$

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

$$\begin{aligned} \rho(A) & = \langle \psi \vert \, A \, \vert \psi \rangle \\ & \coloneqq \langle \psi, \pi(A) \psi \rangle \end{aligned}$$

with respect to some star-representation

$$\pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})$$

of $\mathcal{A}$ 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 $\mathcal{A}$ (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 $\mathcal{A}$ plays the role of an algebra of observables and $\rho \colon \mathcal{A} \to \mathbb{C}$ 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 $C^\ast$-algebras

Theorem

Given

1. a C*-algebra $\mathcal{A}$;

2. a state $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$ in the sense of

   1. a linear map on the underlying vector space

      (1)$$\rho \,\colon\, \mathcal{A} \to \mathbb{C} \,,$$

   2. which takes star-involution to complex conjugation

      (2)$$A \in \mathcal{A} \;\;\; \vdash \;\;\; \rho(A^\ast) = \overline{\rho(A)} \,,$$

   3. and satisfies “positivity in the sense that

      (3)$$A \in \mathcal{A} \;\;\; \vdash \;\;\; \rho(A^\ast A) \in \mathbb{R}_{\geq 0} \subset \mathbb{C} \,,$$

there exists

1. a C*-representation

   $$\pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})$$

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

2. a cyclic vector $\psi_\rho \in \mathcal{H}$

such that $\rho$ is the pure state corresponding to $\psi_\rho$, in that

$$\begin{aligned} \rho(A) & = \langle \psi_\rho \vert\, A \, \vert \psi_\rho \rangle \\ & \coloneqq \langle \psi_\rho , \pi(A) \psi_\rho \rangle \end{aligned}$$

for all $A \in \mathcal{A}$.

Proof

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

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

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

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

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

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

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)$$\left. \begin{array}{l} N \in \mathcal{N} \\ A \in \mathcal{A} \end{array} \right\} \;\;\; \vdash \;\;\; \left\{ \begin{array}{l} \big\vert \rho\big( (A N)^\ast A N \big) \big\vert^2 \\ \;=\; \big\vert \rho\big( (N^\ast A^\ast A) N \big) \big\vert^2 \\ \;\leq\; \rho\big( (N^\ast A^\ast A) (N^\ast A^\ast A)^\ast \big) \, \underset{ = 0 }{ \underbrace{ \rho\big( N^\ast N \big) \,. } } \end{array} \right.$$

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

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

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

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

because

$$\begin{array}{l} \big\langle A_1 ,\, A_2 + N_2 \big\rangle \\ \;\equiv\; \rho\big( A_1^\ast (A_2 + N_2) \big) \\ \;=\; \rho\big( A_1^\ast A_2 \big) + \rho\big( A_1^\ast N_2 \big) \\ \;=\; \rho\big( A_1^\ast A_2 \big) \,, \end{array}$$

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

$$\begin{array}{l} \big\langle A_1 + N_1 ,\, A_2 \big\rangle \\ \;=\; \overline{ \big\langle A_2 ,\, A_1 + N_1 \big\rangle } \\ \;=\; \overline{ \big\langle A_2 ,\, A_1 \big\rangle } \;=\; \big\langle A_1 ,\, A_2 \big\rangle \,, \end{array}$$

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 $\mathcal{A}$ on itself also descends to an action on the quotient Hilbert space (9):

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

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

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

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

$$\left\vert \psi_\rho \right\rangle \left\langle \psi_\rho \right\rangle \;=\; {\big\vert [1] \big\rangle} {\big\langle [1] \big\vert} \,.$$

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

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

For $C^\ast$-categories

The GNS construction for $C^\ast$-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 $C^\ast$-algebras):

Theorem

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

$$\rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}$$

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

$$\sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.$$

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

Functorial Aspects

See Functorial Aspects of the GNS Representation.

Related concepts

• Riesz representation theorem

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

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.), $C^\ast$-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:

• Gerard Murphy, Section 3.4 of: $C^\ast$-algebras and Operator Theory, Academic Press 1990 (doi:10.1016/C2009-0-22289-6)

• Konrad Schmüdgen, Section 8.3 of: Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) (doi:10.1007/978-3-0348-7469-4)

• Kehe Zhu, Section 14 of: An Introduction to Operator Algebras, CRC Press 1993 (ISBN:9780849378751)

• Richard V. Kadison, John R. Ringrose, Theorem 4.5.2 in: Fundamentals of the theory of operator algebras – Volume I: Elementary Theory, Graduate Studies in Mathematics 15, AMS 1997 (ISBN:978-0-8218-0819-1, ZMATH)

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 Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer (2017) [doi:10.1007/978-3-319-70706-8]

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

• Wikipedia, Gelfand-Naimark-Segal construction

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

• P. Ghez, Ricardo Lima, John E. Roberts, Prop. 1.9 in: $W^\ast$-categories, Pacific J. Math. 120 1 (1985) 79-109 [euclid:pjm/1102703884]

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) [doi:10.1007/978-3-319-21353-8]

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 $\mathbb{C}[ [ \hbar ] ]$:

• Martin Bordemann, Stefan Waldmann, Formal GNS Construction and States in Deformation Quantization, Commun. Math. Phys. (1998) 195: 549. (arXiv:q-alg/9607019, doi:10.1007/s002200050402)

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

• Bart Jacobs, Involutive Categories and Monoids, with a GNS-correspondence, Foundations of Physics, July 2012, Volume 42, Issue 7, pp 874–895 (arXiv:1003.4552)

• Arthur Parzygnat, From observables and states to Hilbert space and back: a 2-categorical adjunction (arXiv:1609.08975)

category: operator algebras