algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
quantum mechanical system, quantum probability
interacting field quantization
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
by a vector $\psi \in \mathcal{H}$ in a complex Hilbert space $\mathcal{H}$ as the “expectation value”
with respect to some star-representation
of $\mathcal{A}$ 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 $\mathcal{A}$ (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 $\mathcal{A}$ 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.
under construction
Given
a C*-algebra, $\mathcal{A}$;
a state, $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$
there exists
of $\mathcal{A}$ on some Hilbert space $\mathcal{H}$
a cyclic vector $\psi \in \mathcal{H}$
such that $\rho$ is the state corresponding to $\psi$, in that
for all $A \in \mathcal{A}$.
The GNS construction for $C^\ast$-algebras is a special case of a more general construction of Ghez, Lima and Roberts applied to C*-categories (horizontal categorification of $C^\ast$-algebras).
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
together with a cyclic vector $\xi \in \rho_\sigma(A)$ such that for all $x \in \mathcal{C}(A,A)$,
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.
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
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:
For formal power series algebras over $\mathbb{C}[ [ \hbar ] ]$:
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)