Gelfand-Naimark-Segal construction

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

*under construction*

The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic $*$-representations of $C^*$-algebras and certain linear functionals (usually called *states*) on those same $C^*$-algebras. The correspondence comes about from an explicit construction of the *-representation from one of the linear functionals (states).

The GNS construction (as outlined above) is a special case of a more general construction of Ghez, Lima and Roberts applied to $C^\ast$-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

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

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

Given a state, $\rho$, on some C*-algebra, $A$, there is a $*$-representation $\pi$ of $A$ with a cyclic vector $\xi$ whose associated state is $\rho$. In other words,

$\rho(x)= \langle \xi, \pi(x)\xi \rangle$

for every $x$ in $A$.

The GNS construction is a central ingredient that translates between the Heisenberg picture and the Schrödinger picture of quantum mechanics: the AQFT and the FQFT picture of quantum field theory. In the former one considers $C^\ast$-algebras of observables, in the latter the spaces of states. Given a $C^\ast$-algebra of observables, the corresponding space of state can be taken to be that given by the GNS construction.

category: operator algebras

Revised on August 16, 2016 09:19:37
by Todd Trimble
(67.81.95.215)