nLab
Gelfand-Naimark-Segal construction
Context
Measure and probability theory
AQFT
AQFT and operator algebra

Definitions
Theorems
States and observables
Operator algebra
Local QFT
Euclidean QFT
Contents
under construction

Idea
The Gelfand–Naimark–Segal (GNS) construction establishes a correspondence between cyclic $*$ -representation s 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).

GNS construction
Theorem
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 $A$ .

and this needs to be finished. I don’t have the head for this right now.

Applications
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.

Revised on September 12, 2013 17:29:17
by

Zoran Škoda
(161.53.130.104)