Observable algebra - Imported Knowlpedia Content

Let $\mathcal H$ be a finite-dimensional complex Hilbert space. An observable algebra is a unital $*$ -subalgebra

\mathcal A\subseteq \mathcal B(\mathcal H),

where $\mathcal B(\mathcal H)$ is the algebra of operators on $\mathcal H$ . Thus $\mathcal A$ contains the identity and is closed under linear combinations, products, and adjoints. The physical observables are the self-adjoint elements of $\mathcal A$ .

States

A state on $\mathcal A$ is a positive normalized linear functional $\omega:\mathcal A\to\mathbb C$ :

\omega(A^*A)\ge 0, \qquad \omega(I)=1.

When $\mathcal A=\mathcal B(\mathcal H)$ , there is a unique density operator $\rho$ such that

\omega(A)=\operatorname{Tr}(\rho A).

Remarks

Commutative observable algebras model compatible families of observables. Noncommutative algebras allow incompatible observables and are the natural setting for quantum dynamics and the quantum Gibbs state.