AQFT and operator algebra
This entry describes a concrete formalization of the general notion of state in the context of AQFT and operator algebra.
The Heisenberg picture of quantum mechanics is sometimes formalized by describing the observables of a quantum system by an operator algebra, and the state of the system as a state of the algebra.
The first “state” in the preceding sentence is the state the described physical system is in, the second one is the mathematical counterpart we are about to define, that inherited its name from the physical concept. This is the viewpoint of the AQFT approach to quantum field theory, especially of the Haag-Kastler approach.
We provide the definition of states and several associated properties and explain the physical interpretation and motivation of these concepts from the viewpoint of AQFT.
The definition of a state relies on the notion of positivity of the elements of an operator algebra, for which we need the structure of a C-star algebra.
An element $A$ of an (abstract) $C^*$-algebra is called positive if it is self-adjoint and its spectrum is contained in $[0, \infinity)$. We write $A \ge 0$ and say that the set of all positive operators is the positive cone (of a given $C^*$-algebra).
This definition is motivated by the Hilbert space situation, where an operator $A \in \mathcal{B} (\mathcal{H})$ is called positive if for every vector $x \in \mathcal{H}$ the inequality $\langle x, A x \rangle \ge 0$ holds. If the abstract $C^*$-algebra of the definition above is represented on a Hilbert space, then we see that by functional calculus we can define a self adjoint operator $B$ by $B \coloneqq f(A)$ with $f(t) := t^{1/2}$ and get $\langle x, A x \rangle = \langle B x, B x \rangle \ge 0$. This shows that the positive elements of the abstract algebra, if represented on a Hilbert space, become positive operators as defined here in the Hilbert space setting.
A linear functional $\rho$ on an $C^*$-algebra is positive if $A \ge 0$ implies that $\rho(A) \ge 0$.
A state of a unital $C^*$-algebra is linear functional $\rho$ such that $\rho$ is positive and $\rho(1) = 1$.
Though the mathematical notion of state is already close to what physicists have in mind, they usually restrict the set of states further and consider normal states only. We let $\mathcal{R}$ be an $C^*$-algebra and $\pi$ an representation of $\mathcal{R}$ on a Hilbert space $\mathcal{H}$.
A normal state $\rho$ is a state that satisfies one of the following equivalent conditions:
$\rho$ is weak-operator continuous on the unit ball of $\pi(\mathcal{R})$.
$\rho$ is strong-operator continuous on the unit ball $\pi(\mathcal{R})$.
$\rho$ is ultra-weak continuous.
There is an operator $A$ of trace class of $\mathcal{H}$ with $tr(A) = 1$ such that $\rho(\pi(R)) = tr(A \pi(R))$ for all $R \in \mathcal{R}$.
This apears as KadisonRingrose, def. 7.1.11, theorem 7.1.12
This list is not complete, there are more commonly used equivalent characterizations of normal states.
The last one is most frequently used by physicists, in that context the operator $A$ is also called a density matrix or density operator.
Sometimes the observables of a system are described by an abstract $C^*$-algebra, in this case an important notion is the folium:
The folium of a representation $\pi$ of an $C^*$-algebra $\mathcal{R}$ on a Hilbert space is the set of normal states of $\pi(\mathcal{R})$.
A state $\rho$ of a representation is called a vector state if there is a $x \in \mathcal{H}$ such that $\rho(\pi(R)) = \langle \pi(R)x, x \rangle$ for all $R \in \mathcal{R}$.
Normal states are vector states if $\mathcal{R}$ is a von Neumann algebra with a separating vector. More precisely: Let $\mathcal{R}$ be a von Neumman algebra acting on a Hilbert space $\mathcal{H}$, let $\rho$ be a normal state of $\mathcal{R}$ and $x \in \mathcal{H}$ be a separating vector for $\mathcal{R}$, then there is a $y \in \mathcal{H}$ such that $\rho(R) = \langle Ry, y \rangle$ for all $R \in \mathcal{R}$.
This appears as KadisonRingrose, theorem 7.2.3.
The set of states of an $C^*$-algebra is sometimes called the state space.
The state space is non-empty (define a state on the subalgebra $\mathbb{C} 1$ and extend it to the whole $C^*$-algebra via the Hahn-Banach theorem), convex and weak$^*$-compact, so it has extreme points. By the Krein-Milman theorem? (see Wikipedia: Krein-Milman theorem) it is the weak$^*$-closure of it’s extreme points.
A pure state is a state that is an extreme point of the state space.
The term “pure” originates from the notion of entanglement, a pure state is not a mixture of two distinct other states.
An important theorem for the physical interpretation of states is Fell's theorem.
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For more references see operator algebra.