4 Time Evolution of States

[210.2.1] In general, the set of observables A of a physical system is not only a Banach space, but forms an algebra, more specifically, a C*-algebra [30]. [210.2.2] In classical physics this algebra is commutative. [210.2.3] The states μ of a physical system are normalized, positive linear functionals on its algebra of observables [30]. [210.2.4] As such they are elements of the dual space A*. [210.2.5] The notation μ,a is used for the value μ⁢a of the observable a in the state μ. [210.2.6] Convex combinations of states are again states. [210.2.7] If a state cannot be written as a convex combination of other states, it is called pure. [page 211, §0]    [211.0.1] Because the observable algebra A is a subset of its bidual, A⊂A*⁣*, its elements can be considered as functions on the set X⁢A of its characters^d (This is a footnote:) ^d A character is an algebraic *-homomorphism from a commutative C*-algebra to C ., i.e. a⁢χ=χ,a for a∈A,χ∈X⁢A. [211.0.2] By virtue of this correspondence, known as the Gelfand isomorphism [30, 31], a commutative C*-algebra is isomorphic to the algebra C0⁢X⁢A of continuous functions on the set X⁢A of its characters equipped with the weak* topology. [211.0.3] Characters are pure states.

[211.1.1] The time evolution of states is obtained from the time evolution of observables by passing to adjoints [28, 32]. [211.1.2]  The adjoint time evolution A*⁢Tt˙:A*→A* with t˙∈R consists of all adjoint operators A⁢Tt˙* on the dual space A* [33, 28]. [211.1.3] If μ⁢t˙1 denotes the state at time t˙1∈R, then equation (8) implies

where t˙2=t˙1+t˙∈R is arbitrary. [211.1.4] For left translations the adjoint group

is the group of right translations with s˙,t˙∈R. [211.1.5] The adjoint semigroup is weak* continuous, but in general not strongly continuous, unless the Banach space A is reflexive [32].