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4 Time Evolution of States

[210.2.1] In general, the set of observables \mathcal{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 \mu 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 \mathcal{A}^{*}. [210.2.5] The notation \langle\mu,a\rangle is used for the value \mu(a) of the observable a in the state \mu. [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 \mathcal{A} is a subset of its bidual, \mathcal{A}\subset\mathcal{A}^{{**}}, its elements can be considered as functions on the set X(\mathcal{A}) of its charactersd (This is a footnote:) d A character is an algebraic *-homomorphism from a commutative C{}^{*}-algebra to \mathbb{C} ., i.e. a(\chi)=\langle\chi,a\rangle for a\in\mathcal{A},\chi\in X(\mathcal{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 C_{0}(X(\mathcal{A})) of continuous functions on the set X(\mathcal{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 {\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}^{*}}}\! T}^{{\dot{t}}}\colon\mathcal{A}^{*}\to\mathcal{A}^{*} with {\dot{t}}\in\mathbb{R} consists of all adjoint operators ({\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}}}\! T}^{{{\dot{t}}}})^{*} on the dual space \mathcal{A}^{*} [33, 28]. [211.1.3] If \mu({{\dot{t}}_{1}}) denotes the state at time {{\dot{t}}_{1}}\in\mathbb{R}, then equation (8) implies

\displaystyle\langle\mu({{\dot{t}}_{1}}),{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}}}\! T}^{{{\dot{t}}}}a({{\dot{t}}_{1}})\rangle \displaystyle= \displaystyle\langle\mu({{\dot{t}}_{1}}),a({{\dot{t}}_{1}}+{\dot{t}})\rangle (9)
\displaystyle= \displaystyle\langle\mu({{\dot{t}}_{2}}-{\dot{t}}),a({{\dot{t}}_{2}})\rangle=\langle{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}^{*}}}\! T}^{{\dot{t}}}\mu({{\dot{t}}_{2}}),a({{\dot{t}}_{2}})\rangle

where {{\dot{t}}_{2}}={{\dot{t}}_{1}}+{\dot{t}}\in\mathbb{R} is arbitrary. [211.1.4] For left translations the adjoint group

{\rule[-2.0pt]{0.0pt}{10.0pt}_{{\mathcal{A}^{*}}}\! T}^{{\dot{t}}}\mu({\dot{s}})=\mu({\dot{s}}-{\dot{t}}) (10)

is the group of right translations with {\dot{s}},{\dot{t}}\in\mathbb{R}. [211.1.5] The adjoint semigroup is weak{}^{*} continuous, but in general not strongly continuous, unless the Banach space \mathcal{A} is reflexive [32].