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3 Almost invariant states

[631.2.1] Strictly stationary or invariant states [15] are an idealization. [631.2.2] In experiments stationarity is never ideal, but only approximate. [631.2.3] Expectation values are uncertain within the accuracy of the experiment. [631.2.4] Experimental accuracy depends on the response and integration times of the experimental apparatus.

[631.3.1] These experimental restrictions suggest to focus on a class of states that are stationary (invariant) only up to a given experimental accuracy \varepsilon. [631.3.2] To do so, recall the definition of invariant (stationary) states. [631.3.3] A state \mathsf{z}\in\mathfrak{A}^{*} is called invariant, if

\displaystyle\left\langle\mathsf{z},T^{t}A\right\rangle=\left\langle\mathsf{z},A\right\rangle (15)

holds for all A\in\mathfrak{A} and t\in\mathbb{R}, i.e. if the expectation values \left\langle A\right\rangle _{{\mathsf{z}}}(t)=\langle\mathsf{z},A\rangle of all observables A\in\mathfrak{A} are constant. [631.3.4] The set of invariant states {\mathsf{B}_{0}}\subset\mathfrak{A}^{*} over \mathfrak{A} is convex and compact in the weak*-topology [6]. [631.3.5] The same holds for the set of all states \mathsf{Z}\supset{\mathsf{B}_{0}}. [631.3.6] Invariant states are fixed points of the adjoint time evolution T^{{*t}} as seen from eq. (8). [631.3.7] Because invariant states are fixed points of T^{{*t}}, they are of limited benefit for a proper mathematical formulation of the problems discussed above. [631.3.8] Once an orbit in state space reaches an invariant state, it remains forever in that state and cannot leave it.

[631.4.1] Almost invariant states are based on states whose expectation values are of bounded mean oscillation (BMO). [631.4.2] A state \mathsf{z}\in\mathsf{Z} is called a BMO-state if all maps \left\langle A\right\rangle _{{\mathsf{z}}}\colon\mathbb{R}\to\mathbb{R} have bounded mean oscillation for all A\in\mathfrak{A}. [631.4.3] The Banach space {\rm BMO}(\mathbb{R}) of functions with bounded mean oscillation on \mathbb{R} is defined as the linear space

\displaystyle{\rm BMO}(\mathbb{R})=\{ f\in L^{1}_{{\rm loc}}(\mathbb{R}),\left\| f\right\| _{{{\rm BMO}}}<\infty\} (16)

where L^{1}_{{\rm loc}}(\mathbb{R}) is the space of locally integrable functions f\colon\mathbb{R}\to\mathbb{R} . [631.4.4] The BMO-norm is defined as

\displaystyle\left\| f\right\| _{{{\rm BMO}}}=\inf _{C}\left\{\int\limits _{I}|f(x)-f_{I}|\mathrm{d}x\leq C|I|,\text{for all~}I\right\} (17)

where I\subset\mathbb{R} denotes intervals of length |I| and

\displaystyle f_{I}=\frac{1}{|I|}\int\limits _{I}f(x)\mathrm{d}x (18)

denotes the average of f over the interval I. [631.4.5] The set of all BMO-states

\displaystyle\mathsf{B}=\left\{\mathsf{z}\in\mathsf{Z}:\left\|\left\langle A\right\rangle _{{\mathsf{z}}}\right\| _{{{\rm BMO}}}<\infty\text{~~for all~}A\in\mathfrak{A}\right\} (19)

is convex by linearity. [631.4.6] As a subset \mathsf{B}\subset\mathsf{Z} of a weak* compact set it is itself weak* compact. [631.4.7] Hence a decomposition theory into extremal BMO-states exists by virtue of the Krein-Milman theorem. [631.4.8] The set of invariant states is identified through

\displaystyle{\mathsf{B}_{0}}=\left\{\mathsf{z}\in\mathsf{B}:\left\|\left\langle A\right\rangle _{{\mathsf{z}}}\right\| _{{{\rm BMO}}}=0\text{~~for all~}A\in\mathfrak{A}\right\} (20)

as a subset {\mathsf{B}_{0}}\subset\mathsf{B}.

[page 632, §1]    [632.1.1] A BMO-state will be called \varepsilon-almost invariant or almost invariant with accuracy \varepsilon if the expectation of all observables are stationary to within experimental accuracy \varepsilon. [632.1.2] More precisely, the set \mathsf{B}_{\varepsilon} of all \varepsilon-almost invariant states is defined as

\displaystyle\mathsf{B}_{\varepsilon}=\left\{\mathsf{z}\in\mathsf{B}:\left\|\left\langle A\right\rangle _{{\mathsf{z}}}\right\| _{{{\rm BMO}}}<\varepsilon\text{~~for all~}A\in\mathfrak{A}\right\} (21)

as a family of subsets of \mathsf{B}. [632.1.3] For small \varepsilon\to 0 these states are almost invariant. [632.1.4] The accuracy \varepsilon measures temporal fluctuations away from the time average.

[632.2.1] The following inclusions of classes of states used in the following are summarized for orientation and convenience

\displaystyle\mathsf{K}_{\beta}\subset{\mathsf{B}_{0}}\subset\mathsf{B}_{\varepsilon}\subset\mathsf{B}_{\infty}=\mathsf{B}\subset\mathsf{Z}\subset\mathfrak{A}^{*} (22)

where 0<\varepsilon<\infty and the set of KMS-states \mathsf{K}_{\beta} at inverse temperature \beta>0 are defined as states \mathsf{z}\in\mathsf{Z} such that the KMS-condition[16]

\displaystyle\left\langle\mathsf{z},T^{{t/\tau}}(A)B\right\rangle=\left\langle\mathsf{z},BT^{{t/\tau+\mathrm{i}\epsilon\beta}}(A)\right\rangle (23)

holds for all t/\tau\in\mathbb{R} and A,B\in\mathfrak{A}. [632.2.2] The KMS-states are invariant states for all \beta\geq 0, but KMS-states for different \beta are disjoint [16]. [632.2.3] For \beta=0 the KMS-states are trace states, i.e. \langle\mathsf{z},AB\rangle=\langle\mathsf{z},BA\rangle holds for all A,B\in\mathfrak{A}. [632.2.4] Because KMS-states are Gibbs states they are usually interpreted as equilibrium states with extremal states corresponding to pure thermodynamic phases [16].