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 ε.
[631.3.2] To do so, recall the definition of invariant (stationary) states.
[631.3.3] A state z∈A* is called invariant, if
holds for all A∈A and t∈R,
i.e. if the expectation values Azt=z,A
of all observables A∈A are constant.
[631.3.4] The set of invariant states B0⊂A* over A
is convex and compact in the weak*-topology [6].
[631.3.5] The same holds for the set of all
states Z⊃B0.
[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 z∈Z is called a BMO-state if
all maps Az:R→R have bounded mean oscillation
for all A∈A.
[631.4.3] The Banach space BMOR of functions with bounded mean oscillation
on R is defined as the linear space
| BMO(R)={f∈Lloc1(R),∥f∥BMO<∞} | | (16) |
where Lloc1R is the space of locally
integrable functions f:R→R .
[631.4.4] The BMO-norm is defined as
| ∥f∥BMO=infC{∫I|f(x)-fI|dx≤C|I|,for all I} | | (17) |
where I⊂R denotes intervals of length I and
denotes the average of f over the interval I.
[631.4.5] The set of all BMO-states
| B=z∈Z:AzBMO<∞ for all A∈A | | (19) |
is convex by linearity.
[631.4.6] As a subset B⊂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
| B0=z∈B:AzBMO=0 for all A∈A | | (20) |
as a subset B0⊂B.
[page 632, §1]
[632.1.1] A BMO-state will be called ε-almost invariant
or almost invariant with accuracy ε
if the expectation of all observables are
stationary to within experimental accuracy ε.
[632.1.2] More precisely, the set Bε of all ε-almost
invariant states is defined as
| Bε=z∈B:AzBMO<ε for all A∈A | | (21) |
as a family of subsets of B.
[632.1.3] For small ε→0 these states are almost invariant.
[632.1.4] The accuracy ε 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
where 0<ε<∞ and the set of
KMS-states Kβ at inverse temperature β>0 are
defined as states z∈Z such that the KMS-condition[16]
| z,Tt/τAB=z,BTt/τ+iϵβA | | (23) |
holds for all t/τ∈R and A,B∈A.
[632.2.2] The KMS-states are invariant states for all β≥0,
but KMS-states for different β are disjoint [16].
[632.2.3] For β=0 the KMS-states are trace states,
i.e. z,AB=z,BA
holds for all A,B∈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].
