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2 Problems and Objective

[629.2.1] There are several unsolved problems with the mathematical framework described in the introduction, particularly when the system is infinite. [629.2.2] The following examples of open problems stem from three different areas of theoretical physics.

  1. Open quantum systems: [629.2.3] For open quantum systems with infinite reservoirs \mathfrak{A}=\mathfrak{B}(\mathfrak{H}) is the C{}^{*}-algebra of bounded operators on the Hilbert space \mathfrak{H} of the system S\cup R in a suitably chosen GNS-representation. [629.2.4] The states \langle\rho,A\rangle=Tr\rho A can be identified with trace class operators on \mathfrak{H} if Tr denotes the trace operation. [629.2.5] The unitary time-automorphisms T^{t}\colon\mathfrak{B}(\mathfrak{H})\to\mathfrak{B}(\mathfrak{H}) of S\cup R are generated by a Hamiltonian H of S\cup R

    H=H_{S}\otimes\mathbf{1}+\mathbf{1}\otimes H_{R}+H_{{SR}}=-\mathrm{i}\mathscr{L} (12)

    written formally in terms of Hamiltonians H_{S} of S, H_{R} of R and their interaction H_{{SR}}. [629.2.6] It is well known[2] that the automorphisms T^{t} on \mathfrak{B}(\mathfrak{H}) do not induce automorphisms on the algebra \mathfrak{B}(\mathfrak{H}_{S}) of the open subsystem, because the time evolution will mix the Hilbert spaces \mathfrak{H}_{S} and \mathfrak{H}_{R} of the system and the reservoir. [629.2.7] Even if the initial state is prepared as a product state \rho _{S}\otimes\rho _{R} the subsystem evolutions defined by

    \displaystyle T^{t}_{S}A \displaystyle=Tr_{R}(\mathbf{1}\otimes\rho _{R})T^{{-t}}(A\otimes\mathbf{1})T^{t} (13a)
    \displaystyle T^{{*\; t}}_{S}\rho _{S} \displaystyle=Tr_{R}\; T^{t}(\rho _{S}\otimes\rho _{R})T^{{-t}} (13b)

    do not form groups

    \displaystyle T^{{a_{1}}}_{S}T^{{a_{2}}}_{S} \displaystyle\neq T^{{a_{1}+a_{2}}}_{S} (14a)
    \displaystyle T^{{*\; a_{1}}}_{S}T^{{*\; a_{2}}}_{S} \displaystyle\neq T^{{*\; a_{1}+a_{2}}}_{S} (14b)

    [page 630, §0]    because of memory effects accumulating from the mixing of the system and the reservoir whenever there is nonvanishing interaction [2]. [630.0.1] Here \rho _{S},\rho _{R} denote the density matrices of the system and the rservoir. [630.0.2] The trace Tr_{R} integrates out the reservoir degrees of freedom.

  2. Classical dynamical systems: [630.0.3] In classical systems it is well known [14] that the orbits in the abelian algebra \mathfrak{A} of functions on phase space cannot always be defined for all t\in\mathbb{R} and for all initial conditions A_{{0}} in the thermodynamics limit. [630.0.4] The integration of eq. (1) does not generally give a dynamical flow of time for all initial conditions and the problem is to find sufficiently large subsets of \mathfrak{A} such that catastrophic behavior is absent and a unique orbit exists for all t\in\mathbb{R}.

  3. Quantum field theory: [630.0.5] For quantum field theories or infinite systems the Stone-von Neumann uniqueness breaks down. [630.0.6] Haags theorem shows that the determination of a suitable representation of the canonical commutation relations becomes a dynamical problem, if the vacuum states for different couplings are different. [630.0.7] Non-normal states arise that yield representations assigning different values to global observables like densities. [630.0.8] Due to the problem of inequivalent representations it is not possible to represent the time evolution as a group of unitary transformation within a single representation, because the representation algebra may change into an inequivalent representation as time evolves.

[630.0.9] One objective of this paper is to suggest that these three open problems are, in fact, related to each other, even though they seem to be unrelated at first sight.

[630.1.1] The present article suggests that the common denominator of problems 2-2 associated with the mathematical framework described in the introduction is the concept of time flow as a translation, implicitly assumed on the left hand side of Equation (1). [630.1.2] The common origin of problems 2-2 emerges from studying the following two general questions associated with Equation (1).

Problem 1.

[630.1.3] Are there global solutions of equation (1), i.e. solutions for all t\in\mathbb{R}?

[630.1.4] If global solutions and hence a group of *-automorphisms on \mathfrak{A} exist, then this implies a continuous time evolution for all states \mathsf{z}\in\mathsf{Z}. [630.1.5] This means a time evolution independent of the state, which is not to be expected for general infinite systems without rescaling time. [630.1.6] Rescaling of time is also expected to be necessary for establishing hydrodynamic limits governing invariant states.

Problem 2.

[630.1.7] If global solutions of equation (1) exist, how can invariant solutions still change with time?

[630.1.8] Local stationarity (invariance) in time arises from the underlying dynamics. [630.1.9] Local stationarity in time is necessary, if thermodynamic observables such as temperature, pressure or densities are to provide an approximate representation of the physical system that changes slowly on long time scales. [630.1.10] Hence one has to study the set of stationary states that are invariant under the time evolution. [630.1.11] If the thermodynamic observables change then there must exist many invariant states and many possible time averages, i.e. the time averages are not unique.

[page 631, §1]    [631.1.1] The objective of this paper is to introduce a framework in which questions concerning the abundance of time-invariant states and their embedding in the set of all states can be posed mathematically in a proper way.