Abstract
Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.
Highlights
Applications of fractional time derivatives in physics and engineering assume the existence of a physical time automorphism of observables, which for closed quantum many-body systems is usually given as a Hamiltonian-generated one-parameter group of unitary operators on a Hilbert space
Dissipative processes, irreversible phenomena, the decay of unstable particles, the approach to thermodynamic equilibrium or quantum measurement processes are difficult to accommodate within this traditional mathematical framework [1,2,3]
The introduction of the sets Bε of ε-almost invariant bounded mean oscillation (BMO)-states with 0 ≤ ε ≤ ∞ has provided a mathematical 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
Summary
Applications of fractional time derivatives in physics and engineering assume the existence of a physical time automorphism (time evolution) of observables, which for closed quantum many-body systems is usually given as a Hamiltonian-generated one-parameter group of unitary operators on a Hilbert space. Dissipative processes, irreversible phenomena, the decay of unstable particles, the approach to thermodynamic equilibrium or quantum measurement processes are difficult to accommodate within this traditional mathematical framework [1,2,3]. It has remained difficult to find physical conditions that rigorously imply irreversibility for the time evolution of the subsystem [4,5]. Relaxation processes in the reservoir R are usually much faster than the characteristic time scale for the evolution of the system S of interest
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