Explaining quantum many-body dynamics is a long-held goal of physics. A
rigorous operator algebraic theory of dynamics in locally interacting systems
in any dimension is provided here in terms of time-dependent equilibrium
(Gibbs) ensembles. The theory explains dynamics in closed, open and
time-dependent systems, provided that relevant pseudolocal quantities can be
identified, and time-dependent Gibbs ensembles unify wide classes of quantum
non-ergodic and ergodic systems. The theory is applied to quantum many-body
scars, continuous, discrete and dissipative time crystals, Hilbert space
fragmentation, lattice gauge theories, and disorder-free localization, among
other cases. Novel pseudolocal classes of operators are introduced in the
process: projected-local, which are local only for some states, crypto-local,
whose locality is not manifest in terms of any finite number of local densities
and transient ones, that dictate finite-time relaxation dynamics. An immediate
corollary is proving saturation of the Mazur bound for the Drude weight. This
proven theory is intuitively the rigorous algebraic counterpart of the weak
eigenstate thermalization hypothesis and has deep implications for
thermodynamics: quantum many-body systems 'out-of-equilibrium' are actually
always in a time-dependent equilibrium state for any natural initial state. The
work opens the possibility of designing novel out-of-equilibrium phases, with
the newly identified scarring and fragmentation phase transitions being
examples.Comment: 30 pages, 6 figures. Detailed examples added. Version as accepted by
PRX. Comments are very welcom