  One of the most profound questions of mathematical physics is that of
establishing from first principles the hydrodynamic equations in large,
isolated, strongly interacting many-body systems. This involves understanding
relaxation at long times under reversible dynamics, determining the space of
emergent collective degrees of freedom (the ballistic waves), showing that
projection occurs onto them, and establishing their dynamics (the hydrodynamic
equations). We make progress in these directions, focussing for simplicity on
one-dimensional systems. Under a model-independent definition of the complete
space of extensive conserved charges, we show that hydrodynamic projection
occurs in Euler-scale two-point correlation functions. A fundamental ingredient
is a property of relaxation: we establish ergodicity of correlation functions
along almost every direction in space-time. We further show that to every
extensive conserved charge with a local density is associated a local current
and a continuity equation; and that Euler-scale two-point correlation functions
of local conserved densities satisfy a hydrodynamic equation. The results are
established rigorously within a general framework based on Hilbert spaces of
observables. These spaces occur naturally in the $C^*$ algebra description of
statistical mechanics by the Gelfand-Naimark-Segal construction. Using Araki's
exponential clustering and the Lieb-Robinson bound, we show that the results
hold, for instance, in every nonzero-temperature Gibbs state of short-range
quantum spin chains. Many techniques we introduce are generalisable to higher
dimensions. This provides a precise and universal theory for the emergence of
ballistic waves at the Euler scale and how they propagate within homogeneous,
stationary states.
