  Symmetries play a pivotal role in our understanding of the properties of
quantum many-body systems. While there are theorems and a well-established
toolbox for systems in thermal equilibrium, much less is known about the role
of symmetries and their connection to dynamics out of equilibrium. This arises
due to the direct link between a system's thermal state and its Hamiltonian,
which is generally not the case for nonequilibrium dynamics. Here we present a
pathway to identify the effective symmetries and to extract them from data in
nonequilibrium quantum many-body systems. Our approach is based on exact
relations between correlation functions involving different numbers of spatial
points, which can be viewed as nonequilibrium versions of (equal-time) Ward
identities encoding the symmetries of the system. We derive symmetry witnesses,
which are particularly suitable for the analysis of measured or simulated data
at different snapshots in time. To demonstrate the potential of the approach,
we apply our method to numerical and experimental data for a spinor Bose gas.
We investigate the important question of a dynamical restoration of an
explicitly broken symmetry of the Hamiltonian by the initial state. Remarkably,
it is found that effective symmetry restoration can occur long before the
system equilibrates. We also use the approach to define and identify
spontaneous symmetry breaking far from equilibrium, which is of great relevance
for applications to nonequilibrium phase transitions. Our work opens new
avenues for the classification and analysis of quantum as well as classical
many-body dynamics in a large variety of systems, ranging from ultracold
quantum gases to cosmology.
