  Recently (PRL 113, 050403 (2014)) the concept of local symmetries in
one-dimensional stationary wave propagation has been shown to lead to a class
of invariant two-point currents that allow to generalize the parity and Bloch
theorem. In the present work we establish the theoretical framework of local
symmetries for higher-dimensional interacting many-body systems. Based on the
Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy we derive the equations
of motion of local symmetry correlators which are off-diagonal elements of the
reduced one-body density matrix at symmetry related positions. The natural
orbital representation yields equations of motion for the convex sum of the
local symmetry correlators of the natural orbitals as well as for the local
symmetry correlators of the individual orbitals themselves. An alternative
integral representation with a unique interpretation is provided. We discuss
special cases, such as the bosonic and fermionic mean field theory and show in
particular that the invariant two-point currents are recovered in the case of
the non-interacting one-dimensional stationary wave propagation. Finally we
derive the equations of motion for anomalous local symmetry correlators which
indicate the breaking of a global into a local symmetry in the stationary
non-interacting case.
