We address the consequences of invariance properties of the free energy of
spatially inhomogeneous quantum many-body systems. We consider a specific
position-dependent transformation of the system that consists of a spatial
deformation and a corresponding locally resolved change of momenta. This
operator transformation is canonical and hence equivalent to a unitary
transformation on the underlying Hilbert space of the system. As a consequence,
the free energy is an invariant under the transformation. Noether’s theorem for
invariant variations then allows to derive an exact sum rule, which we show to
be the locally resolved equilibrium one-body force balance. For the special
case of homogeneous shifting, the sum rule states that the average global
external force vanishes in thermal equilibrium.

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