  We derive a closed equation of motion for the current density of an
inhomogeneous quantum many-body system under the assumption that the
time-dependent wave function can be described as a geometric deformation of the
ground-state wave function. By describing the many-body system in terms of a
single collective field we provide an alternative to traditional approaches,
which emphasize one-particle orbitals. We refer to our approach as continuum
mechanics for quantum many-body systems. In the linear response regime, the
equation of motion for the displacement field becomes a linear fourth-order
integro-differential equation, whose only inputs are the one-particle density
matrix and the pair correlation function of the ground-state. The complexity of
this equation remains essentially unchanged as the number of particles
increases. We show that our equation of motion is a hermitian eigenvalue
problem, which admits a complete set of orthonormal eigenfunctions under a
scalar product that involves the ground-state density. Further, we show that
the excitation energies derived from this approach satisfy a sum rule which
guarantees the exactness of the integrated spectral strength. Our formulation
becomes exact for systems consisting of a single particle, and for any
many-body system in the high-frequency limit. The theory is illustrated by
explicit calculations for simple one- and two-particle systems.
