  We prove the Lieb-Schultz-Mattis (LSM) theorem on the energy spectrum of a
general two or three-dimensional quantum many-body system with the U(1)
particle number conservation and translation symmetry. Especially, it is
demonstrated that the theorem holds in a system with long-range interactions.
To this end, we introduce approximate magnetic translation symmetry under the
total magnetic flux $\Phi=2\pi$ instead of the exact translation symmetry, and
explicitly construct low energy variational states. The energy spectrum at
$\Phi=2\pi$ is shown to agree with that at $\Phi=0$ in the thermodynamic limit,
which concludes the LSM theorem.
