  We discuss conditions for the absence of spontaneous breakdown of continuous
symmetries in quantum lattice systems at $T=0$. Our analysis is based on
Pitaevskii and Stringari's idea that the uncertainty relation can be employed
to show quantum fluctuations. For the one-dimensional systems, it is shown that
the ground state is invariant under the continuous transformation if a certain
uniform susceptibility is finite. For the two- and three-dimensional systems,
it is shown that truncated correlation functions cannot decay any more rapidly
than $|r|^{-d+1}$ whenever the continuous symmetry is spontaneously broken.
Both of these phenomena occur owing to quantum fluctuations. Our theorems cover
a wide class of quantum lattice-systems having not-too-long-range interactions.
