Uncertainty principle, quantum fluctuations, and broken symmetries

We discuss a natural generalization of the uncertainty inequality holding in the case of non-hermitian operators. The inequality is employed to derive useful constraints on the behaviour of quantum fluctuations in problems with continuous group symmetries. Applications to Bose superfluids, antiferromagnets and crystals at zero temperature are discussed. We provide, in particular, a simple and direct proof of the absence of long range order at zero temperature in the1D case. A new inequality involving the spectral function at finite temperature is finally derived.