Uncertainty principle, quantum fluctuations, and broken symmetries

The authors 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 behavior of quantum fluctuations in problems with continuous-group symmetries. Applications to Bose superfluids, antiferromagnets, and crystals at zero temperature are discussed. In particular, a simple and direct proof is provided of the absence of long-range order at zero temperature in the 1D case. A new inequality involving the spectral function at finite temperature is finally derived