A natural generalization of the Heisenberg uncertainty principle inequality holding for non-Hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long-range order in the ground state of several one-dimensional (1D) systems with continuous-group symmetries. In this paper, we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long-range order in this two-dimensional (2D) problem is directly connected with the ${\mathit{q}}^2$ behavior of the static structure function S(q) at small momenta.

• Received 5 October 1992

DOI:https://doi.org/10.1103/PhysRevB.47.10915