Truncated Wigner dynamics and conservation laws

Ultracold Bose gases can be used to experimentally test many-body theory predictions. Here we point out that both exact conservation laws and dynamical invariants exist in the topical case of the one-dimensional Bose gas, and these provide an important validation of methods. We show that the first four quantum conservation laws are exactly conserved in the approximate truncated Wigner approach to many-body quantum dynamics. Center-of- mass position variance is also exactly calculable. This is nearly exact in the truncated Wigner approximation, apart from small terms that vanish as N-3/2 as N -> infinity with fixed momentum cutoff. Examples of this are calculated in experimentally relevant, mesoscopic cases.