The transport of matter is examined in the context of relativistic quantum transport theory for the case of neutral scalar fields. The goal is to formulate a theory valid off mass shell and out of equilibrium. We construct a conserved moment tensor which coincides with the ensemble average of the Noether tensor, or the improved energy‐momentum tensor with the ensemble average of the Noether tensor, or the improved energy‐momentum tensor within an additive constant of the ‘‘improvement’’ term. Conditions for closure of the conservation equation are given for the φ4 coupling.