In the study of quark-gluon systems using transport equations, the production of quarks and antiquarks is often introduced in the form of a source term based on the Schwinger formula for pair production in the presence of a constant (color-)electric field. However, the production of pairs must, in principle, already be present in forms of transport equations based on the Wigner representation, since these start out exactly equivalent to the full quantum dynamics of the system. It is the purpose of the present study to establish the precise nature of the appearance of pair production in transport-equation formulations. This allows for the treatment of this phenomenon as an integral part of the solution of the transport equations. It also assures the avoidance of double counting of part of the effects of a (color-)electric field through its involvement both in the dynamics of the particles and as a source term for their production.