The minimally coupled electromagnetic current Jμ and the symmetric energy-momentum tensor Θμν of the strong interactions are written as the sources of the electromagnetic and gravitational fields, respectively. In a formal sense the canonical commutators for these fields restrict the Schwinger terms in commutators containing Jμ and/or Θμν. The model-independent results are (x0=0):ST(n≥2)[Jμ(x),Jν(0)]=0, ST(n≥4)[Θμν(x),Θρλ(0)]=0, where ST(n) denotes the nth-order Schwinger term (term with the nth derivative of a δ function). In addition, in low-spin models it can be shown that (for spin ≤ 1) ST(n≥4)×[Jλ(x),Θμν(0)]=0, and that (for spin ≤½) ST(n≥3)[Jλ(x),Θμν(0)]=0. These conditions apply when Θμν is defined in the usual way or in the manner prescribed by Callan, Coleman, and Jackiw. Stronger conditions can be derived for a more narrowly defined Θμν and for models with restricted forms of interactions. The formal significance of these results is discussed.