The minimally coupled electromagnetic current $J_{\mu }$ and the symmetric energy-momentum tensor ${\Theta }_{\mu \nu }$ 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 $J_{\mu }$ and/or ${\Theta }_{\mu \nu }$. The model-independent results are $(x^0=0):S_T(n\ge 2)\left[J_{\mu }\mathrm{}\right(x),J_{\nu }\mathrm{}(0\left)\right]=0\mathrm{}$, $S_T(n\ge 4)\left[{\Theta }_{\mu \nu }\mathrm{}\right(x),{\Theta }_{\rho \lambda }\mathrm{}(0\left)\right]=0\mathrm{}$, where $S_T\mathrm{}\left(n\right)\mathrm{}$ denotes the $n\mathrm{th}$-order Schwinger term (term with the $n\mathrm{th}$ derivative of a $\delta$ function). In addition, in low-spin models it can be shown that (for spin ≤ 1) $S_T(n\ge 4)\times \left[J_{\lambda }\mathrm{}\right(x),{\Theta }_{\mu \nu }\mathrm{}(0\left)\right]=0\mathrm{}$, and that (for spin ≤½) $S_T(n\ge 3)\left[J_{\lambda }\mathrm{}\right(x),{\Theta }_{\mu \nu }\mathrm{}(0\left)\right]=0\mathrm{}$. These conditions apply when ${\Theta }_{\mu \nu }$ 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 ${\Theta }_{\mu \nu }$ and for models with restricted forms of interactions. The formal significance of these results is discussed.

• Received 23 October 1970

DOI:https://doi.org/10.1103/PhysRevD.3.1320