Off-Mass-Shell Matrix Element and Jacobi Identity

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Abstract
A study is made on the conventional use of partial integration applied to divergences of currents in the off-mass-shell matrix elements regarding some of the external bosons. It is shown that equal-time commutators among currents and relevant operators must, in general, satisfy certain conditions in order for the above use of partial integration to be justified. In the case of the electromagnetic and/or weak matrix element in which two external bosons are made off the mass shell, the condition is that the Jacobi (commutator) identify holds for the relevant electromagnetic and/or weak operator and the time components of currents. In the case of the strong matrix element in which three external bosons are off the mass shell, the condition is that the Jacobi identity should be valid for some components of currents including both time and space components, and also for divergences. Explicit expressions are given for these off-mass-shell matrix elements when these identities are satisfied.