The occurrence of a 3-cocycle in quantum mechanics or quantum field theory has been interpreted somewhat paradoxically as a breakdown of the Jacobi identity. The main result of this paper is that the 3-cocycle in chiral QCD arises as an obstruction which prevents the existence of a certain extension of one Lie algebra by another. This obstruction may be avoided by constructing a modified Lie algebra extension consisting of derivations on the algebra generated by the fields. However, the 3-cocycle then appears when an attempt is made to implement these derivations by commutation with unbounded operators in the canonical equal-time formalism. Assuming the existence of these unbounded operators is what leads to the violation of the Jacobi identity.
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