This question is about free field theories. One usually derives Ward identity from the path integral by considering the variation of the path integral under a symmetry. See for example page 41 of volume 1 of Polchinski. You obtain some relation of divergence of the current with other operator insertions. However, through the usual operator correspondence, this current operator can be divergent, for example energy-momentum tensor in free bosonic theory, because it includes multiplication of two fields at the same point. However, these relations are then used with currents and are replaced with their normal ordered forms. My question is, why do normal ordered currents satisfy the Ward identity?
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