Integration by parts changes a switched density

Suppose two local densities obey

Then

Thus equality of unswitched actions modulo a total derivative does not identify two arbitrarily switched density prescriptions unless the switch-dependent term is also included.

The Action Ward Identity is relevant to formulating this dependence correctly DF; it is not permission to discard the last term in (eq:switched-IBP).

Figure fig:VI-switching-audit collects the switching terms that must be retained in the local and retarded identities.