  We present a formal analysis of nonlinear response functions in terms of
correlation functions in real- and imaginary-time domains. In particular, we
show that causal nonlinear response functions, expressed in terms of nested
commutators in real time, can be obtained from the analytic continuation of
time-ordered response functions, which are more easily amenable to diagrammatic
calculation. This generalizes the well-known result of linear response theory.
We then use gauge invariance arguments to derive exact relations between
second-order response functions in density and current channels. These
identities, which are non-perturbative in the strength of inter-particle
interactions, allow us to establish exact connections between nonlinear optics
calculations done in different electromagnetic gauges.
