  A complete, rigorous relativistic field-theory formulation of the
nucleon-nucleon (NN) bremsstrahlung reaction is presented. The resulting
amplitude is unitary as a matter of course and it is gauge invariant, i.e., it
satisfies a generalized Ward-Takahashi identity. The novel feature of this
approach is the consistent microscopic implementation of local gauge invariance
across all interaction mechanisms of the hadronic systems, thus serving as a
constraint for all subprocesses. The formalism is quite readily adapted to
approximations and thus can be applied even in cases where the microscopic
dynamical structure of the underlying interacting hadronic systems is either
not known in detail or too complex to be treated in detail. We point out how
the interaction currents resulting from the photon being attached to
nucleon-nucleon-meson vertices can be treated by phenomenological four-point
contact currents that preserve gauge invariance. In an advance application of
the present formalism [K. Nakayama and H. Haberzettl, PRC 80, 051001(R)
(2009)], such interaction currents were found to contribute significantly in
explaining experimental data. In addition, we provide a scheme that permits the
approximate treatment of current contributions resulting from pieces of the NN
interaction that cannot be incorporated exactly. In each case, the
approximation procedure ensures gauge invariance of the entire bremsstrahlung
amplitude. We also discuss the necessary modifications when taking into account
baryonic states other than the nucleon N; in detail, we consider the
$Delta(1232)$ resonance by incorporating the couplings of the $NN$ to the
$N\Delta$ and $\Delta\Delta$ systems, and the $\gamma N \to \Delta$
transitions. We apply the formalism to the 280-MeV bremsstrahlung data from
TRIUMF [PRD 41, 2689 (1990)] incorporating $\gamma N\Delta$ transition currents
and find good agreement with the data.
