  Polaritonic chemistry is an interdisciplinary emerging field that presents
several challenges and opportunities in chemistry, physics, and engineering. A
systematic review of polaritonic response theory is presented, following a
chemical perspective based on molecular response theory. We provide the reader
with a general strategy for developing response theory for ab initio cavity
quantum electrodynamics (QED) methods and critically emphasize details that
still need clarification and require cooperation between the physical and
chemistry communities. We show that several well-established results can be
applied to strong coupling light-matter systems, leading to novel perspectives
on the computation of matter and photonic properties. The application of the
Pauli-Fierz Hamiltonian to polaritons is discussed, focusing on the effects of
describing operators in different mathematical representations. We thoroughly
examine the most common approximations employed in ab initio QED, such as the
dipole approximation. We introduce the polaritonic response equations for
recently developed ab initio QED Hartree-Fock and QED coupled cluster methods.
The discussion focuses on the similarities and differences from standard
quantum chemistry methods, providing practical equations for computing the
polaritonic properties.
