Abstract: Forbidden lines are almost invisible in laboratory spectra but are ubiquitous in active galactic nuclei (AGN), especially in the narrow-line region (NLR), where they appear exceptionally bright and can even surpass the permitted hydrogen lines in intensity. This phenomenon has long been a central issue in astrophysical research. In this paper, we analyze how transitions forbidden by the electric dipole selection rules can proceed via higher-order channels such as magnetic dipole and electric quadrupole radiation, based on the multipole radiation theory of atomic physics. By combining estimates of typical spontaneous emission lifetimes, collisional timescales, and critical densities, we explain why forbidden lines can be efficiently excited and radiated in the low-density, highly ionized environment of the NLR. Through a discussion of representative forbidden lines in composite AGN spectra, we further demonstrate their unique role in probing the gas conditions and ionization structure of AGN, thereby providing a physical basis for future diagnostic studies based on emission lines.

Key words: active galactic nuclei, forbidden transitions, narrow-line region, critical density, atomic spectra