  A new path-integral theory is developed to calculate the photoemission
spectra (PES) of correlated many-electron systems. The application to the study
on Bi2Sr2CaCu2O8 (Bi2212) and boron-doped diamond (BDD) is discussed in
details. It is found that the isotopic shift in the angle-resolved
photoemission spectra of Bi2212 is due to the off-diagonal quadratic
electron-phonon (e-ph) coupling, whereas the presence of electron-electron
repulsion partially suppresses this effect. For the BDD, a semiconductor-metal
phase transition, which is induced by increasing the e-ph coupling and dopant
concentration, is reproduced by our theory. Additionally, the presence of Fermi
edge and phonon step-like structure in PES is found to be due to a co-existence
of itinerant and localized electronic states in BDD.
