Participant Prods., responsible for “An Inconvenient Truth,” turns to another Democratic leader who has rebounded from a stinging political defeat with a new documentary on Jimmy Carter. Jonathan Demme has been following the ex-president during his book tour for “Palestine Peace Not Apartheid,” a new tome that has inspired its share of protest. In his doc titled “He Comes in Peace,” Demme says he hopes to capture the spontaneous moments in verite style, rather than rely on talking heads. (He even gets Carter swimming at New York’s Peninsula Hotel. Demme certainly got that with a recent Carter appearance on C-SPAN’s “Book Notes,” in which one caller labeled him an anti-Semite.