Is a man permitted to kill his wife if he finds her committing adultery and has no witnesses, and is there any proof for this in the Quran or Sunnah?

Islamic Sharia aims to establish a stable and organized Muslim society. Therefore, it has made the execution of punishments (hudud) the prerogative of the ruler, not of the general public. Thus, it is not permissible for a man to kill his wife if he finds her committing adultery. Rather, he must follow the Sharia in dealing with such a situation. There is no legal evidence (daleel shar'i) that permits a man to kill his adulterous wife.