The event of someone appearing after the death to a loved one, or loved ones, is common in human history. many people experience it. My own mother had such an appearance happen after my father died, and he spoke to her and she touched him and he felt solid. Now, you would have to know my mother to realize how unlike her it was to admit to anything like this – she was pragmatic, hated religious preaching and histrionics, and was the last person to put any validity to ghosts, ESP, or anything of the like. But there he was and she saw him. 

Now, does that mean my father was resurrected from his grave? I don’t think the apostles necessarily had that notion. That’s a Pauline notion that came later. In any case, speculating over who believed what seems irrelevant to me. Those inclusions in Mark and the other Gospels were made significantly after the fact. Peter and James are supposed to have acknowledged the resurrection, but are those their actual beliefs? We have no way of knowing. I think the idea that any of the apostles believed in a resurrected Jesus is highly suspect. The resurrected Jesus is the main Pauline idea. Everything else falls second to it for Paul. Peter and James were opposed to Paul’s teaching, and so we have to suspect that, whatever words have been inserted into the Letters or Gospels, Peter, James and the Jerusalem church began preaching Jesus as a wise man with an apocalyptic message and a spiritual code of how to live, how to be in the moment of the “life breath”, and how to honor God, the Jewish God. The resurrected Jesus was Paul’s God.  

Well Paul was the one who equated his own experience of the risen Jesus with that of the disciples.  In some instances Paul is in conflict with them but he never denies their witness or their authority.  Paul claims to be equal to them.  The “bereavement hallucination” hypothesis is interesting but hardly conclusive.  I’ve had dreams of departed love ones and even of Jesus myself and in an ancient context they would qualify as perfectly valid resurrection appearances.  What would be more natural than for some of Jesus’ close followers to dream about him after his death?   

But are we to believe Paul on that? He’s telling his followers that the apostles agree with him, with followers who have no way to confirm that, with the center of the Petrine church in Jerusalem. Could that have been one of the issues Peter had with Paul? There’s no documentation during Paul’s life that says Peter believed in the resurrection. That comes later in the synoptics, which were hugely influenced by the writings of Paul, the sequence of which is largely accepted in modern scholarship. Paul wasn’t referring to the Gospels, they didn’t exist. It’s the other way around.