Firstly, he is an Orthodox Jew and experienced a fair bit of anti-semetic abuse when he was young, and he was also a target of a white supremacist terror plot in 2002 (this is info from wikipedia so I need to back that up...). This may well explain his interest and involvement in Schindler's list among other films and he even produced a few documentaries (these films and documentaries include Band of Brothers, Broken Silence, Price for Peace, Eyes of the Holocaust, Survivors of the Holocaust, Letters from Iwo Jima, Flags of our Fathers). It's mostly the documentaries he produced that focus on the holocaust (the operative word being mostly), even his other WW2 films show his religious and personal background influencing his work. Anyway, what I was trying to get at is, that is a fairly obvious influence which will be good to include once I back it up.
He uses children in danger a fair bit, and as Paul pointed out he uses families with divorced parents or with the father not being around, the latter may well be a reflection of his childhood. He also shows alot of unremarkable people doing or discovering remarkable things (eg. Jurassic park, E.T., Casper, Men in black, back to the future etc etc). which might be a reflection on how he felt about himself, he was an unremarkable kid who has gone on to be this amazing director and producer.
However, NONE of this is psychoanalysis. It's difficult to describe on here, but Freud's theories are all about how your upbringing affects the development of your personality, and fair enough what I've said above is about how his upbringing affected his career. However, because we don't know anything more than basics like the fact that his dad wasn't around much...we can't say much more than that.
Freud is all about the id, ego and super ego being developed in the oral, anal and phallic stages when you're a baby....then defences such as repression, projection, sublimation and reaction formation are developed by your ego to protect your id....none of which really apply.
Anyway, I'm rambling....my point is, I've found a bit but possibly not what you were after. Is there anything else anyone thinks is relevant that I need to look into further etc? And is any of this stuff even useful??!
I'm also a little worried we're not looking into the whole time machine aspect of this enough....
aaaaaanyway let me know what you think :D
