In The Road to Reality by Roger Penrose, projective geometry as developed during the Renaissance is framed as (eventually) playing a pivotal role in quantum mechanics. (In fact, Penrose seems enamored with the idea that there is some connection between painting and physics, particularly where twistor theory is concerned. The book's epilogue is the most blatant example of this, though other examples abound.)
Not having studied quantum mechanics, I can't really imagine how projective spaces would be used to formalize what I know intuitively about the way QM works.
Can someone provide a very simple example?