(Note Scott Morrison is speaking this week.)
I'll explain the foundations of noncommutative probability, as a common generalisation of classical probability and quantum mechanics. The mathematical prerequisites are slight: we'll be talking about *-algebras, but not need to go much beyond simple examples such as 2x2 complex matrices, with *-operation being conjugate transpose.
I'll define events, observables, states, measurements and Bayes law, and show how these definitions
- simplify to the usual notions in the commutative case, and
- provide the quantum mechanical notions of observables, states, measurements and "wave-function collapse" in the non-commutative case.
This perspective on quantum mechanics goes back to von Neumann (indeed, he invented von Neumann algebras precisely to generalise this story to the infinite dimensional setting). I find it very useful, but unfortunately it's not often explained except in the context of operator algebra theory. Two important payoffs of this approach are:
- Clarifying the role of classical probability in quantum mechanics.
- Explaining where Hilbert spaces come from: in this perspective it is a theorem that you can describe states using vectors in Hilbert spaces, rather than a postulate.