Quantum foundations (noncommutative probability)

(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

  1. simplify to the usual notions in the commutative case, and
  2. 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.