And non-commutative ops can be supported by adding derived total order.

| physics/mathematics | Classical Physics | Quantum Mechanics |<br /> |---|---|---|<br /> | State Space | fields satisfying equations of laws<br>- the state is given by a point in the space | vector in a complex vector space with a Hermitian inner product (wavefunctions) |<br /> | Observables | functions of fields<br>- usually differential equations with real-valued solutions | self-adjoint linear operators on the state space<br>- some confusion may result when operators don't commute; there are usually no simple (real-valued) numerical solutions |