He has some lectures on quantum field theory online. I decided it may be fun to read some of them since: it's Ed Witten! On lecture one he decides to define what he means by a quantum theory:
We will define a quantum theory to be a pair (A,H) where A is a *-algebra (not necessarily commutative), and H is a selfadjoint element of A, defined up to a real number. The algebra A is called the algebra of quantum observables (operators), The element H, as before, is called the Hamiltonian...Now, it turns out if you have taken enough math, quantum mechanics and read through the above slowly you can see that what he is saying does make sense. (Obviously, as if he would be wrong.)
By a realization (or solution) of a quantum theory (A,H) we will mean an irreducible *-representation of the algebra A in some Hilbert space H, such that the spectrum of the operator H is bounded from below (representations are considered up to an isomorphism that preserves H). We will always normalize H so that the lowest point of a spectrum is zero.
But come on! I don't think any normal human thinks about quantum principles as deeply and rigorously as this. Edward Witten is definitely very special and part of me wishes I could understand physics at this level.
