Can We Obtain Quantum Theory From Reasonable Axioms?

The usual formulation of quantum theory is rather abstract (it employs complex Hilbert spaces, Hermitian operators, the trace rule, and unitary or superoperator evolution). It is natural to ask why the formalism is like this. In this paper we will show that, for finite dimensional systems, this rather abstract formalism follows from a set of reasonable axioms. These axioms might well have been postulated without reference to any experimental data. If one of the axioms is dropped then, rather than getting quantum theory, we get classical probability theory. In developing the axioms we obtain a representation of quantum theory entirely in terms of real numbers and, correspondingly, a generalization of the Bloch sphere to arbitrary dimension.