  The usual formulation of quantum theory is based on rather obscure axioms
(employing complex Hilbert spaces, Hermitean operators, and the trace rule for
calculating probabilities). In this paper it is shown that quantum theory can
be derived from five very reasonable axioms. The first four of these are
obviously consistent with both quantum theory and classical probability theory.
Axiom 5 (which requires that there exists continuous reversible transformations
between pure states) rules out classical probability theory. If Axiom 5 (or
even just the word "continuous" from Axiom 5) is dropped then we obtain
classical probability theory instead. This work provides some insight into the
reasons quantum theory is the way it is. For example, it explains the need for
complex numbers and where the trace formula comes from. We also gain insight
into the relationship between quantum theory and classical probability theory.
