Quantum Theory From Five Reasonable Axioms

The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace formula 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 axioms are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exist continuous and reversible transformations between pure states) rules out classical probability theory. If Axiom 5 is substituted by another axiom, Axiom 5C, then we obtain classical probability theory instead. This work provides 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.