Glafka 2004: Some Remarks on the Role of Complex Numbers in Quantum Theory

Abstract

We consider the issue of whether complex numbers are necessary in the formulation of quantum theory. We argue that their introduction is not forced on us by the dynamics, but it is absolutely necessary in order to incorporate the observable properties of the geometric phases that appear in interference experiments. This remark provides the motivation for the construction of a histories-based axiomatic scheme for quantum theory, in which phases are considered as primitive elements, on equal footing with probabilities. This scheme reduces to standard quantum theory for systems characterised by a background causal structure–however it may lead to a different description in the domain of quantum gravity.