Quantum theory in complex Hilbert space

Abstract

The theory of complexification of a real Hilbert space as developed by the author is scrutinized with the aim of explaining why quantum theory should be done in a complex Hilbert space in preference to a real Hilbert space. It is suggested that, in order to describe periodic motions in stationary states of a quantum system, the mathematical object modelling a state of a system should have enough points in it to be able to describe explicit time dependence of a periodic motion without affecting the probability distributions of observables. Heuristic evidence for such an assumption comes from Dirac’s theory of interaction between radiation and matter. If the assumption is adopted as a requirement on the mathematical model for a quantum system, then a real Hilbert space is ruled out in favour a complex Hilbert space for a possible model for such a system.