Quantum optical phase, rigged Hilbert spaces and complementarity.

We place Loudon's quantum treatment of optical phase in in its historical context, and outline research that it inspired. We show how it led Pegg and Barnett to their quantum phase formalism, explaining the challenges that they overcame to define phase operators and phase eigenstates rigorously. We show how the formalism essentially constructs an extended rigged Hilbert space that supports strong limits of the phase operators and includes their eigenstates. We identify the complementarity structure (consisting of mutually unbiased bases and generators of cyclical permutations) underpinning Pegg and Barnett's general approach that gives a quantum-classical correspondence free of the ambiguity of Dirac's commutator-Poisson bracket correspondence.This article is part of the theme issue 'The quantum theory of light'.