On the quantum phase problem

Based on the phase operator introduced by Turski we present a formalism for phase that passes Barnett-Pegg's acid test giving the correct phase variance for a number state. We show that this formalism is in fact the radially integrated Q-function formalism that is used to obtain phase properties. It is also shown that depending on the commutation relation used for phase and number, the phase fluctuations for a coherent state obtained from the integrated Q-function tend to the 1/2ρ2 limit while for the Pegg-Barnett formalism they tend to 1/(4ρ2+3/π2) just like the fluctuations from the integrated Wigner function, where ρ is the amplitude of the coherent state