Quantum action-angle variables for harmonic oscillators

The well-known difficulties of defining a phase operator of an oscillator, caused by the lower bound on the number operator, is overcome by enlarging the physical Hilbert space by means of a spin-like, two-valued quantum number. On the enlarged space a phase representation exists on which trigonometric functions of the phase are numbers, and the “number of quanta” is a differential operator. Physical results are recovered by projection on the “upper components.” Coherent states, indeterminacy relations, as well as generalizations to other Hamiltonians, including the quantum analog of the quasi-periodic case, are discussed.