Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR
The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).