Phase operator for the photon , an index theorem , and quantum anomaly †

The quantum phase operator has been studied by various authors in the past[1-4]. We here remark on the absence of a hermitian phase operator and the lack of a mathematical basis for ∆N∆φ ≥ 1/2, on the basis of a notion of index or an index theorem. To be specific, we study the simplest one-dimensional harmonic oscillator defined by h = aa+ 1/2 where a and a stand for the annihilation and creation operators satisfying the standard commutator [a, a] = 1. The vacuum state |0〉 is annihilated by a, a|0〉 = 0 which ensures the absence of states with negative norm. The number operator defined by N = aa then has non-negative integers as eigenvalues, and the annihilation operator a is represented by a = |0〉〈1|+ |1〉〈2| √ 2 + |2〉〈3| √ 3 + .... (1)