Hilbert space phase properties, operators

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

We also note that, in contrast to the Pegg-Bamett formalism [45], we consider an extended space of states, including the Hilbert-Fock state of photons as well as the space of atomic states [36,46,53,54]. The quantum phase of radiation is defined, in this case, by mapping of corresponding operators from the atomic space of states to the whole Hilbert-Fock space of photons. This procedure does not lead to any violation of the algebraic properties of multipole photons and therefore gives an adequate picture of quantum phase fluctuations [46],... 

There is a formal similarity between this equation and Eq. (9.32) the Poisson bracket in the latter is replaced by the a commutator in the former. Poisson brackets and commutators have a number of properties in common. They are Lie brackets, which means they are linear, are antisymmetric, and satisfy the Jacobi identity. Both 7f, p) and —j[H, p] can be viewed as an operations on the probability density. But here the similarity ends. The probability densities themselves hve in completely different spaces. The first is a function in classical phase space, and the second is an operator on a Hilbert state space, ft has up to now not been possible to unify those two spaces, so that, for instance, a correct classical limit can be taken. 

---