Hilbert space Hermitian phase operator

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

Polar decomposition of the field amplitude, as in (36), which is trivial for classical fields becomes far from being trivial for quantum fields because of the problems with proper definition of the Hermitian phase operator. It was quite natural to associate the photon number operator with the intensity of the field and somehow construct the phase operator conjugate to the number operator. The latter task, however, turned out not to be easy. Pegg and Barnett [11-13] introduced the Hermitian phase formalism, which is based on the observation that in a finite-dimensional state space, the states with well-defined phase exist [14]. Thus, they restrict the state space to a finite (cr + l)-dimensional Hilbert space H-+ spanned by the number states 0), 1),. .., a). In this space they define a complete orthonormal set of phase states by... 

The generalized phase CS, p,0o)(s), and truncated phase CS, j3,0o)(s), are associated with the Pegg-Bamett formalism of the Hermitian phase operator S. The operators 4>s, Hilbert space Thus the generalized and truncated phase CS are properly defined only in of finite dimension. States p,0o)( and p, 0o)(s), similar to a)(s) and a)(s), approach each other for p 2 = p 2 < C s/n [20]. This can be shown explicitly by calculating the scalar product between generalized and truncated phase CS. We find (p = p)... 

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]. 

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