Hermitian phase operator properties

The Pegg-Barnett Hermitian phase formalism allows for direct calculations of quantum phase properties of optical fields. As the Hermitian phase operator is defined, one can calculate the expectation value and variance of this operator for a given state /). Moreover, the Pegg-Barnett phase formalism allows for the introduction of the continuous phase probability distribution, which is a representation of the quantum state of the field and describes the phase properties of the field in a very spectacular fashion. For so-called physical states, that is, states of finite energy, the Pegg-Barnett formalism simplifies considerably. In the limit as a —> oo one can introduce the continuous phase distribution... 

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]. 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

Also, since S + is a raising operator, we must have / — ca, where c is some constant. We can evaluate c by use of the normalization of the spin functions and the Hermitian property of Sx and Sy one finds (Problem 1.8) c = h. Choosing the phase of c as zero, we have... 

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