Weyl-Heisenberg algebra

The canonical quantization of the field has introduced by Dirac [1] (see also Refs. 2-4,10,11,14,15,26,27) is provided by the substitution of the photon operators, forming a representation of the Weyl-Heisenberg algebra, into the... 

SU(2) subalgebra (58) in the Weyl-Heisenberg algebra of electric dipole photons cannot be constructed in the way discussed in Section III.B. 

Hence, the operators a in (67) also form a representation of the Weyl-Heisenberg algebra of the electric dipole photons. Employing this transformation (67) then gives the diagonal representation of the operator (63)... 

The preceding results lead to the conclusion that the radiation phase states (72) are dual to the conventional Fock number states nm). In turn, the operators (67) form the representation of the Weyl-Heisenberg algebra of the electric dipole photons dual to the operators am and [46]. 

The Weyl-Heisenberg algebra of multipole photons allows the dual representation in which we deal with the photons with given radiation phase (the 517(2) phase of angular momentum) instead of standard photons with given projection of the angular momentum. 

Now consider the set of Stokes operators that can be obtained by canonical quantization of (132). On the other hand, the Stokes operators should by definition represent the complete set of independent Hermitian bilinear forms in the photon operators of creation and annihilation. It is clear that such a set is represented by the generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of electric dipole photons. The nine generators have the form [46]... 

Let us stress a very important difference between the representations of Stokes operators (137) and (157). If the former is valid only for the electric dipole photons, the latter describes an arbitrary multipole radiation with any X and j. The similarity in the operator structure and quantum phase properties is caused by the same number of degrees of freedom defining the representation of the SU(2) subalgebra in the Weyl-Heisenberg algebra. 

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