they do not generate rotations, it is suggested that they do correspond to observable continuous values of the OAM and SAM of a non-paraxial transverse field [@EN].

Remarkably, in the helicity representation the matrix components of the operators (\[eqn:2\]) and (\[eqn:3\]) become diagonal. We introduce spherical coordinates $\left(\theta,\phi,k\right)$ with basic vectors $\left( {{\bf e}_\theta  ,{\bf e}_\phi  ,{\bm \kappa }} \right)$ in ${\bf k}$-space, so that the free electric field has only $\left( {{\bf e}_\theta  ,{\bf e}_\phi} \right)$-components. The helicity basis of circular polarizations corresponds to the basic vectors ${\bf e}^ \pm   = e^{\pm im\phi } \left({{\bf e}_{\theta} \pm i{\bf e}_{\phi} }\right)/{\sqrt 2 }$, where $e^{\pm im\phi }$ is an arbitrary gauge factor [@Hawton]. Transition from the global Cartesian field components $( {\tilde E_x ,\tilde E_y ,\tilde E_z } )^T$ to the helicity amplitudes $( {\tilde E^ +  ,\tilde E^ -  ,\tilde E_\parallel  } )^T$ is realized via the local unitary transformation $\hat U\left( {\theta ,\phi } \right) = \hat R_z \left( { - \phi } \right)\hat R_y \left( { - \theta } \right)\hat R_z \left( {m\phi } \right)\hat V $, where $\hat R_a \left( \alpha  \right) = e^{i\alpha \hat S_a }$ is the matrix of rotation by an angle $\alpha$ 