position operators describe spin-dependent part of OAM (responsible for spin-to-orbital AM conversion) and spin-dependent shift of the center of gravity of light (i.e., the spin-Hall effect of light). We have applied the general theory to symmetric and asymmetric vector Bessel beams and found that our non-canonical operators indeed correspond to the observable quantities. The obtained Bessel-beam intensity distributions exhibit fine SOI splitting of caustics and Hall effects of light in perfect agreement with the derived OAM and position operators. These effects can be observed experimentally in tightly focused fields.

This work was supported by the European Commission (Marie Curie Action), Science Foundation Ireland (Grant No. 07/IN.1/I906), the Australian Research Council (ARC), and von Humboldt foundation. We are grateful to A. Y. Bekshaev and M. V. Berry for fruitful discussions.

Operator formalism for wave packets and beams
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One can separate two basic situations, for which the operator formalism of Section II can be adopted in a slightly different way. The first one is evolution of a wave-packet-like field localized in 3D space. Obviously, such field is nonmonochromatic and time-dependent. The plane-wave Fourier decomposition of the complex electric field can be written as $$\label{eqn:A1}
{\bf E}\left( {\bf r},t \right) = \frac{g}{(2\pi)^{3/2} 