  The central role played by pseudodifferential operators in relativistic
dynamics is very well know. In this work, operators as the Schrodinger one
(e.g: square root) are treated from the point of view of the non-local
pseudodifferential Green functions. Starting from the explicit construction of
the Green (semigroup) theoretical kernel, a theorem linking the integrability
conditions and their dependence on the spacetime dimensions is given.
Relativistic wave equations with arbitrary spin and the causality problem are
discussed with the algebraic interpretation of the radical operator and their
relation with coherent and squeezed states. Also we perform by mean of pure
theoretical procedures (based in physical concepts and symmetry) the
relativistic position operator which satisfies the conditions of integrability
: it is non-local, Lorentz invariant and does not have the same problems as the
"local"position operator proposed by Newton and Wigner. Physical examples, as
Zitterbewegung and rogue waves, are presented and deeply analysed in this
theoretical framework.
