Relativistic internal time operator

Abstract

We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincare group. The Lie algebra generated by the time operator and the generators of the Poincare group turns out to be an infinitedimensional extension of the Poincare algebra. The internal time operator generates two new entities, namely the velocity operator and the internal position operator. The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory. This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator. The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold.