Separable Four-dimensional Harmonic Oscillators and Representations of the Poincaré Group

Abstract

It is possible to construct representations of the Lorentz group using four-dimensional harmonic oscillators. This allows us to construct three-dimensional wave functions with the usual rotational symmetry for space-like coordinates and one-dimensional wave function for time-like coordinate. It is then possible to construct a representation of the Poincaré group for a massive particles having the O(3) internal space-time symmetry in its rest frame. This oscillator can also be separated into two transverse components and the two-dimensional world of the longitudinal and time-like coordinates. The transverse components remain unchanged under Lorentz boosts, while it is possible to construct the squeeze representation of the $O(1,1)$ group in the space of the longitudinal and time-like coordinates. While the squeeze representation forms the basic language for squeezed states of light, it can be combined with the transverse components to form the representation of the Poincarè group for relativistic extended particles.