Separable Four-dimensional Harmonic Oscillators and Representations of the Poincar\'e 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 Poincar\'e 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 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 Poincar\`e group for relativistic extended particles.
Cite
@article{arxiv.math-ph/9811013,
title = {Separable Four-dimensional Harmonic Oscillators and Representations of the Poincar\'e Group},
author = {Y. S. Kim},
journal= {arXiv preprint arXiv:math-ph/9811013},
year = {2007}
}
Comments
LaTex 10 pages, no figures, presented at the 3rd International Workshop on Classical and Quantum Integrable Systems (Yerevan, Armenia, 1998), to be published in the proceedings