Separable Four-dimensional Harmonic Oscillators and Representations of the Poincar\'e Group

TL;DR

This paper develops a novel framework using four-dimensional harmonic oscillators to construct Poincaré group representations, enabling a unified description of relativistic particles with internal symmetries and squeezed states.

Contribution

It introduces a new oscillator-based method to represent the Poincaré group, incorporating internal symmetries and Lorentz transformations for relativistic particles.

Findings

Constructed Lorentz group representations with 4D harmonic oscillators.

Separated oscillators into transverse and longitudinal components under Lorentz boosts.

Connected squeezed states of light with relativistic particle representations.

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 $O(1,1)$ 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 Poincar\`e group for relativistic extended particles.