Where do bosons actually belong?

TL;DR

This paper advocates for using the su(1,1) algebra instead of the traditional h(1) for describing boson systems, highlighting its advantages in quantization, symmetry, and modeling interactions.

Contribution

It demonstrates that su(1,1) provides a more natural and effective framework for boson systems, including quantization of Maxwell's equations and the Jaynes-Cummings model.

Findings

su(1,1) correctly reproduces Bose-Einstein statistics

su(1,1) emerges naturally from Maxwell's equations quantization

su(1,1) offers advantages in modeling interacting bosons

Abstract

We explore a variety of reasons for considering su(1,1) instead of the customary h(1) as the natural unifying frame for characterizing boson systems. Resorting to the Lie-Hopf structure of these algebras, that shows how the Bose-Einstein statistics for identical bosons is correctly given in the su(1,1) framework, we prove that quantization of Maxwell's equations leads to su(1,1), relativistic covariance being naturally recognized as an internal symmetry of this dynamical algebra. Moreover su(1,1) rather than h(1) coordinates are associated to circularly polarized electromagnetic waves. As for interacting bosons, the su(1,1) formulation of the Jaynes-Cummings model is discussed, showing its advantages over h(1).