The Mass Hyperboloid as a Poisson-Lie Group

TL;DR

This paper explores the mass hyperboloid as a Poisson-Lie group, revealing a novel perspective on Lorentz invariance where the mass shell's structure encodes symmetries through non-abelian group operations and Poisson brackets.

Contribution

It introduces the interpretation of the mass hyperboloid as a Poisson-Lie group, connecting Lorentz invariance with Drinfel'd's duality and Lie bi-algebra structures.

Findings

Mass hyperboloid has a Poisson-Lie group structure.

Lorentz invariance is realized through non-abelian group multiplication.

Rotations form the dual group of the hyperboloid in the Drinfel'd sense.

Abstract

The light cone formalism of a massive scalar field has been shown by Dirac to have many advantages. But it is not manifestly Lorentz invariant. We will show that this is a feature not a bug: Lorentz invariance is indeed a symmetry, but in a different sense defined by Drinfel'd. The key idea is that the mass shell (mass hyperboloid) is a Poisson-Lie group: there is a non-abelian group multiplication and non-zero Poisson brackets between components of four-momentum. Rotations form the dual group of the hyperboloid in the sense of Drinfel'd. Infinitesimal Lorentz transformations form a Lie bi-algebra.