Galilei particles revisited

Abstract

We revisit the classifications of classical and quantum galilean particles: that is, we fully classify homogeneous symplectic manifolds and unitary irreducible projective representations of the Galilei group. Equivalently, these are coadjoint orbits and unitary irreducible representations of the Bargmann group, the universal central extension of the Galilei group. We provide an action principle in each case, discuss the nonrelativistic limit, as well as exhibit, whenever possible, the unitary irreducible representations in terms of fields on Galilei spacetime. Motivated by a forthcoming study of planons we pay close attention to the mobility of the less familiar massless Galilei particles.

Figures (1)

• Figure 1: This figure contrasts the energy-momentum orbits $(E,\boldsymbol{p})$ of the Poincaré (left) and Galilei (right) group. For further details we refer to Section \ref{['sec:particle-dynamics']}, cf., also with Table \ref{['tab:coadjoint-orbits']}. The Poincaré orbits are foliated by hypersurfaces of the form $E^{2}- \Vert\boldsymbol{p}\Vert^2 =m^{2}$. For $m^{2}>0$ this leads to the massive orbits with positive and negative energy (green), for $m=0$ to the massless orbits (yellow) and for $m^{2}<0$ to tachyonic orbits (gray). For the case of Galilei the energy-momentum orbits depend on the mass $m$ and we picture three plots for fixed positive, vanishing and negative mass. For positive and negative $m$ the energy-momentum orbits are hypersurfaces $E-\frac{\Vert\boldsymbol{p}\Vert^2}{2m}=E_{0}$, where $E_{0}$ shifts the parabolas along the energy axis. For vanishing mass $m=0$ the Galilei orbits are foliated by cylinders $\|\boldsymbol{p}\|=p_0>0$ and when $\|\boldsymbol{p}\|=0$ the orbits consist of disjoint points $E=E_{0}$ (pictured as a black line).