Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity

TL;DR

The paper investigates the quantum mechanics of a point particle in 2+1-dimensional gravity by performing a Hamiltonian reduction that yields the finite-dimensional phase space $\mathcal{P} \simeq T_*\mathsf{SL}(2)$. Momentum becomes group-valued on $\mathsf{SL}(2)$, while position coordinates do not commute, producing a noncommutative spacetime and a discretized notion of time with a Planck-scale step $\ell_{\mathrm{P}}$. The dynamics are governed by a deformed Poincaré symmetry with deformed translations, encapsulated by a mass-shell constraint that leads to a discretized Klein–Gordon equation on a lattice-like time evolution. The analysis, supported by harmonic analysis on $\mathsf{SL}(2)$, reveals a semi-discrete spacetime structure and a spectrum of spacetime observables, highlighting gravitational corrections to the relativistic point-particle quantum mechanics.

Abstract

We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation.

Figures (3)

• Figure 1:
• Figure 2: The embedding of a neighbourhood of the cylindrical boundary of spacetime (to the left) into the background Minkowski space (to the right). Thereby, the boundary at $r=0$ is mapped onto the world line pointing into the direction of $\mathb{p}$, the two sides of the cut at $\varphi=\pm\pi$ are mapped onto the surfaces $\theta_\pm$, and the ADM surface of constant $t$ is mapped onto a surface of constant $f_0$.
• Figure 3: The spectrum of quantum spacetime. Each circle in this diagram corresponds to a tower of states labeled by the spatial angular momentum quantum number $n$. The position of the circles in Minkowski space are determined by the quantum numbers $t$ and $s^2$.