Noncommutative spaces as quantized constrained Hamiltonian systems

TL;DR

This paper explores how the strong-field limit of a charged particle in an electromagnetic field can be modeled as a constrained Hamiltonian system, revealing a connection to noncommutative geometry and fuzzy spheres.

Contribution

It establishes a novel link between constrained Hamiltonian dynamics and noncommutative geometry using a toy model of a charged particle in an electromagnetic field.

Findings

Constraints depend on the rank of the field strength tensor.

Quantization leads to noncommuting coordinate operators.

Magnetic monopole fields produce fuzzy sphere geometries.

Abstract

We investigate the strong-field limit of a charged particle in an electromagnetic field as a toy model for general covariant systems, establishing a novel connection between constrained Hamiltonian dynamics and noncommutative geometry. Starting from the action $S=\int d\tau \, \dot{x}^i A_i(x)$, which represents the holonomy of the particle's path with respect to the electromagnetic potential $A_i$, we analyze the resulting general covariant system with vanishing Hamiltonian. The equations of motion $F_{ij}\dot{x}^j=0$ confine the particle to leaves of a singular foliation defined by the field strength tensor $F_{ij}=\partial_i A_j -\partial_j A_i$. We show that the physical state space corresponds to the space of leaves of this foliation, with points connected by field lines being gauge equivalent. The Hamiltonian analysis reveals constraints $\kappa_i=p_i-A_i$ that are locally classified as first-class or second-class depending on the rank of the field strength tensor. Upon quantization, this leads to noncommuting coordinate operators, establishing the physical state space as a noncommutative geometry. We provide explicit examples and show in particular that the magnetic monopole field strength yields a fuzzy sphere.