Noncommutative spaces as quantized constrained Hamiltonian systems

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.