Fermionisation of the Aharonov--Bohm Phase on the Lightfront

TL;DR

This work analyzes the Maxwell phase space on a lightlike hypersurface and constructs the covariant phase-space Poisson algebra of $U(1)$ holonomies (Wilson lines) on null initial data. A ribbon regularisation reveals that holonomies along hoops intersecting the same light ray have geometry-dependent structure constants set by the conformal class of the boundary metric, in contrast to the spacelike case where holonomies commute. The authors propose a lattice-type quantisation in which holonomies become anti-commuting Grassmann variables, with a vacuum that depends on the chosen framing and with a continuous interpolation between fermionic and bosonic relations for different intersections. This fermionisation hints at deep links between null surface holography, edge modes, and two-dimensional conformal structures on light rays, and it provides a concrete path to nonperturbative quantisation in a null setting.

Abstract

We consider the phase space of the Maxwell field as a simplified framework to study the quantisation of holonomies (Wilson line operators) on lightlike (null) surfaces. Our results are markedly different from the spacelike case. On a spacelike surface, electric and magnetic fluxes each form a commuting subalgebra. This implies that the holonomies commute. On a lightlike hypersurfaces, this is no longer true. Electric and magnetic fluxes are no longer independent. To compute the Poisson brackets explicitly, we choose a regularisation. Each path is smeared into a thin ribbon. In the resulting holonomy algebra, Wilson lines commute unless they intersect the same light ray. We compute the structure constants of the holonomy algebra and show that they depend on the geometry of the intersection and the conformal class of the metric at the null surface. Finally, we propose a quantisation. The resulting Hilbert space shows a number of unexpected features. First, the holonomies become anti-commuting Grassmann numbers. Second, for pairs of Wilson lines, the commutation relations can continuously interpolate between fermionic and bosonic relations. Third, there is no unique ground state. The ground state depends on a choice of framing of the underlying paths.

Figures (2)

• Figure 1: We consider $U(1)$ Wilson lines along elementary hoops on a null hypersurface $\mathcal{N}$. Any such hoop $\gamma$ consists of two lightlike segments and a spacelike path $\hat{\gamma}$. The corners $p_0$ and $p_1$ are where the different segments join. The projection of the path onto $\mathcal{C}_-$ is $\check{\gamma}=\pi\circ\hat{\gamma}$, where $x_{0,1}=\pi(p_{0,1})$. The images under the projection $\pi$ are what we call shadows, i.e. $\check{\gamma}=\pi(\hat{\gamma})$ is the shadow of $\hat{\gamma}$. Dotted lines are light rays tangential to $\mathcal{N}$.
• Figure 2: The Poisson brackets between two $U(1)$ holonomies for hoops $\gamma_+,\gamma_-\subset\mathcal{N}$ vanish unless the paths $\gamma_\pm$ intersect with the same light ray (light rays are dashed lines in the left panel). To regularise the Poisson brackets between $U(1)$ holonomies, we extend every path into a two-dimensional ribbon (right panel). The calculation of the Poisson algebra for the smeared $U(1)$ parallel transport reduces to a two-dimensional integral at the base manifold, i.e. the initial cut $\mathcal{C}_-$. The integral vanishes unless the shadows of the paths $\hat{\gamma}_\pm$ intersect.