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.
There has been recently renewed interest in the quantisation of gravity by considering local subsystems on light-like hypersurfaces. The main purpose of this paper is to present a theory-independent perspective on these developments assuming only basic knowledge of quantum theory and general relativity. In addition, we present a top-down approach to constructing local amplitudes in causal diamonds. The fundamental building block is a slab of light-like geometry (e.g. a segment of a light cone embedded into spacetime). Each slab is a three-dimensional light-like hypersurface bounded by two cuts, its past and future corners. After briefly reviewing the timeless formalism of quantum theory, we equip each null slab a with a kinematical Hilbert space that factorizes into constituents associated to the three-dimensional interior and its two corners. Upon assuming the existence of vacuum states for the bulk and boundary symmetries and a fundamental projector onto physical states, we explain how to introduce local amplitudes by contracting boundary states according to the pattern of a causal diamond. Finally, we show that the resulting local transition amplitudes satisfy Ward identities and charge conservation for the underlying symmetries. The paper closes with a summary and discussion for how different approaches to quantum gravity can realise our proposal in practice.