On-shell actions with lightlike boundary data

TL;DR

The paper advocates formulating finite-region quantum gravity using boundary data on null hypersurfaces, arguing that such a null-first viewpoint naturally suppresses transverse diffeomorphism redundancies and aligns with holographic expectations while forfeiting ordinary global charges. It develops the on-shell action formalism for classical field theory in null-bounded regions, deriving closed-form expressions for a free scalar and for Maxwell theory, and analyzes gravity to reveal a simple area-based boundary term plus an imaginary corner contribution that reproduces the Bekenstein–Hawking entropy in Lorentzian signature. The results show inherent non-locality and, in specific cases, a holographic reduction of degrees of freedom, suggesting a deep connection between null-boundary observables and finite-region quantum gravity. Together with a discussion of discrete geometries and the implications for quantum gravity, the work provides a concrete classical foundation for exploring finite-region QG and its possible quantum realizations.

Abstract

We argue that finite-region observables in quantum gravity are best approached in terms of boundary data on null hypersurfaces. This has far-reaching effects on the basic notions of classical and quantum mechanics, such as Hamiltonians and canonical conjugates. Such radical properties are not unexpected in finite-region quantum gravity. We are thus motivated to reformulate field theory in terms of null boundary data. As a starting point, we consider the on-shell action functional for classical field theory in finite null-bounded regions. Closed-form results are obtained for free scalars and for Maxwell fields. The action of classical gravity is also discussed, to the extent possible without solving the field equations. These action functionals exhibit non-locality and, in special cases, a "holographic" reduction of the degrees of freedom. Also, they cannot be used to define global charges. Whereas for ordinary field theory these are just artifacts of a restrictive formalism, in quantum gravity they are expected to be genuine features. This further supports a connection between quantum gravity and null-boundary observables. In our treatment of the GR action, we identify a universal imaginary term that reproduces the Bekenstein entropy formula.

Figures (3)

• Figure 1: Some terminology for null-bounded regions. The lines represent codimension-1 lightsheets; the dots represent codimension-2 spacelike surfaces. The arrows indicate the "outgoing" null normal that corresponds to the outgoing covector, for a mostly-plus metric signature. In $d>2$ dimensions with trivial topology (but not in the simple examples in section \ref{['sec:class']}), the left and right sides of the figure are understood to be connected.
• Figure 2: An assignment of boost angles $\eta$ in the Lorentzian plane, according to eq. \ref{['eq:eta']}. This is one of two complex-conjugate choices, distinguished by the sign convention below \ref{['eq:eta']}. The horizontal and vertical axes are $x$ and $t$, respectively. The angles are defined up to integer multiples of $2\pi i$.
• Figure 3: The boost angles from figure \ref{['fig:angles_plane']}, as applied to the null boundary normals from figure \ref{['fig:boundary']}. For clarity, spacelike and timelike normals are included at the equator and tips. They arise if one replaces the corners with smooth curves. Empty circles denote the "signature flip" surfaces, where the boost angle crosses into a neighboring quadrant. The path integral favors placing them infinitesimally close to the tips. The "corner" contributions to the action can be read off by traveling counter-clockwise and picking up angle differences times $A/(8\pi G)$ factors.