Open-Channel Operator Closure of the Finite-Cutoff JT Gravity Disk Amplitude

TL;DR

This paper develops an operator-level open-channel formulation of the finite-cutoff disk amplitude in JT gravity, clarifying its structure and representations.

Contribution

It provides the first complete operator closure of the finite-cutoff JT gravity disk amplitude, separating geometric data from auxiliary spectral structures.

Findings

Reproduces the known finite-cutoff disk amplitude as a boundary-state matrix element.

Shows the geodesic sector is bandlimited and admits sampled, discrete representations.

Demonstrates the amplitude is not a thermal trace of any single bounded Hamiltonian.

Abstract

The finite-cutoff disk amplitude of Jackiw-Teitelboim (JT) gravity is known from closed-channel spectral methods and finite-cutoff trumpet/cap gluing, while its complete open-channel operator formulation has remained incomplete. In this paper, we provide an operator-level open-channel closure of this known result. More precisely, we separate the data imported from finite-cutoff geometry -- the rigid length --momentum kernel, the disk-trumpet gluing relation, and hence the target cap overlap -- from the structures derived within the parity-even auxiliary problem, namely the Neumann vacuum sector, the generalized eigenbasis, and the branch-projecting spectral functional. When these ingredients are combined, the known finite-cutoff disk amplitude is reproduced as a boundary-state matrix element. We further show that the induced finite-cutoff geodesic sector is bandlimited and therefore admits sampled and branch-doubled discrete representations of the same physical sector, rather than an independent microscopic lattice model. Finally, we show that the resulting compact-support branch-difference amplitude is not the ordinary thermal trace of any single lower-bounded self-adjoint $\beta$-independent Hamiltonian.