Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Pauli Operators

The finite entropy of black holes suggests that local regions of spacetime are described by finite-dimensional factors of Hilbert space, in contrast with the infinite-dimensional Hilbert spaces of quantum field theory. With this in mind, we explore how to cast finite-dimensional quantum mechanics in a form that matches naturally onto the smooth case, especially the recovery of conjugate position/momentum variables, in the limit of large Hilbert-space dimension. A natural tool for this task are the Generalized Pauli operators (GPO). Based on an exponential form of Heisenberg's canonical commutation relation, the GPO offers a finite-dimensional generalization of conjugate variables without relying on any a priori structure on Hilbert space. We highlight some features of the GPO, its importance in studying concepts such as spread induced by operators, and point out departures from infinite-dimensional results (possibly with a cutoff) that might play a crucial role in our understanding of quantum gravity. We introduce the concept of "Operator Collimation," which characterizes how the action of an operator spreads a quantum state along conjugate directions. We illustrate these concepts with a worked example of a finite-dimensional harmonic oscillator, demonstrating how the energy spectrum deviates from the familiar infinite-dimensional case.