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

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