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

Abstract

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.

arXiv comment23 pages, 5 figures, comments welcome. v2 contains some change in terminology, added references

References in corpus (23)

• The World as a Hologram
• A bound on chaos
• Encoding a qubit in an oscillator
• A Generalized Uncertainty Principle in Quantum Gravity
• A new proof for the existence of mutually unbiased bases
• Quantum Gravity In De Sitter Space
• Discrete phase space based on finite fields
• On the Consistent Histories Approach to Quantum Mechanics
• Localization techniques in quantum field theories
• Entropy Bounds and Black Hole Remnants
• Cosmological Breaking of Supersymmetry?
• Consciousness as a State of Matter
• The Hilbert Space of Quantum Gravity Is Locally Finite-Dimensional
• Quantum-first gravity
• Glimmers of a pre-geometric perspective
• Cosmic Equilibration: A Holographic No-Hair Theorem from the Generalized Second Law
• Superintegrability of Geodesic Motion on the Sausage Model
• Quantum mechanics in finite dimensional Hilbert space
• General phase spaces: from discrete variables to rotor and continuum limits
• Optimal uncertainty relations in a modified Heisenberg algebra
• Geometric formulation of quantum mechanics
• Notes on the Qubit Phase Space and Discrete Symplectic Structures
• Quantum gravity tomography

Cited by in corpus (6)

• Flow of time during energy measurements and the resulting time-energy uncertainty relations
• Toolkit for Scalar Fields in Universes with finite-dimensional Hilbert Space
• Noncommutative Gauge Theories and Gravity
• Experimental characterization of quantum processes: a selective and efficient method in arbitrary finite dimension
• Spacetime granularity from finite-dimensionality of local observable algebras
• Gauge Theories on Fuzzy Spaces and Gravity