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Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra

Singh, Ashmeet and Carroll, Sean M. (2018) Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra. . (Submitted)

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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 is the generalized Clifford algebra (GCA). Based on an exponential form of Heisenberg's canonical commutation relation, the GCA offers a finite-dimensional generalization of conjugate variables without relying on any a priori structure on Hilbert space. We highlight some features of the GCA, its importance in studying concepts such as locality of 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 "Schwinger locality," 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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Singh, Ashmeet0000-0002-4404-1416
Carroll, Sean M.0000-0002-4226-5758
Additional Information:We would like to thank Anthony Bartolotta, ChunJun (Charles) Cao, Aidan Chatwin-Davies, Swati Chaudhary, Prof. R. Jagannathan and Jason Pollack for helpful discussions during the course of this project. This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech and by DOE grant DE SC0011632.
Group:Walter Burke Institute for Theoretical Physics
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Department of Energy (DOE)DE-SC0011632
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Record Number:CaltechAUTHORS:20180710-134008015
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87711
Deposited By: Joy Painter
Deposited On:10 Jul 2018 20:51
Last Modified:03 Oct 2019 19:58

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