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Towards Space from Hilbert Space: Finding Lattice Structure in Finite-Dimensional Quantum Systems

Pollack, Jason and Singh, Ashmeet (2019) Towards Space from Hilbert Space: Finding Lattice Structure in Finite-Dimensional Quantum Systems. Quantum Studies: Mathematics and Foundations, 6 (2). pp. 181-200. ISSN 2196-5609. doi:10.1007/s40509-018-0176-8.

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Field theories place one or more degrees of freedom at every point in space. Hilbert spaces describing quantum field theories, or their finite-dimensional discretizations on lattices, therefore have large amounts of structure: they are isomorphic to the tensor product of a smaller Hilbert space for each lattice site or point in space. Local field theories respecting this structure have interactions which preferentially couple nearby points. The emergence of classicality through decoherence relies on this framework of tensor-product decomposition and local interactions. We explore the emergence of such lattice structure from Hilbert-space considerations alone. We point out that the vast majority of finite-dimensional Hilbert spaces cannot be isomorphic to the tensor product of Hilbert-space subfactors that describes a lattice theory. A generic Hilbert space can only be split into a direct sum corresponding to a basis of state vectors spanning the Hilbert space; we consider setups in which the direct sum is naturally decomposed into two pieces. We define a notion of direct-sum locality which characterizes states and decompositions compatible with Hamiltonian time evolution. We illustrate these notions for a toy model that is the finite-dimensional discretization of the quantum-mechanical double-well potential. We discuss their relevance in cosmology and field theory, especially for theories which describe a landscape of vacua with different spacetime geometries.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Pollack, Jason0000-0003-4754-4905
Singh, Ashmeet0000-0002-4404-1416
Additional Information:© 2018 Chapman University. Received: 17 July 2018; Accepted: 22 November 2018; First Online: 04 December 2018. We are thankful to ChunJun (Charles) Cao, Sean Carroll, Tamir Hemo, and Ingmar Saberi for helpful discussions. J.P. is supported in part by the Simons Foundation and in part by the Natural Sciences and Engineering Research Council of Canada. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632, as well as by the Walter Burke Institute for Theoretical Physics at Caltech and the Foundational Questions Institute.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Simons FoundationUNSPECIFIED
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Foundational Questions Institute (FQXI)UNSPECIFIED
Subject Keywords:Quantum mechanics; Emergent spacetime; Decoherence; Quantum field theory; Quantum gravity
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Issue or Number:2
Record Number:CaltechAUTHORS:20180228-095233162
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Official Citation:Pollack, J. & Singh, A. Quantum Stud.: Math. Found. (2019) 6: 181.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:84999
Deposited By: Joy Painter
Deposited On:28 Feb 2018 18:05
Last Modified:15 Nov 2021 20:24

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