A model of quantum gravity with emergent spacetime
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Springer
Received: February 2, 2020 Accepted: May 15, 2020 Published: June 9, 2020
A model of quantum gravity with emergent spacetime
Department of Physics & Astronomy, McMaster University, Hamilton ON, Canada Perimeter Institute for Theoretical Physics, Waterloo ON, Canada
E-mail: [email protected] Abstract: We construct a model of quantum gravity in which dimension, topology and geometry of spacetime are dynamical. The microscopic degree of freedom is a real rectangular matrix whose rows label internal flavours, and columns label spatial sites. In the limit that the size of the matrix is large, the sites can collectively form a spatial manifold. The manifold is determined from the pattern of entanglement present across local Hilbert spaces associated with column vectors of the matrix. With no structure of manifold fixed in the background, the spacetime gauge symmetry is generalized to a group that includes diffeomorphism in arbitrary dimensions. The momentum and Hamiltonian that generate the generalized diffeomorphism obey a first-class constraint algebra at the quantum level. In the classical limit, the constraint algebra of the general relativity is reproduced as a special case. The first-class nature of the algebra allows one to express the projection of a quantum state of the matrix to a gauge invariant state as a path integration of dynamical variables that describe collective fluctuations of the matrix. The collective variables describe dynamics of emergent spacetime, where multi-fingered times arise as Lagrangian multipliers that enforce the gauge constraints. If the quantum state has a local structure of entanglement, a smooth spacetime with well-defined dimension, topology, signature and geometry emerges at the saddle-point, and the spin two mode that determines the geometry can be identified. We find a saddle-point solution that describes a series of (3 + 1)-dimensional de Sitter-like spacetimes with the Lorentzian signature bridged by Euclidean spaces in between. The phase transitions between spacetimes with different signatures are caused by Lifshitz transitions in which the pattern of entanglement is rearranged across the system. Fluctuations of the collective variables are described by bi-local fields that propagate in the spacetime set up by the saddle-point solution. Keywords: AdS-CFT Correspondence, Models of Quantum Gravity, Space-Time Symmetries ArXiv ePrint: 1912.12291
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)070
JHEP06(2020)070
Sung-Sik Lee
Contents 1 Introduction 1.1 Conceptual overview 1.2 Outline
1 3 7 9 9 9 11
3 Gauge symmetry 3.1 Review of the Hamiltonian formalism of the general relativity 3.2 Momentum constraint 3.3 Hamiltonian constraint 3.4 First-class constraint algebra
13 13 15 17 20
4 Path integral representation of state projection 4.1 Projection 4.2 Gauge invariant local structure 4.3 Path integration of collective variables 4.4 Gauge invariance of the action
22 22 23 24 26
5 Spacetime diffeomorphism and emergent g
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