Emergence of Riemannian Quantum Geometry

Haggard, Hal M.; Lewandowski, Jerzy; Sahlmann, Hanno

doi:10.1007/978-981-19-3079-9_96-1

Cited by 1 publication

(3 citation statements)

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“…The non-trivial step ahead is to identify the secondary constraints necessary to guarantee the stability of the primary conditions under the Hamiltonian flow, i.e. imposing {H, Im(E i I )} = 0 at the discretized level 17 . What makes these ideas appealing is that they strongly resonate with the construction of the spinfoam vertex amplitudes, which, at least in three-dimensions, are discrete solutions to the Wheeler-De Witt equation on an elementary simplex [37,46].…”

Section: Outlook and Discussionmentioning

confidence: 99%

“…No new data were created or analysed in this study. 17 In the continuum, the reality conditions impose constraints on both Ashtekar the electric field and the connection. The condition on the electric field is that the imaginary (boost) part of Ẽi a vanishes, the condition on the Ashtekar connection is more intricate.…”

Section: Data Availability Statementmentioning

confidence: 99%

“…The physical Hilbert space is the kernel of the now discretized constraints. A concrete realization of this idea can be found in the loop gravity and spinfoam approaches to quantum gravity [2,3,[14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Simplicial graviton from selfdual Ashtekar variables

Wieland

2023

Class. Quantum Grav.

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In perturbative gravity, it is straight-forward to characterize the two local degrees of freedom of the gravitational field in terms of a mode expansion of the linearized perturbation. In the non-perturbative regime, we are in a more difficult position. It is not at all obvious how to construct Dirac observables that can separate the gauge orbits. Standard procedures rely on asymptotic boundary conditions or formal Taylor expansions of relational observables. In this paper, we lay out a new non-perturbative lattice approach to tackle the problem in terms of Ashtekar's self-dual formulation. Starting from a simplicial decomposition of space, we introduce a local kinematical phase space at the lattice sites. At each lattice site, we introduce a set of constraints that replace the generators of the hypersurface deformation algebra in the continuum. We show that the discretized constraints close under the Poisson bracket. The resulting reduced phase space describes two complex physical degrees of freedom representing the two radiative modes at the discretized level. The paper concludes with a discussion of the key open problems ahead and the implications for quantum gravity.

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Section: Outlook and Discussionmentioning

confidence: 99%

Section: Data Availability Statementmentioning

confidence: 99%