Abstract The authors previously introduced a diffeomorphisminvariant definition of a homogeneous and isotropic sector of loop quantum gravity (LQG), along with a program to embed loop quantum cosmology (LQC) into it. The present paper works out that program in detail for the simpler, but still physically nontrivial, case where the target of the embedding is the homogeneous, but not isotropic, Bianchi I model. The diffeomorphisminvariant conditions imposing homogeneity and isotropy in the full theory reduce to conditions imposing isotropy on an already homogeneous Bianchi I spacetime. The reduced conditions are invariant under the residual diffeomorphisms still allowed after gauge fixing the Bianchi I model. We show that there is a unique embedding of the quantum isotropic model into the homogeneous quantum Bianchi I model that (a) is covariant with respect to the actions of such residual diffeomorphisms, and (b) intertwines both the (signed) volume operator and at least one directional Hubble rate. That embedding also intertwines all other operators of interest in the respective loop quantum cosmological models, including their Hamiltonian constraints. It thus establishes a precise equivalence between dynamics in the isotropic sector of the Bianchi I model and the quantized isotropic model, and not just their kinematics. We also discuss the adjoint relationship between the embedding map defined here and a projection map previously defined by Ashtekar and WilsonEwing. Finally, we highlight certain features that simplify this reduced embedding problem, but which may not have direct analogues in the embedding of homogeneous and isotropic LQC into full LQG.
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What Are Observables in Hamiltonian EinsteinMaxwell Theory?
Is change missing in Hamiltonian Einstein–Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each firstclass constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (‘Stueckelberg’) are inequivalent to those of massive electromagnetism without gauge freedom (‘Proca’). The alternative Pons–Salisbury–Sundermeyer definition of observables, aiming for Hamiltonian–Lagrangian equivalence, uses the gauge generator G, a tuned sum of firstclass constraints, rather than each firstclass constraint separately, and implies equivalent observables for equivalent massive electromagnetisms. For General Relativity, G generates 4dimensional Lie derivatives for solutions. The Lie derivative compares different spacetime points with the same coordinate value in different coordinate systems, like 1 a.m. summer time versus 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4dimensional Lie derivative (not 0) for observables. These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein–Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein–Yang–Mills. For Einstein–Maxwell, both 𝐹𝜇𝜈 and 𝑔𝜇𝜈 are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under 4dimensional coordinate transformations. Using the bifurcated definition, these quantities count as observables, as one would expect on nonHamiltonian grounds.
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 Award ID(s):
 1734402
 NSFPAR ID:
 10209267
 Date Published:
 Journal Name:
 Foundations of physics
 Volume:
 49
 ISSN:
 15729516
 Page Range / eLocation ID:
 786796
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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