Abstract
The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic cosmology is a canonical realization, while the group-field version is a bosonic realization. The existence of a large number of bosonic realizations suggests generalizations of models in group field cosmology.
Highlights
Consider a dynamical system given by a real variable, V, and a complex variable, J, with
We conclude that the paradigmatic bounce model of loop quantum cosmology, analyzed numerically in [19], is a canonical realization of harmonic cosmology
In loop quantum cosmology [15], the Hubble parameter is obtained from a component of the Ashtekar connection, which, as in the full theory of loop quantum gravity, is not represented directly as an operator, but only indirectly through holonomies [16,17]
Summary
Consider a dynamical system given by a real variable, V, and a complex variable, J, with. The third (real) variable, ReJ, is not independent provided we fix the value of the Casimir R = V 2 − | J |2 of the Lie algebra su(1, 1) given by the brackets (1). Displays the paradigmatic behavior of the volume of a bouncing universe model This construction defines harmonic cosmology [1,2]. The Hamiltonian (5) allows us to derive a relationship between φ and proper time τ, measured by co-moving observers in an isotropic cosmological model. Proper-time equations of motion are determined by Poisson brackets with the Hamiltonian constraint, to which (5) provides the matter contribution. Which, upon relating P = ȧ/(4πGa) to the Hubble parameter and V to the scale factor cubed, is equivalent to the Friedmann equation of an isotropic, spatially-flat model sourced by a free, massless scalar field with momentum pφ : ρφ
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