Martín-Benito–Mena Marugán–Olmedo prescription for the Dapor-Liegener model of loop quantum cosmology

Abstract

Recently, an alternative Hamiltonian constraint for loop quantum cosmology has been put forward by Dapor and Liegener, inspired by previous work on regularization due to Thiemann. Here, we quantize this Hamiltonian following a prescription for cosmology proposed by Martín-Benito, Mena Marugán, and Olmedo. To this effect, we first regularize the Euclidean and Lorentzian parts of the Hamiltonian constraint separately in the case of a Bianchi I cosmology. This allows us to identify a natural symmetrization of the Hamiltonian which is apparent in anisotropic scenarios. Preserving this symmetrization in isotropic regimes, we then determine the Hamiltonian constraint corresponding to a Friedmann-Lemaître-Robertson-Walker cosmology, which we proceed to quantize. We compute the action of this Hamiltonian operator in the volume eigenbasis and show that it takes the form of a fourth-order difference equation, unlike in standard loop quantum cosmology, where it is known to be of second order. We investigate the superselection sectors of our constraint operator, proving that they are semilattices supported only on either the positive or the negative semiaxis, depending on the triad orientation. Remarkably, the decoupling between semiaxes allows us to write a closed expression for the generalized eigenfunctions of the geometric part of the constraint. This expression is totally determined by the values at the two points of the semilattice that are closest to the origin, namely the two contributions with smallest eigenvolume. This is in clear contrast with the situation found for the standard Hamiltonian of loop quantum cosmology, where only the smallest value is free. This result indicates that the degeneracy of the new geometric Hamiltonian operator is equal to two, doubling the possible number of solutions with respect to the conventional quantization considered until now.

Dapor-Liegener formalism of loop quantum cosmology for Bianchi I spacetimes

“…In contrast with the situation found in Ref. [39], there are now terms with a shift of (plus or minus) four units in v generated by the Lorentzian contribution. The origin of this discrepancy is the different factor ordering chosen for the powers of the volume.…”

“…(A8) and show that the main results of Ref. [39] are robust against this change. Furthermore, we will also consider a general factor ordering for the powers of the volume operator of the kind discussed in the beginning of Sec.…”

“…In this Appendix, we present a reformulation of the Dapor-Liegener model of flat FLRW LQC, originally quantized adopting the MMO prescription in Ref. [39]. The factor ordering of this reformulation makes it directly comparable with the analysis developed in the present article.…”

“…However, all the studies cited above (with the exception of some parts of Ref. [39]) have been performed in homogeneous and isotropic spacetimes. For this reason, it is interesting to try and generalize this formalism to less symmetric scenarios such as anisotropic cosmologies.…”

“…Although these spacetimes have been extensively studied within the framework of LQC, the Lorentzian term of the corresponding Hamiltonian constraint had not been discussed before Ref. [39], where a preliminary analysis of Bianchi I cosmologies was used as the starting point to identify a natural symmetrization that could be respected upon taking the isotropic limit, so as to motivate a preferred factor ordering prescription in flat FLRW spacetimes. In that reference, the Lorentzian part was only regularized classically.…”