Algorithmic approach to Cosmological Coherent State Expectation Values in LQG

Abstract

In the lattice approach to Loop Quantum Gravity on a fixed graph computations tend to be involved and are rarely analytically manageable. But, when interested in the expectation values of coherent states on the lattice which are sharply peaked on isotropic, flat cosmology several simplifications are possible which reduce the computational effort. We present a step-bystep algorithm resulting in an analytical expression including up to first order corrections in the spread of the state. The algorithm is developed in such a way that it makes the computation straightforward and easy to be implementable in programming languages such as Mathematica.Exemplarily, we demonstrate how the algorithm streamlines the road to obtain the expectation value of the euclidean part of the scalar constraint and as a consistency check perform the analytic computation as well. To showcase further applications of the algorithm, we investigate the fate of the effective dynamics program custom in Loop Cosmology and find that the next-to-leading order corrections can not be used as corrections for an effective Hamiltonian.

Quantum speed limit and stability of coherent states in quantum gravity

“…Building on this result, a feasibly implementable algorithm which allows to compute expectation values in Ψ t z including the next-to-leading order corrections for arbitrary (polynomial) operators has very recently been introduced and tested [19]. With this algorithm it is possible to compute the expectation values of complex and nested operators, such as the Thiemannregularization of the Euclidean part of the scalar constraint…”

“…While numerous applications of QSL are known in the literature [25][26][27][28][29][30][31][32], this tool has never before been introduced to Quantum Gravity. However, as our algorithm [19] in principle allows to compute the variance of the Hamiltonian constraint, Eq. ( 12) becomes practically meaningful.…”

“…Before the era of quantum simulations in LQG, preparatory work aiming at making complex objects in LQG computable (by both classical and quantum machines) shall be in focus. As a part of efforts directed towards this objective, we have recently established an algorithm [19] offering a severe simplification of the computations with the coherent states in LQG. In this letter we employ these complex techniques to explore new physics within LQG.…”