Deriving loop quantum cosmology dynamics from diffeomorphism invariance

We use the requirement of diffeomorphism invariance in the Bianchi I context to derive the form of the quantum Hamiltonian constraint. After imposing the correct classical behavior and making a certain minimality assumption, together with a certain restriction to "planar loops", we then obtain a unique expression for the quantum Hamiltonian operator for Bianchi I to both leading and subleading orders in $\hbar$. Specifically, this expression is found to exactly match the form proposed by Ashtekar and Wilson-Ewing in the loop quantum cosmology (LQC) literature. Furthermore, by using the projection map from the quantum states of the Bianchi I model to the states of the isotropic model, we constrain the dynamics also in the homogeneous isotropic case, and obtain, again to both leading and subleading order in $\hbar$, a quantum constraint which exactly matches the standard `improved dynamics' of Ashtekar, Pawlowski and Singh. This result in the isotropic case does not require a restriction to planar loops, but only the minimality assumption. Our results strengthen confidence in LQC dynamics and its observational predictions as consequences of more basic fundamental principles. Of the assumptions made in the isotropic case, the only one not rigidly determined by physical principle is the minimality principle, our work also shows the exact freedom allowed when this assumption is relaxed.