Hamiltonian Analysis of f ( Q ) $f(\mathbb {Q})$ Gravity and the Failure of the Dirac–Bergmann Algorithm for Teleparallel Theories of Gravity

Abstract

In recent years, $f\left(Q\right)$ gravity has enjoyed considerable attention in the literature and important results have been obtained. However, the question of how many physical degrees of freedom the theory propagates—and how this number may depend on the form of the function f—has not been answered satisfactorily. In this article it is shown that a Hamiltonian analysis based on the Dirac-Bergmann algorithm—one of the standard methods to address this type of question—fails. The source of the failure is isolated and it is shown that other commonly considered teleparallel theories of gravity are affected by the same problem. Furthermore, it is pointed out that the number of degrees of freedom obtained in Phys. Rev. D 106 no. 4, (2022) by K. Hu, T. Katsuragawa, and T. Qui (namely eight), based on the Dirac-Bergmann algorithm, is wrong. Using a different approach, it is shown that the upper bound on the degrees of freedom is seven. Finally, a more promising strategy for settling this important question is proposed.