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, 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.

Hamiltonian analysis of metric-affine-R 2 theory

“…Motivated by the question of the number and character of degrees of freedom in general metric-affine theories, and the connection to projective-and Weyl-symmetry, in this work we worked out the technical details for the Hamiltonian constraint analysis necessary to address this question. It is particularly relevant to demonstrate that this approach works given the recent report claiming that the Dirac-Bergmann algorithm, that the Hamiltonian constraint analysis is based on, does not work for all field theories [38]. We encounter no obstructions to this algorithm.…”

“…Progress has recently been made towards automating this full nonlinear analysis of such theories using computational algebra [32]. Hamiltonian analysis has also been applied to so-called teleparallel theories that do not contain the metric, but are either built only out of torsion [33][34][35][36] (dubbed f (T ) theories), or only out of non-metricity [37][38][39][40][41] (dubbed f (Q) theories).…”

Cosmological teleparallel perturbations

“…Phys. 71 (2023) 2300185 [42]. The apparent loss of these additional degrees of freedom in both flat and curved de Sitter space indicates a strong coupling problem of symmetric teleparallism in those backgrounds.…”

“…Phys. 71 (2023) 2300185 [42]. Note that with our choice of solving for perturbations it is precisely the spatial metric (with six components) and the lapse (one component) that are dynamical.…”

“…To cover a wide range of cosmological backgrounds we consider both spatially flat and curved geometries, as well as pure vacuum with the cosmological expansion driven by gravity alone, as well as with a matter source in the form of a scalar field with arbitrary potential. A particular focus is placed on discerning the number of propagating gravitational perturbation variables, which for f (T) and f (Q) gravity are strongly suspected to be more than the two tensor modes represented in GR [40][41][42]. Moreover, f (T) gravity seems to exhibit a strong coupling problem in Minkowski space [43,44] and cosmological backgrounds [45], and we wish to extend this analysis to f (G) and f (Q) gravity and spatially curved cosmologies including not only the metric perturbations but the connection perturbations as well.…”