A canonical formalism of f ( R )-type gravity in terms of Lie derivatives

Abstract

A canonical formalism of f (R)-type gravity is proposed, resolving the problem in the formalism of Buchbinder and Lyakhovich(BL). The new coordinates corresponding to the time derivatives of the metric are taken to be its Lie derivatives which is the same as in BL. The momenta canonically conjugate to them and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that our method surely resolves the problem of BL.

Hamiltonian formulation of Palatinif(R)theories à la Brans-Dicke theory

“…This situation changed recently [9], when it was shown that a certain Palatini Lagrangian could faithfully reproduce the effective dynamics of loop quantum cosmology (LQC) [10], a Hamiltonian-based approach to quantum cosmology inspired by the nonperturbative quantization techniques of loop quantum gravity [11]. Though the Hamiltonian description of metric fðRÞ theories has been discussed several times and from different points of view in recent literature [12][13][14], a discussion of the Palatini case is still missing. The main purpose of this paper, therefore, will be the study of the Hamiltonian formulation of Palatini fðRÞ theories.…”

Cosmological constraints on f(R) gravity theories within the Palatini approach

“…These models are called nonlinear theories of gravity (Magnano & Sokolowski 1994;Allemandi et al 2006) or f (R) theories (Bronnikov & Chernakova 2005;Cognola et al 2005;Nunez & Solganik 2004;Ezawa et al 2003;Barraco et al 2000;Schmidt 1998;Rippl et al 1996;Barrow & Ottewill 1983), since the scalar curvature R in the EinsteinHilbert Lagrangian is replaced by a general function f (R). The main motivation for this generalization is that higher-order terms in curvature invariants (such as R 2 , R µν R µν , R µναβ R µναβ , etc.)…”

The Equivalence Theorem in the Generalized Gravity of f(R)-Type and Canonical Quantization

“…Since, in that case, changes of variables are restricted to those along the paths of motions, which does not fit to Poisson brackets which use changes in any direction. Nevertheless, we can show, using (6.30), that assumption that all of the equations (6.10), (6.23a,b) hold leads to contradiction (Ezawa et al, 2006(Ezawa et al, , 2010. In other words, two frames are not related by a canonical transformation.…”

“…This simplicity could be added to the advantages of f (R)-type gravity. Quantum aspects of the theorem are then summarized when we quantize the theory canonically in the framework of the generalized Ostrogradski formalism (Ezawa et al, 2006) which is a natural generalization to the system in a curved spacetime. The main result is that if the f (R)-type theory is quantized canonically, Einstein gravity with a scalar field has to be quantized non-canonically.…”