Formalismo Hamiltoniano no contexto da quebra de difeomorfismos espontânea

Abstract

The aim of this work is, starting from the fact that General Relativity is a classical theory of constraints, to develop the Hamiltonian formulation for an effective model of modified gravity that incorporates a dynamic scalar field, denoted as u(x), contained in the minimal gravitational sector of the Standard Model Extension. The presence of such a scalar field spontaneously breaks one of the fundamental symmetries of General Relativity, namely diffeomorphism symmetry. This, in turn, is considered one of the most prominent signals for physical effects at the Planck scale. To construct the Hamiltonian, we will make use of the (3 + 1) decomposition, developed by Arnowitt-Deser-Misner, which consists of foliating the four-dimensional spacetime into three-dimensional spacelike hypersurfaces of constant time. This foliation is governed by the lapse functions N and the shift vectors Ni , which represent gauge degrees of freedom. On each hypersurface, a set of canonical variables is defined, to which we associate a set of conjugate momenta, leading to the

derivation of the canonical Hamiltonian through the Legendre transformation, which al- lows for a description of the gravitational phase space. Once the Hamiltonian is obtained,

it is necessary to analyze constraints and, consequently, obtain the equations of motion associated with the background scalar field, the induced metric on the hypersurface, and their corresponding canonical moments. These latter equations make it possible to study the dynamics of the theory. We will see that, similar to the Einstein-Hilbert theory, our model leads to the propagation of two degrees of freedom, consistent with spontaneous breaking of diffeomorphism invariance. By investigating the dynamics and constraints of this modified gravity model, this work aims to provide insights into the behavior and implications of theories that go beyond General Relativity, shedding light on the possible effects of diffeomorphism symmetry breaking at the Planck scale.