Exploring Cartan gravity with dynamical symmetry breaking
Classical and quantum gravity
It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an SO(1,4) or SO(2,3) gauge field A^a_abμ and Higgs field V^a which acts to break the symmetry down to that of the Lorentz group SO(1,3). This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field Φ of electroweak theory is taken as a genuine dynamical
... eld satisfying a Klein-Gordon equation, the gauge independent norm V^2≡η_abV^aV^b of the Higgs-type field V^a is typically regarded as non-dynamical. Instead, in many treatments V^a does not appear explicitly in the formalism or is required to satisfy V^2 = const.≠ 0 by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection A^a_abμ and Higgs field V^a as genuine dynamical fields with no ad-hoc constraints imposed. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. One simple choice leads to the extensively studied Peebles-Ratra rolling quintessence model. Another choice yields a scalar-tensor symmetry broken phase of the theory with positive cosmological constant and an effective mass M of the gravitational Higgs field ensuring the constancy of V^2 at low energies and agreement with empirical data if M is sufficiently large. More general cases are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity, each of which yield propagating torsion.