Poincar\'e gauge gravity: an emergent scenario

Abstract

The Poincar\'e gauge gravity (PGG) with the underlying vector fields of tetrads and spin-connections is perhaps the best theory candidate for gravitation to be unified with the other three elementary forces of nature. There is a clear analogy between local frame in PGG and local internal symmetry space in the Standard Model. As a result, the spin-connection fields, gauging the local frame Lorentz symmetry group SO(1,3)_{LF}, appear in PGG much as photons and gluons appear in SM. We propose that such an analogy may follow from their common emergent nature allowing to derive PGG in the same way as conventional gauge theories. In essence, we start with an arbitrary theory of some vector and fermion fields which possesses only global spacetime symmetries, such as Lorentz and translational invariance, in flat Minkowski space. The two vector field multiplets involved are proposed to belong, respectively, to the adjoint (A_{{\mu}}^{ij}) and vector (e_{{\mu}}^{i}) representations of the starting global Lorentz symmetry. We show that if these prototype vector fields are covariantly constrained, A_{{\mu}}^{ij}A_{ij}^{{\mu}}=M_{A} and e_{{\mu}}^{i}e_{i}^{{\mu}}=M_{e}, thus causing a spontaneous violation of the accompanying global symmetries (M_{A,e} are their proposed violation scales), then the only possible theory compatible with these length-preserving constraints is turned out to be the gauge invariant PGG, while the corresponding massless (pseudo)Goldstone modes are naturally collected in the emergent gauge fields of tetrads and spin-connections. In a minimal theory case being linear in a curvature we unavoidably come to the Einstein-Cartan theory. The extending theories with propagating spin-connection and tetrad modes are also considered and their possible unification with the Standard Model is briefly discussed.