Paths to gravitation via the gauging of parametrized field theories

Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

“…The models that we will look at are models where g µν is recovered from the combination (10) with θ I µ = 0 and the dynamical variables of the theory will be {ϕ I , ω IJ µ }. This suggests that ultimately we should identify e I µ as being recovered from the object D µ ϕ I = ∂ µ ϕ I + ω I Jµ ϕ J and so, as in the case of Einstein-Cartan theory and Ashtekar's chiral theory we can look to construct Lagrangian densities which are quadratic in this field and linear in the curvature of ω IJ µ , anticipating that this may be the simplest action giving non-trivial gravitational dynamics [5]:…”

“…where η IJ = diag(−1, 1, 1, 1) and x µ are coordinates in spacetime. Analogously to the model (3), actions with the replacement (5) with the promotion of ϕ I to dynamical fields then possess a four-dimensional spacetime diffeomorphism symmetry despite not including the gravitational interaction. If the gauge ϕ I * = x I (x µ ) is accessible, where x I are fields playing the role of Minkowski coordinates in spacetime, then special-relativistic physics is recovered with the ϕ I equations of motion corresponding to equations expressing conservation of stress energy.…”

“…Can the gravitational interaction be recovered from this starting point and, if so, is the resulting theory general relativity? To take steps toward this, we note that actions built using (5) have an additional symmetry which corresponds to:…”

“…where Λ I J ∈ SO (1,3) and P I are independent of coordinates x µ and hence (6) can be interpreted as a global Poincaré transformation acting on the fields ϕ I . If some of {Λ I J , P I } do depend on position then the ordinary derivative ∂ µ ϕ I in (5) no longer transforms homogeneously under the local generalization of (6) and so actions containing (5) will not be invariant under such transformations. This can be remedied by the introduction of fields {ω I Jµ , θ I µ } such that an operator D µ can be constructed, acting on ϕ I as:…”

“…when β = ±i [3][4][5][6], where D µ ϕ I = ∂ µ ϕ I + ω I Jµ ϕ I is the SO(1, 3)-covariant derivative of ϕ I and R IJ αβ is the curvature two-form of the field ω I Jµ . As such, the action (11) is manifestly invariant under local SO (1,3) transformations.…”