Quantum cosmology of pure connection general relativity

Abstract

We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Pleba'{n}ski formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.

Overall signature of the metric and the cosmological constant

“…Classical signature change is not optional but mandatory here. The fact that the same theory (with fixed but arbitrary non-zero Λ) always contains both adS and dS solutions was observed in [25].…”

“…There is nothing exuberant about negative electric fields. This is forced upon us in the quantum theory [28][29][30] (see also [25]), should we require unitarity and starting from the connection representation. It is interesting that even classically we can find new solutions exhibiting what presumably is the peak of the wave packets moving from the a 2 < 0 to the a 2 > 0 region.…”

Unimodular Plebański gravity

“…This is the unimodular analogue of the first order chiral connection theory discussed in [2,22,24]. The field equations…”

“…The trace Einstein equation R µν = Λg µν is equivalent to Rµν = σΛg µν where Rµν is the Ricci tensor of the Lorentzian metric gµν , which is real. Thus, if Λ is a fixed real parameter, there are no dynamical solutions with σ = ±i (but both σ = ±1 sectors are always present) [22]. In the unimodular approach Λ cannot be fixed and so 'Lorentzian' solutions for all four possible 'signatures' σ are present.…”

“…where ξ and η are arbitrary vector fields and i ξ Σ i := Σ i (ξ, •) denotes the contraction of a 2-form with the vector field ξ; see [2,22] for more details. The fact that the metric can be reconstructed explicitly from the Σ i allows defining arbitrary matter couplings at least in principle, see, e.g.…”