Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity

AWheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b) the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincaré group including the quantum ADM energy. Attempts at defining an operator which corresponds to the Hamiltonian constraint of four-dimensional Lorentzian vacuum canonical gravity [1] have first been made within the framework of the ADM or metric variables (see, for instance, [2]) This formulation of the theory seemed hopelessly difficult because of the complicated algebraic nature of the Hamiltonian (or Wheeler-DeWitt) constraint. It was therefore thought to be mandatory to first cast the Hamiltonian constraint into polynomial form by finding better suited canonical variables. That this is indeed possible was demonstrated by Ashtekar [3]. There are two, a priori, problems with these Ashtekar connection variables for Lorentzian gravity : 1) they are complex valued and are therefore subject to algebraically highly complicated reality conditions, difficult to impose on the quantum level, which make sure that we are still dealing with real general relativity and 2) the Hamiltonian constraint is polynomial only after rescaling it by a non-polynomial function, namely a power of the square root of the determinant of the three-dimensional metric, that is, the original Wheeler-DeWitt constraint has actually been altered. A solution to problem 1) has been suggested in [4] (see also [5]) : namely, one can define real Ashtekar variables [6] which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to ∗[email protected]