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

Abstract

A Wheeler-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 * thiemann@math.harvard.edu 1

Spin foam quantization and anomalies

“…Being path integrals of a gauge theory, they have to deal with the anomaly issue: the path integral measure has to be invariant under transformations generated by the constraints. Sometimes it is claimed that a covariant quantization avoids the issue of anomalies which plagued canonical approaches for a long time (but see [6]). However, it is well known that there is also an anomaly problem in path integral quantizations which in spin foam quantizations has just been ignored in most of the existing literature (see, however, [7] for a recent paper which discusses this issue independently in the example of 3-dimensional BFtheory).…”

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

“…As shown in [33,38], the commutator of two Hamiltonians indeed vanishes on this 'habitat', and one is therefore led to conclude that the full constraint algebra closes 'without anomalies'. The same conclusion was already arrived at in an earlier computation of the constraint algebra in [30,37], which was done from a different perspective (no 'habitats'), and makes essential use of the space of diffeomorphism invariant states H diff , the 'natural' kernel of the r.h.s. of (21).…”

“…Unfortunately, to the best of our knowledge, the 'off-shell' calculation of the commutator of two Hamiltonian constraints in LQG -with an explicit operatorial expression as the final result -has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted 'habitat' of states [33,37,38], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [8] that this 'on shell closure' is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes 'off shell', i.e.…”

“…of (21). Here the idea is to verify that [30,37] 6 For reasons of space, we here restrict attention to the bracket between two Hamiltonian constraints, because the remainder of the algebra involving the kinematical constraints is relatively straightforward to implement. for all |X ∈ H diff , and for all |Ψ in the space of finite linear combinations of spin network states.…”

Curvature operator for loop quantum gravity

“…In this article we want to pursue this perspective as a tool to build a curvature operator in LQG, fundamental to solve the most challenging issue in the canonical approach: the quantum dynamics related to the Hamiltonian constraint. The Hamiltonian Constraint has been quantized by Thiemann [11,12] improving several previous proposals [13] and finally succeeding in defining an anomaly free operator. It is defined employing a regularization procedure with specific rules that might be changed to bring it closer to the spinfoam formalism [17,18] (but till now spoiling the anomaly free property).…”