The classical limit and the form of the hamiltonian constraint in non-perturbative quantum general relativity

It is argued that some approaches to non-perturbative quantum general relativity lack a sensible continuum limit that reproduces general relativity. This may be true in spite of their being mathematically well defined diffeomorphism invariant quantum field theories that result from applying canonical quantization to general relativity. The basic problem is that generic physical states lack long ranged correlations, because the form of the state allows a division into spatial regions, such that no change in the physical state in one region can be measured by observables restricted to another. These disconnected regions have generically finite expectation value of physical volume, which means that the theory has no long ranged correlations or massless particles. One consequence of this is that the ADM energy is unbounded from below, at least when that is defined with respect to a natural notion of quantum asymptotic flatness and a corresponding definition of an operator that measures EADM (which is given here). These problems occur in Thiemann’s new formulation of quantum gravity. Related issues arise in some other approaches such as that of Borissov, Rovelli and Smolin. A new approach to the Hamiltonian constraint, which may avoid the problem of the lack of long ranged correlations, is proposed. ∗ [email protected]