Lessons for Loop Quantum Gravity from Parametrised Field Theory

Abstract

In a series of seminal papers, Laddha and Varadarajan have developed in depth the quantisation of Parametrised Field Theory (PFT) in the kind of discontinuous representations that are employed in Loop Quantum Gravity (LQG). In one spatial dimension (circle) PFT is very similar to the closed bosonic string and the constraint algebra is isomorphic to two mutually commuting Witt algebras. Its quantisation is therefore straightforward in LQG like representations which by design lead to non anomalous, unitary, albeit discontinuous representations of the spatial diffeomorphism group. In particular, the complete set of (distributional) solutions to the quantum constraints, a preferred and complete algebra of Dirac observables and the associated physical inner product has been constructed. On the other hand, the two copies of Witt algebras are classically isomorphic to the Dirac or hypersurface deformation algebra of General Relativity (although without structure functions). The question we address in this paper, also raised by Laddha and Varadarajan in their paper, is whether we can quantise the Dirac algebra in such a way that its space of distributional solutions coincides with the one just described. This potentially teaches us something about LQG where a classically equivalent formulation of the Dirac algebra in terms of spatial diffeomorphism Lie algebras is not at our disposal. We find that, in order to achieve this, the Hamiltonian constraint has to be quantised by methods that extend those previously considered. The amount of quantisation ambiguities is somewhat reduced but not eliminated. We also show that the algebra of Hamiltonian constraints closes in a precise sense, with soft anomalies, that is, anomalies that do not cause inconsistencies. We elaborate on the relevance of these findings for full LQG.