  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.
