SL(2, R) model with two Hamiltonian constraints

General relativity (GR) has a characteristic gauge invariance, which implies that its canonical Hamiltonian vanishes weakly. As a consequence, its dynamics is not governed by a genuine Hamiltonian, but rather by a “Hamiltonian constraint”. This peculiar feature of the theory has a crucial physical significance, connected to the relational nature of the general-relativistic spatiotemporal notions [1–3], and raises a number of important conceptual as well as technical problems, particularly in relation to the quantization of the theory [4]. In the past, much clarity has been shed on these problems by studying finite dimensional models mimicking the constraint structure of the theory, and in particular, having weakly vanishing Hamiltonian [3]. There is an aspect of the constraint structure of GR, however, which, as far as we are aware, has not been analyzed with the use of constrained models. In GR, there isn’t just a single Hamiltonian constraint, but rather a family of Hamiltonian constraints, one, so to say, for each coordinate-space point. Furthermore, the Hamiltonian constraints do not commute with each other (have nonvanishing Poisson brackets with each other). Indeed, the constraint algebra of GR has the well known structure