Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory

Finding classical canonical observables consists of taking a function space over phase space. For constrained theories, these functions must form zero brackets with a closed algebraic structure of first-class constraints. This brackets condition can moreover be recast as a first-order PDE system, to be treated as a free characteristic problem. We explore explicit observables equations and their concrete solutions for a translation- and reparametrization-invariant action, thereby populating the following variety of notions of observables with examples. 1) The brackets can be strongly or weakly zero in Dirac's sense, i.e.\ a linear combination of constraints. 2) Observables can admit pure-configuration and pure-momentum restrictions. 3) Our model provides the translation constraint P$_i$ encoding zero total momentum of the model universe and depending homogeneous-linearly on momenta, and the Chronos constraint equation of time reinterpretation of the `constant-energy condition' and depending quadratically on momenta. These are mechanical analogues of GR's momentum M$_i$ and Hamiltonian H constraints respectively. P$_i$ and Chronos moreover algebraically close separately (only M$_i$ does for GR). Our model thus supports translation gauge-observables $G$ and Chronos observables $C$, as well as unrestricted observables $U$ and Dirac observables $D$ brackets-commuting with neither and both respectively. We relate the strong to properly-weak split of weak observables to the complementary-function to particular-integral split of the complete solution, with the properly-weak rendered of measure-0 relative to the strong by the latter's free characteristicness. The closed algebraic structures form a bounded lattice and the corresponding notions of observables a dual lattice, with the observables themselves forming a presheaf of function spaces thereover.