Exactly soluble sector of quantum gravity

Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally covariant Galilean-relativistic limit form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate ``metric'' structure of spacetime remains fixed with two mutually orthogonal nondynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the nonmetric gravitational connection field is dynamical in the sense that it is determined by matter distributions. Here, this generally covariant but Galilean-relativistic theory of gravity with a nonzero cosmological constant, viewed as a parametrized gauge theory of a gravitational vector potential minimally coupled to a complex Schr\"odinger field (bosonic or fermionic), is successfully cast---for the first time---into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1 dimensions and into a manifestly covariant reduced phase-space form with nondegenerate symplectic structure in 4 dimensions. Next, this Newton-Cartan-Schr\"odinger system is nonperturbatively quantized using the standard C^*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally covariant, but also exactly soluble and---thanks to the immutable causal structure of the Newton-Cartan spacetime---free of all conceptual and mathematical difficulties usually encountered in quantizing Einstein's theory of gravity. Consequently, the resulting theory of the quantized Newton-Cartan-Schr\"odinger system constitutes a perfectly consistent Galilean-relativistic sector of the elusive full quantum theory of gravity coupled to relativistic matter, regardless of what ultimate form the latter theory eventually takes.