Exactly Soluble Sector of Quantum Gravity

Cartan’s spacetime reformulation of the Newtonian theory of gravity is a generallycovariant 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 non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric 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 non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schrödinger-field (bosonic or fermionic), is successfully cast — for the first time — into a manifestly covariant Lagrangian form. Then, exploiting the fact that NewtonCartan 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 non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schrödinger system is non-perturbatively 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 generallycovariant, 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 quantized Newton-Cartan-Schrödinger 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. Journal-ref: Physical Review D56, No. 8, 15 Oct. 1997, pp 4844-4877