Exactly Soluble Sector of Quantum Gravity

TL;DR

This paper reformulates Newton-Cartan gravity as a covariant gauge theory coupled to quantum matter, deriving an exactly soluble quantum field theory that unifies classical and quantum descriptions of Newtonian gravity.

Contribution

It provides the first covariant Lagrangian and Hamiltonian formulations of Newton-Cartan gravity coupled to quantum matter, leading to an exactly soluble quantum field theory.

Findings

Successfully cast Newton-Cartan gravity into covariant Lagrangian and Hamiltonian forms.

Performed non-perturbative quantization yielding an exactly soluble quantum gravity-matter system.

Established a generally-covariant quantum field theory of Newtonian gravity coupled to matter.

Abstract

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 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 possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schroedinger-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 non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger 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 generally-covariant, but also exactly soluble.