From 3-geometry transition amplitudes to graviton states

TL;DR

This paper analyzes the propagation kernel in Lorentzian quantum gravity, demonstrating its relation to quantum states, constraints, and graviton states, and explores its implications for the spinfoam approach.

Contribution

It explicitly computes the propagation kernel in Lorentzian quantum gravity, showing how it projects onto physical states and encodes graviton interactions.

Findings

Kernel projects onto solutions of quantum constraints

Reproduces vacuum and n-graviton states

Encodes Newtonian interaction in the propagator

Abstract

In various background independent approaches, quantum gravity is defined in terms of a field propagation kernel: a sum over paths interpreted as a transition amplitude between 3-geometries, expected to project quantum states of the geometry on the solutions of the Wheeler-DeWitt equation. We study the relation between this formalism and conventional quantum field theory methods. We consider the propagation kernel of 4d Lorentzian general relativity in the temporal gauge, defined by a conventional formal Feynman path integral, gauge fixed a\' la Fadeev--Popov. If space is compact, this turns out to depend only on the initial and final 3--geometries, while in the asymptotically flat case it depends also on the asymptotic proper time. We compute the explicit form of this kernel at first order around flat space, and show that it projects on the solutions of all quantum constraints, including the Wheeler-DeWitt equation, and yields the correct vacuum and n-graviton states. We also illustrate how the Newtonian interaction is coded into the propagation kernel, a key open issue in the spinfoam approach.