Path Integrals and Instantons in Quantum Gravity

TL;DR

The paper derives a path integral for quantum gravity in minisuperspace from refined algebraic quantization, yielding a gauge-fixed inner product expressed as a path integral over lapse and phase-space variables with a well-defined measure. Analytically solvable models demonstrate that Euclidean segments contribute exponentially damped factors $e^{-|S_E|/\hbar}$ and that Euclidean instantons with negative action do not enhance the semiclassical amplitude; a Baierlein–Sharp–Wheeler-like reduction further clarifies the configuration-space structure. The results indicate a robust semiclassical picture where Lorentzian and positive-action Euclidean paths contribute, while problematic unbounded Euclidean actions do not destabilize the integral, suggesting these features may extend to full quantum gravity. This provides a controlled framework for interpreting gravitational instantons and has potential implications for black hole production, the cosmological constant problem, and other global issues in quantum cosmology.

Abstract

While there does not at this time exist a complete canonical theory of full 3+1 quantum gravity, there does appear to be a satisfactory canonical quantization of minisuperspace models. The method requires no `choice of time variable' and preserves the systems' explicit reparametrization invariance. In the following study, this canonical formalism is used to derive a path integral for quantum minisuperspace models. As it comes from a well-defined canonical starting point, the measure and contours of integration are specified by this construction. The properties of the resulting path integral are analyzed, both exactly and in the semiclassical limit. Particular attention is paid to the role of the (unbounded) Euclidean action and Euclidean instantons are argued to contribute as $e^{-|S_E|/\hbar}$.