It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics.
More precisely, it substitutes a mathematical problem in Minkowski space into a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.
For example, a Wick rotation could be used to relate a macroscopic event temperature diffusion (like in a bath) to the underlying thermal movements of molecules.
The path integral formulation is the conceptual tool used to describe the movements of this unique molecule, and Wick rotation is one of the mathematical tools that are very useful to analyse a path integral problem.
It differs clearly to the movement of a classical object (e.g. a billiard ball), since in this case a single path with precise position and speed can be described.
In 1966 an explicitly gauge invariant functional-integral algorithm was found by DeWitt, which extended Feynman's new rules to all orders.
What is appealing in this new approach is its lack of singularities when they are unavoidable in general relativity.
Another operational problem with general relativity is the computational difficulty, because of the complexity of the mathematical tools used.
Path integrals in contrast have been used in mechanics since the end of the nineteenth century and is well known.
[citation needed] In addition, the path-integral formalism is used both in classical and quantum physics so it might be a good starting point for unifying general relativity and quantum theories.
Following the usual quantum field-theoretic formulation, the vacuum to vacuum amplitude is written as a functional integral over the metric tensor, which is now the quantum field under consideration.
be independent of the lapse function and shift vector at the boundaries, so we obtain where