Another operational problem with general relativity is the difficulty to do calculations, because of the complexity of the mathematical tools used. Integral of path in contrast has been used in mechanics since the end of the 19th century and is well known. In addition Path integral is a formalism used both in mechanics and quantum theories so it might be a good starting point for unifying general relativity and quantum theories. Some quantum features like the and the are also related by Wick rotation. So the Wick relation is a good tool to relate a classical phenomenon to a quantum phenomenon. The ambition of Euclidean quantum gravity is to use the Wick rotation to find connections between a macroscopic phenomenon, gravity, and something more microscopic.
Euclidean quantum gravity refers to a version of , formulated as a . The that are used in this formulation are 4-dimensional instead of . It is also assumed that the manifolds are , and (i.e. no ). Following the usual quantum field-theoretic formulation, the to vacuum amplitude is written as a over the , which is now the quantum field under consideration.
One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over Euclidean geometries can be performed constructively by the method of dynamical triangulations. One can define a proper-time propagator. This propagator can be used to calculate generalized Hartle-Hawking amplitudes and it can be used to understand the the fractal structure of quantum geometry. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from two-dimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of three-dimensional Euclidean quantum gravity. General properties of the four-dimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and asymptotic safety is till awaiting.