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The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the (− + + +) metric signature , the gravitational part of the action is given as [ 1 ]
Action principles are "integral" approaches rather than the "differential" approach of Newtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their ...
where is the Euclidean action and are Euclidean coordinates. This form is closely connected to the partition function in statistical mechanics, especially since the Euclidean Lagrangian is usually bounded from below in which case it can be interpreted as an energy density. It also allows for the interpretation of the exponential factor as a ...
More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. [4] In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points.
Euclidean quantum gravity refers to a Wick rotated version of quantum gravity, formulated as a quantum field theory. The manifolds that are used in this formulation are 4-dimensional Riemannian manifolds instead of pseudo Riemannian manifolds. It is also assumed that the manifolds are compact, connected and boundaryless (i.e. no singularities).
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be