Sum over histories

 Many possible worlds in superposition

The Hartle-Hawking proposal includes several ingredients. First it uses Richard Feynman's path integral formulation of quantum mechanics. In this approach every possible path a particle can take through spacetime contributes to the solution with its own amplitude and phase. Technical challenges with those sums lead to the second ingredient, a transformation to Euclidean space-time: a geometry which combines 3 space dimensions with an imaginary time dimension.^[6]: 172 This is related to the Wick rotation, ${\displaystyle \tau =it}$, and it converts the spacetime metric in to a Euclidean metric, ${\displaystyle d{s}^2=d{\tau }^2+d{x}^2+d{y}^2+d{z}^2}$. In Hawking's approach, this rotation is applied to every path, not to the background space of the paths as in Wick's approach, and therefore the sum of histories becomes a quantum superposition of spacetimes.^[2]: 769 This curved Euclidean spacetime can be analogous to a sphere in being both finite in extent and yet have no boundary.