哈特尔－霍金无边界条件1983

Ingredients

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. This is related to the Wick rotation, τ =it, and it converts the spacetime metric into an Euclidean metric, ds²=dτ ²+dx²+dy²+dz². 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. This curved Euclidean spacetime can be analogous to a sphere in being both finite in extent and yet have no boundary.

History

The original 1983 paper by Hartle and Hawking grew out of a summer visit by Hawking to UC Santa Barbara where Hartle worked. Hawking was exploring the idea that the boundary condition for space time was simply no-boundary at all.

With Hartle this idea was converted in to a proposal and published.

In 1998 Hawking worked with Neil Turok to expand the Hartle-Hawking concept to include a hyperbolic or open geometry.