The Hartle–Hawking state, also known as the no-boundary wave function, is a cosmological model that applies quantum mechanics to the Big Bang.^[2]: 769 It is named after James Hartle and Stephen Hawking, who first proposed it in 1983.^[3][4] The concept can also be considered as an initial condition for models of quantum cosmology.^[5]: 14

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 an 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, ${\displaystyle ds^{2}=d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}}$. 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.^[6]: 174

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.^[6]: 175 In 1998 Hawking worked with Neil Turok to expand the Hartle-Hawking concept to include a hyperbolic or open geometry.^[7][2][8][1]

• Imaginary time
• Multiple histories
• Signature change