Hartle–Hawking proposal

Extrapolation of a Big Bang model to time zero (left) compared to and Hartle–Hawking state concept (right). Diagram shows two space dimensions horizontally and one time dimension vertically.^[1]

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

Ingredients

The Hartle-Hawking proposal includes several ingredients. First it uses Richard Feynman's sum over histories approach to 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 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

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.^[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]

See also

• Physics portal

• Imaginary time
• Multiple histories
• Signature change

References

1. "Centre for Theoretical Cosmology: The Origins of the Universe: General open inflation". www.ctc.cam.ac.uk. Retrieved 2025-08-23.
2. Penrose, Roger (2005). The road to reality: a complete guide to the laws of the universe (1st American ed.). New York: A.A. Knopf. ISBN 978-0-679-45443-4.
3. Hartle, J.; Hawking, S. (1983). "Wave function of the Universe". Physical Review D. 28 (12): 2960. Bibcode:1983PhRvD..28.2960H. doi:10.1103/PhysRevD.28.2960. S2CID 121947045.
4. Lehners, Jean-Luc (June 2023). "Review of the no-boundary wave function". Physics Reports. 1022: 1–82. arXiv:2303.08802. Bibcode:2023PhR..1022....1L. doi:10.1016/j.physrep.2023.06.002.
5. Weinberg, Steven (1989-01-01). "The cosmological constant problem". Reviews of Modern Physics. 61 (1): 1–23. Bibcode:1989RvMP...61....1W. doi:10.1103/RevModPhys.61.1. hdl:2152/61094. ISSN 0034-6861.
6. Hawking, Stephen (1996). The illustrated A brief history of time (Updated and expanded ed.). New York: Bantam Books. ISBN 978-0-553-10374-8.
7. Hawking, S. W.; Turok, Neil (1998). "Open inflation without false vacua". Physics Letters B. 425 (1): 25–32. arXiv:hep-th/9802030. Bibcode:1998PhLB..425...25H. doi:10.1016/S0370-2693(98)00234-2. ISSN 0370-2693.
8. Hawking, S. W.; Turok, Neil (April 16, 1998). "Open inflation without false vacua". Physics Letters B. 425 (1): 25–32. arXiv:hep-th/9802030. Bibcode:1998PhLB..425...25H. doi:10.1016/S0370-2693(98)00234-2. ISSN 0370-2693.