2 + 1 dimensional gravity as an exactly soluble system
Edward Witten
Abstract and summary · read the original at the source
In one page
Quantum gravity is hard because in four dimensions Einstein’s equations refuse to be quantised — the infinities will not cancel. Edward Witten’s 1988 paper takes the problem to a world with two dimensions of space and one of time and shows that there the whole thing comes apart cleanly. By disentangling the Hamiltonian constraints, the equations that say how space at one instant has to fit together, he shows that gravity in 2 + 1 dimensions, with or without a cosmological constant, is exactly soluble both classically and quantum-mechanically. And it turns out not to be a separate subject at all: it is closely related to Yang-Mills theory carrying only the Chern-Simons action — the gauge theory that had just been shown to define a soluble quantum field theory. Its renormalized perturbation expansion is straightforward and its beta function vanishes. Witten offers the result as a laboratory: a place to ask what a classical singularity becomes once gravity is quantum, and a tool for three-dimensional geometry.
Why it matters hereChapter 13 is the unified picture, and this is the cleanest published demonstration that gravity and gauge theory can be the same object written two ways — a metric theory rewritten exactly as a Chern-Simons gauge theory, with the divergences gone. Chapter 4 is what happens to a spacetime you engineer, and Witten’s own stated use for the model is to find out what a classical singularity turns into when the theory is quantum.
What it claims
01By disentangling the Hamiltonian constraint equations, 2 + 1 dimensional gravity — with or without a cosmological constant — is shown to be exactly soluble at the classical and the quantum levels. Not approximated, not perturbatively tamed: solved.Abstract, sentence 1; Nuclear Physics B 311, 46–78
Settled physics02The theory is closely related to Yang-Mills theory with purely the Chern-Simons action, which had itself recently turned out to define a soluble quantum field theory. Gravity in three dimensions is a gauge theory written in other letters.Abstract, sentence 2
Settled physics032 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function — the coupling does not run, and the obstruction that defeats the quantisation of gravity in four dimensions is simply absent here.Abstract, sentence 3
Settled physics042 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics — a solved model in which the question of what becomes of a singularity can actually be answered rather than argued.Abstract, final sentence, first clause
What to watch05It may be related to the discrete series of Virasoro representations in 1 + 1 dimensions, tying three-dimensional gravity to the conformal field theories of the string worldsheet.Abstract, final sentence, second clause
What to watch06What to watch: it may be a useful tool in studying three-dimensional geometry — physics handed back to mathematics as a method. The programme this opens is whether an exactly soluble gravity in one dimension fewer teaches anything transferable about the four-dimensional case, and that is still the open question the model is used to attack.Abstract, final sentence, third clause
What to watch
Read it · abstract
Abstract
By disentangling the hamiltonian constraint equations, 2 + 1 dimensional gravity (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels. Indeed, it is closely related to Yang-Mills theory with purely the Chern-Simons action, which recently has turned out to define a soluble quantum field theory. 2 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function. 2 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics, may be related to the discrete series of Virasoro representations in 1 + 1 dimensions, and may be a useful tool in studying three-dimensional geometry.
Edward Witten, 2 + 1 dimensional gravity as an exactly soluble system, Nuclear Physics B 311, 46–78 (December 1988); preprint IASSNS-HEP-88-32, School of Natural Sciences, Institute for Advanced Study, Princeton. The published paper is at doi.org/10.1016/0550-3213(88)90143-5.
(Abstract only — no other text of the paper is reproduced here; see the rights note above. On this site, the string-geometry route to the same unification is at /library/stm-3f6aec7115; Sakharov’s argument that gravity is induced rather than fundamental is at /library/stm-aef0021a1e; Verlinde’s emergent-gravity programme is at /library/stm-601c8315b2; Maldacena, Milekhin and Popov’s traversable wormhole in four dimensions is at /library/stm-d0d2a8e701; and Banks on the quantum theory of de Sitter space is at /library/stm-aabd243c91.)
The way in
https://doi.org/10.1016/0550-3213(88)90143-5SOURCE NOT REACHED IN FULL. Nuclear Physics B volume 311, pages 46 to 78, December 1988, written at the School of Natural Sciences, Institute for Advanced Study, Princeton, and issued as preprint IASSNS-HEP-88-32. Elsevier holds the article closed and no open copy was reached: the INSPIRE-HEP record for this DOI, control number 264816, carries the full bibliographic entry and the abstract but its documents field is empty, so INSPIRE has no attached scan to read; a guessed address for a KEK scanned-preprint copy answered with an HTML page rather than a PDF and was not pursued further on a guess; and the paper predates the arXiv, which opened in 1991, so no preprint exists there. THE ABSTRACT below is the authors’ own as deposited by Elsevier and carried by INSPIRE-HEP, which labels its source as Elsevier. It is reproduced exactly as INSPIRE holds it, with no repair needed. Every locator points to a clause of that abstract or to the bibliographic record. Crossref returns an empty abstract for this DOI, and the page range and December 1988 date are Crossref’s.
How to cite it
Edward Witten (1988) 2 + 1 dimensional gravity as an exactly soluble system. doi:10.1016/0550-3213(88)90143-5
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