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Physics LibreTexts

Quantum Geometry

Loop quantum gravity is the leading purely gravitational approach to quantizing general relativity ("purely gravitational" as opposed to string theory, in which quantum gravity comes as one piece of a larger structure). Its aim is to construct a theory of "quantum geometry" in which geometrical objects such as length and areas -- the metrical information fundamental to general relativity -- appear as operators acting on quantum states. This will be a very incomplete summary; for more, see this paper and the articles, talks, and videos under "Outreach" at the Penn State gravity center Web site. For a technical review, see Carlo Rovelli's article in Living Reviews.

The states of loop quantum gravity are described by "spin networks," graphs whose edges are labeled by spins and whose vertices are labeled by "intertwiners" (think Clebsch-Gordan coefficients) that tell how to combine spins. Geometric operators like area, constructed from the spacetime metric, act on these states, changing the network. The dynamics of general relativity comes in through the Hamiltonian constraint, a not-fully-understood operator condition that determines the admissible spin networks. An alternative view describes three-dimensional spin networks as tracing out spin foams in (3+1)-dimensional spacetime, providing a setting for a path integral approach. The (2+1)-dimensional version of this spin foam picture, the Turaev-Viro model, is a well-understood quantization of (2+1)-dimensional gravity.

Loop quantum gravity has had important successes in black hole physics and in quantum cosmology. Furthermore, unlike many past attempts to quantize general relativity, the loop approach is known to be mathematically well-defined, and it is "background-free," avoiding some of the conceptual problems of other methods. But although it is a quantum theory based on general relativity, it is not entirely clear that it is really a "quantum theory of gravity" -- the time evolution remains only partially understood, and we do not yet know how to recover a good classical limit that looks like classical general relativity.

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