**Quantum gravity** is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. The ultimate goal is a unified framework for all fundamental forces—a theory of everything.

Much of the difficulty in merging these theories comes from the radically different assumptions that these theories make on how the universe works. Quantum field theory depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as mass moves. The most obvious ways of combining the two (such as treating gravity as simply another particle field) run quickly into what is known as the renormalization problem. Gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results. This is in contrast with quantum electrodynamics where the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization.

Both quantum mechanics and general relativity have been highly successful. Unfortunately, the energies and conditions at which quantum gravity effects are likely to be important are inaccessible to current laboratory experiments. The result is there are no experimental observations which would provide any hints as to how to combine the two.

The general approach taken in deriving a theory of quantum gravity is to assume that the underlying theory will be simple and elegant and then to look at current theories for symmetries and hints for how to combine them elegantly into an overarching theory. One problem with this approach is that it is not known if quantum gravity will be a simple and elegant theory.

Such a theory is required in order to understand those problems involving the combination of very large mass or energy and very small dimensions of space, such as the behaviour of black holes, and the origin of the universe.

## Historical perspective

Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity.

The first is that the geometric interpretation of general relativity is not fundamental, but just an emergent quality of some background-dependent theory. This is explicitly stated, for example, in Steven Weinberg's classic *Gravitation and Cosmology* textbook.

The opposing view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time. The geometric point of view is expounded in the classic text *Gravitation*, by Misner , Wheeler and Thorne.

The two books by giants of theoretical physics expressing completely opposite views of the meaning of gravitation were published almost simultaneously in the early 1970s. The reason was that an impasse had been reached, a situation which led Richard Feynman (who himself had made important attempts at understanding quantum gravity) to write, in desperation, *"Remind me not to come to any more gravity conferences"* in a letter to his wife in the early 1960's.

Progress has been made on both fronts, leading (as of 2004) to string theory on the one hand and loop quantum gravity on the other.

## The incompatibility of quantum mechanics and general relativity

At present, one of the deepest problems in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, planets, galaxies), with quantum mechanics, which describes the other three fundamental forces acting on the microscopic scale.

A fundamental lesson of general relativity is that there is no fixed spacetime background, as found in Newtonian mechanics and special relativity; the spacetime geometry is dynamic. While easy to grasp in principle, this is the hardest idea to understand about general relativity, and its consequences are profound and not fully explored, even at the classical level. To a certain extent, general relativity can be seen to be a relational theory, in which the only physically relevant information is the relationship between different events in space-time.

On the other hand, quantum mechanics has depended since its invention on a fixed background (non-dynamical) structure. In the case of quantum mechanics, it is time that is given and not dynamic, just as in Newtonian classical mechanics. In relativistic quantum field theory, just as in classical field theory, Minkowski spacetime is the fixed background of the theory. Finally, string theory started out as a generalization of quantum field theory where instead of point particles, string-like objects propagate in a fixed spacetime background. Although string theory had its origins in the study of quark confinement and not of **quantum gravity**, it was soon discovered that the string spectrum contains the graviton, and that "condensation" of certain vibration modes of strings is equivalent to a modification of the original background.

Quantum field theory on curved (non-Minkowskian) backgrounds, while not a quantum theory of gravity, has shown that some of the core assumptions of quantum field theory cannot be carried over to curved spacetime, let alone to full-blown quantum gravity. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see Unruh effect). Also, the field concept is seen to be fundamental over the particle concept (which arises as a convenient way to describe localized interactions). This latter point is not uncontroversial, as it is contrary to the way quantum field theory on Minkowski space is developed by Steven Weinberg's book *Quantum Field Theory*.

Loop quantum gravity is the fruit of the effort to formulate a background-independent quantum theory. Topological quantum field theory provided an example of background-independent quantum theory, but with no local degrees of freedom, and only finitely many degrees of freedom globally. This is inadequate to describe gravity in 3+1 dimensions, which even in vacuum has local degrees of freedom according to general relativity. In 2+1 dimensions, however, gravity is a topological field theory and it has been successfully quantized in several different ways, including spin networks.

## Theories

There are a number of proposed quantum gravity theories:

The "direct" way of quantizing gravity comes with many choices. Do we use functional integrals over Wick rotated Riemannian metrics (e.g. by Hawking)? See Euclidean path integral approach. Do we use the covariant Peierls bracket? Do we use BRST/Batalin-Vilkovisky formalism or gauge fixing or gauge factoring? If we pick canonical quantization, do we use the Einstein-Hilbert action with only the metric as dynamical to get the Wheeler-deWitt equation? Or do we treat the metric and the affine connection independently? Or do we have the whole Poincaré group as the gauge group starting with the Einstein-Cartan theory? Or do we use the Cartan method of moving frames with the Palatini action to get second class constraints? Do we eliminate the second class constraints using the Ashtekar variables to get loop quantum gravity or do we do something else? The existence of spinor fields may force us to work with the Cartan formalism or something comparable. Or maybe we should look at representations of the diffeomorphism group just as Wigner looked at representations of the Poincaré group.

## Quantum gravity theorists

*See list of quantum gravity researchers*

## See also

## External links

The famous spoof of postmodernism by Alan Sokal (see Sokal Affair) was entitled *Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity*.

Last updated: 05-07-2005 17:28:17

Last updated: 05-13-2005 07:56:04