Quantum gravity

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 of some is a unified framework for all fundamental forces—a "theory of everything".

Overview

Much of the difficulty in merging these theories comes from the 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 (still, the series don't converge).

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. Thus, 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 behavior of black holes, and the origin of the universe.

Historical perspective

Historically, there have been many reactions to the apparent inconsistency of quantum theories with general relativity.

The first is that the geometric interpretation of general relativity is not fundamental. This possibility is mentioned, for example, in Steven Weinberg's classic Gravitation and Cosmology textbook.

Another view is that background independence is fundamental, and quantum mechanics needs to be generalized to settings where there is not a priori specified time. The geometric point of view is expounded in the classic text Gravitation, by Misner, Wheeler and Thorne.

Others believe that understanding a quantum theory of gravity will lead to a radically new view of space and time, and that geometry will only emerge in a semi-classical limit.

Both string theory and loop quantum gravity, while having different origins, appear to fall into the third category.

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.

General relativity, like electromagnetism, is a classical field theory. Naively one expects that, as with electromagnetism, there should be a corresponding quantum field theory. However, one runs into a serious problem: gravity is nonrenormalizable. For a quantum field theory to be well-defined, it must be asymptotically free or asymptotically safe. In less technical language, this has a simple meaning: the theory must be characterized by a choice of finitely many parameters, which could in principle be set by experiment. For example, in quantum electrodynamics these parameters are the charge and mass of the electron, as measured at a particular energy scale. On the other hand, when one quantizes gravity, one finds that there are infinitely many independent parameters needed to define the theory. For a given choice of those parameters, one could make sense of the theory, but since we can never do infinitely many experiments to fix the values of every parameter, we do not have a meaningful physical theory. At low energies, the logic of the renormalization group tells us that, despite the unknown choices of these infinitely many parameters, quantum gravity will reduce to the usual Einstein theory of general relativity. On the other hand, if we could probe very high energies where quantum effects take over, then every one of the infinitely many unknown parameters would begin to matter, and we could make no predictions at all.

Thus, any meaningful theory of quantum gravity must have some deep principle that reduces the infinitely many unknown parameters to a finite number that can then be measured. One possibility is that normal perturbation theory is not a reliable guide to the renormalizability of the theory, and that there really is a UV fixed point for gravity. Since this is a question of nonperturbative quantum field theory, it is difficult to find a reliable answer, but some people still pursue this option. Another possibility is that there are new symmetry principles that constrain the parameters and reduce them to a finite set. This is the route taken by string theory, where all of the excitations of the string essentially manifest themselves as new symmetries.

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 inception 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. In this sense, string perturbation theory exhibits exactly the features one would expect of a perturbation theory that may exhibit a strong dependence on asymptotics (as seen, for example, in the AdS/CFT correspondence) which is a weak form of background dependence.

Quantum field theory on curved (non-Minkowskian) backgrounds, while not a quantum theory of gravity, has shown that some of the 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).

Loop quantum gravity is the fruit of an 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 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.

There are three other points of tension between quantum mechanics and general relativity. First, general relativity predicts its own breakdown at singularities, and quantum mechanics becomes inconsistent with general relativity in a neighborhood of singularities. Second, it is not clear how to determine the gravitational field of a particle, if under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty. Finally, there is a tension, but no logical contradiction, between violations of Bell's inequality in quantum mechanics, which imply superluminal influence, and the speed of light as a speed limit in relativity. The resolution of the first two points may come from a better understanding of general relativity ^*.

Theories

There are a number of proposed quantum gravity theories:

Weinberg-Witten theorem

There is a theorem in QFT called the Weinberg-Witten theorem which places some constraints on theories of composite gravity/emergent gravity.

Quantum gravity theorists

See list of quantum gravity researchers

In Popular Culture

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

External links