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Quantum gravity merges two other theories of quantum mechanics.
Quantum gravity is the field of Theoretical Physics attempting to unify 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 of quantum gravity.
Much of the difficulty in merging these theories at all energy scales 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. Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the renormalization problem. In the old-fashioned understanding of renormalization, 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, while the series still don't converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization.
In recent decades, however, this antiquated understanding of renormalization has given way to the modern idea of Effective field theory. All quantum field theories come with some high-energy cutoff, beyond which we do not expect that the theory provides a good description of nature. The "infinities" then become large but finite quantities proportional to this finite cutoff scale, and correspond to processes that involve very high energies near the fundamental cutoff. These quantities can then be absorbed into an infinite collection of coupling constants, and at energies well below the fundamental cutoff of the theory, to any desired precision only a finite number of these coupling constants need to be measured in order to make legitimate quantum-mechanical predictions. This same logic works just as well for the highly successful theory of low-energy Pions as for quantum gravity. Indeed, the first quantum-mechanical corrections to graviton-graviton scattering and Newton's law of gravitation have been explicitly computed (although they are so astronomically small that we may never be able to measure them), and any more fundamental theory of nature would need to replicate these results in order to be taken seriously. In fact, gravity is in many ways a much better quantum field theory than the Standard Model, since it appears to be valid all the way up to its cutoff at the Planck scale. (By comparison, the Standard Model is expected to break down above its cutoff at the much smaller TeV scale.)
While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies (in fact, the complete structure of gravity can be shown to arise automatically from the quantum mechanics of spin-2 massless particles), this way of thinking makes clear that near or above the fundamental cutoff of our effective quantum theory of gravity (the cutoff is generally assumed to be of order the Planck scale), a new model of nature will be needed. That is, in the modern way of thinking, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.
The general approach taken in deriving a theory of quantum gravity that is valid at even the highest energy scales 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 of quantum gravity.
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, albeit in an unsubstantiated statement claiming that "no-one" takes a geometric viewpoint of gravity "seriously", a statement that has been proven inaccurate by research subsequent to the book's publication in the 1970s.
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; this text, however, was written before even the discovery of the semi-classical Hawking radiation which provides perhaps the major, and certainly the first well-known, mixture of general relativity and quantum mechanics.
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. Examples of such an approach include attempts to entirely quantise space and time utilising, for example, mathematical approaches to entirely discrete Riemannian space-times.
While various approaches are currently considered, the two major theories are string theory and loop quantum gravity. String theory is a background-dependent, perturbative theory of gravity, normally formulated on a flat (Minkowski) spacetime. In the time since its conception, string theory has become a major hope towards a "theory of everything" since, within its many limits, it appears to include the symmetry groups of the Standard Model of particle physics. In stark contrast, loop quantum gravity is an attempt merely to directly quantise General Relativity and makes no claims towards being a "theory of everything". In loop quantum gravity, spacetime is redefined in new Ashtekar variables that facilitate the removal of the infinities that plague a more traditional approach to a quantum theory of general relativity. In essence, while string theory is an ambitious attempt to generate a theory of everything from (unsubstantiated) fundamental principles, loop quantum gravity merely seeks to pursue well-established quantisation procedures within the context of curved-spacetimes. Despite this intrinsic (and fundamental) disparity in philosophy, it has been suggested that string theory and loop quantum gravity, to some extents, embody effects of an underlying unity. For all this, proponents of loop quantum gravity will point out that while a free (Imirzi) parameter in LQG can be fixed with reference to the Entropy of black holes, there are few -- if any -- possible observational constraints on string theory.
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. This problem must be put in the proper context, however. In particular, contrary to the popular but erroneous claim that quantum mechanics and general relativity are fundamentally incompatible, one can in fact demonstrate that the structure of general relativity essentially follows inevitably from the quantum mechanics of interacting spin-2 massless particles (called gravitons). Furthermore, recent work has shown that by treating general relativity as an Effective field theory, one can actually make legitimate predictions for quantum gravity, at least for low-energy phenomenology. An example is the well-known calculation of the tiny first-order quantum-mechanical correction to the classical Newtonian gravitational potential between two masses. Such predictions would need to be replicated by any candidate theory of high-energy quantum gravity.
Historically, however, it was believed for a long time that general relativity was in fact fundamentally inconsistent with quantum mechanics. The argument went as follows. 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, according to this now-outdated understanding of the subject, 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.
However, from the perspective of Effective field theory, one sees that all but the first few such parameters are suppressed by huge energy scales and hence can be neglected when computing low-energy effects. Thus, at least in the low-energy regime, the model is indeed a predictive quantum field theory. (A very similar situation occurs for the very similar effective field theory of low-energy pions.) Furthermore, most theorists agree that even the Standard Model should really be regarded as an effective field theory as well, with "nonrenormalizable" interactions suppressed by large energy scales and whose effects have consequently not been observed experimentally.
However, any meaningful theory of quantum gravity that makes sense and is predictive at all energy scales 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 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, some argue that in curved spacetime, the field concept is seen to be fundamental over the particle concept (which arises as a convenient way to describe localized interactions). However, since it appears possible to regard curved spacetime as consisting of a condensate of gravitons, there is still some debate over which concept is truly the more fundamental.
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 two 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 (however, no one is certain that classical general relativity should necessarily be trusted near singularities in the first place). 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. The resolution of these points may come from a better understanding of general relativity.
Weinberg-Witten theorem of quantum gravity.
There is a theorem in Quantum field theory called the Weinberg-Witten theorem which places some constraints on theories of composite gravity/emergent gravity.
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