# Quantum gravity

**Quantum Gravity** is the name given to any theory that describes *gravity* in
the regimes where *quantum* effects cannot be disregarded. At present, there is no
such a theory which is universally accepted and confirmed by experience. Therefore
the term "Quantum Gravity" indicates more an open problem than a specific theory.
Several research lines, at different levels of development, offer tentative solutions
to the problem. These tentative quantum-gravity theories are variously
viewed as competing research directions, or as contributions to the common goal of finding the
physically correct theory. The quest for the good quantum theory of gravity bears on a number of
fundamental issues and it is sometimes presented as the most important open problem in
fundamental physics: the "Holy Grail" of contemporary theoretical physics.

## The problem of quantum gravity

The current theory of gravity is *general relativity*, a theory developed by Albert Einstein at the
beginning of the
XX century, which has changed in depth our understanding of what are space and time.
Similarly, *quantum mechanics* has changed in depth our understanding of matter, energy
and causality. These two theories, today very well empirically supported, have opened a
major conceptual revolution in physics. However, they
appear to be incompatible, at least at a first reading, because each of the
two is formulated on the basis of principles that are are explicitly contradicted by
the other theory. Therefore in spite of the immense development of
scientific knowledge it has delivered, the physics of the XX century has left
us with a major confusion on what are the basic conceptual ingredients for understanding
the physical world. In other words, the scientific revolution opened by general
relativity and quantum mechanics at the beginning of the XX century is not concluded
yet, and a new synthesis is required. Quantum gravity, merging general relativity
and quantum mechanics should be this new synthesis.

### Quantum space and quantum time

Quantum gravity is expected to force us to further modifications of the concepts
of *space* and *time*,
along the direction opened by Einstein's general relativity, in order to make them
fully compatible with quantum theory.
In general relativity,
space and time loose their properties of being a fixed framework in which the
dynamical world is immersed. They are identified with the gravitational field and
acquire dynamical properties: spacetime can "fold" and "strech" like
a rubber sheet. When we take quantum mechanics into account, we realize that
this "rubber sheet" is in fact a quantum field, and therefore, like all
quantum fields, it must have a microscopic granular structure (like the photons
forming the electromagnetic field) and a probabilistic dynamics. Therefore
quantum gravity is likely to be the theory of granular and probabilistic
"quantum space" and "quantum time". Building the mathematical language and
the conceptual structure for making sense of such notions of
quantum space and quantum time is the challenge for a quantum
theory of gravity.

### The Planck scale

Simple dimensional arguments show that the physical phenomena where quantum gravitational effects becomes relevant as those characterized by the length scale \(l_{\rm Planck}=\sqrt{\hbar G/c^3}\sim 10^{-33}cm\ ,\) called the "Planck length". Here \(\hbar\) is the Planck constant that governs the scale of the quantum effects, \(G\) is the Newton constant that governs the strength of the gravitational force, and \(c\) is the speed of light, that governs the scale of the relativistic effects. The Planck length is extremely small. To have an idea, the Planck length is as many times smaller than an atom, as an atom is smaller than the solar system. Current technology is not yet capable of observing physical effects at scales that are so small (although several recent suggestions of how it could be possible to do so have appeared.) Because of this, we have no direct experimental guidance for building a quantum theory of gravity. This is not by itself a complete impediment, because general relativity and quantum mechanics are themselves strong guidances for constructing the theory: several major advances in the history of physics have been obtained in the absence of new experiments, from the effort of merging two empirically supported but apparently contradictory theories. (Examples are Newton's merge of Kepler's and Galileo's theories, Maxwell's merge of electric and magnetic theory, or Einstein's derivation of special relativity from the apparent contradiction between electromagnetism and mechanics.) However, until genuine quantum gravitational phenomena are directly or indirectly observed, we cannot confirm or falsify any of the current tentative theories.

### Physical applications

Some phenomena where quantum gravity plays a major role are the following.

- The microscopic structure of spacetime.

If we could measure the geometry of space and time at the Planck scale, we should be able
to see quantum gravitational effects. Many arguments indicate that the Planck
length may appear
as a limit to the infinite divisibility of space, that is, as a *minimal length*. Intuitively,
any attempt to measure smaller distances would result in the concentration of too much
energy in too small a space, with the result of forming a micro-black-hole, effectively
subtracting the region from observation. A minimal length would complete the tern of
fundamental scales in Nature, together with the speed of light, which is the maximal
velocity of a body, and the Planck constant, which is the minimal amount of action
exchanged between two systems.

- Early cosmology.

According to the currently standard cosmological model,
the Universe was very dense and hot in the past. Extrapolating back the model,
we encounter a singular point of infinite density, temperature and curvature,
conventionally denoted the *Big Bang*. However, this final extrapolation is
certainly incorrect because quantum gravitational phenomena become dominant
when the universe is very dense and hot, and these effects are
not included in the usual model. A quantum theory of gravity is needed in order to
take these effects into account and
study the early instants in the life of our universe. Some current theories
of gravity (in particular loop quantum gravity, see below), indicate
that the singular Big-Bang point is never reached, and the current expansion
of the universe might have been preceded by a collapsing phase, and a
"Big Bounce". One of the major hopes of observing traces of
quantum gravitational phenomena is in this cosmological context, as
traces of early universe phenomena left in the cosmic background radiation
currently under intense observation, or in the background gravitational
wave radiation, which is expected to be observed in the next decade.

Quantum gravity should play a role in several aspects of black-hole physics. First, it should give a complete understanding of the thermal radiation that black holes are expected to produce, first computed by Stephen Hawking. Second, Hawking's analysis shows that black holes carry enormous entropy about 10 to the power 77 for a solar mass black hole. What is the statistical mechanical origin of this number which is enormous even by the standards of thermodynamics? Third, quantum gravity is expected to replace the infinite singularity that general relativity predicts at the center of black holes with a more physically reasonable picture. Finally, the theory should explain what happens at the end of the Hawking evaporation of a black hole.

- Astrophysical effects.

Several astrophysical quantum-gravitational effects have been suggested. None has been observed so far, but different calculations suggest that they might observed be in the near future. An example is a small dependence of the speed of light on the color of the light, caused by the Planck-scale granularity of space. The effect is very small because of the smallness of the Planck scale, but it might become detectable if it is cumulated over a very long intergalactic path traveled by the light. Observations for testing this prediction are ongoing.

## Main approaches

The following approaches to quantum gravity open fascinating perspectives on the structure of Nature, and *might* be correct. But for the moment, none of these offers a *complete* theory of quantum gravity and none of these has been corroborated by experience. Because we do not have experimental data with direct bearing on quantum gravity, of necessity each program singles out certain aspects as being fundamental and focuses on them in the hope that other
problems can be easily sorted out later. Opinions vary widely on the relative merit of these approaches and the polemic, as often in science, is sometimes fiery. The list that follows is roughly ordered according to the number of scientists working in each line of research.
One should however remember that history teaches us that often a large
majority of scientists has been working on the wrong tentative theory,
with only a few working on the theory that ended up to be the correct:
scientific truth is judged by experimental corroboration, not by a democratic
counting of votes! These theories must all be considered tentative, at present.

### String theory

String theory is not just a theory of quantum gravity in the strict sense, because its objective is
wider: the theory aims at giving a unified description of the physical world, where *all* physical
entities are understood as manifestations of the motion of a single object: a string. Gravity
emerges in the theory as one of the aspects of the dynamics of the string. String theory
can be defined in terms of a perturbation expansion around a fixed spacetime. Remarkably, in this formulation certain infinities that plague perturbative quantum general relativity do not appear and there is an ongoing program to establish that each term in the series is in fact finite. However, when summed the entire series appears to be divergent. In any case, a definition of the theory as a perturbation expansion is not sufficient for describing genuine quantum gravitational phenomena, which appear in the nonperturbative regime.
Therefore much of the research in this area in recent years focuses on nonperturbative effects. Numerous indications point out to the existence of a fundamental nonperturbative definition of string theory, to which the various perturbative formulations
should converge, and various partial attempts to define this nonperturbative theory are
ongoing. In this sought-for nonperturbative formulation,the characteristic
features of quantum gravity become manifest: for instance, the lack of a fixed background
space and time, and the resulting conceptual difficulties. Research is currently active to try
to find this fundamental description of string theory.

### Loop quantum gravity and spinfoams

Loop quantum gravity is the most developed attempt to solve the specific problem
of finding a quantum version of general relativity, and thus a genuine merge of the
conceptual novelties represented by general relativity and quantum mechanics.
The theory is consistent with the other fundamental physical theories (such as
the standard model of particle physics), but it does *not* unify gravity and these
theories in the sense of presenting them as manifestation of the dynamics of
a single physical entity. Loop quantum gravity is based in the formulation of
general relativity developed by Abhay Ashtekar, and offers a precise mathematical
description of quantum spacetime. The granular properties of space can be
explicitly computed. In particular, a result of the theory is that the
*area* and the *volume* of any physical surface of region turns out to be
"quantized" (like the energy of an hydrogen atom), and the corresponding
discrete values that area and volume can take have been computed within the
theory. The quantum states of physical space are described in the theory by
labelled graphs called *spin networks*. Each node of a spin network
represents an elementary "quantum of space", and the links between these
indicates who is next to who, building the spacial structure. The main incompleteness
of the theory regards the relation between the Planck scale and macroscopic physics,
and the consistency of its classical limit.

Related to loop quantum gravity is a covariant approach to quantum that
goes under the name of *spinfoam formalism*. This is sometimes presented as
the path-integral, or, covariant (Lagrangian) version of the canonical (Hamiltonian) loop formalism.

### Noncommutative geometry

Einstein's discovery is that gravity, which is a dynamical field, is the geometry of spacetime. Quantum mechanics teaches us that dynamical quantities are noncommutative --meaning that the outcome of the measurement of two of their properties depends on the order in which the measurement is performed. It is therefore natural to suspect that the mathematics needed to describe quantum spacetime is a noncommutative version of geometry. A number of different formulations of such a noncommutative theory of geometry are under study. One line of research sees noncommutative spaces as effective (approximate) descriptions of the deep quantum structure of spacetime, and studies quantum field theory over such spaces. Alain Connes has developed a beautiful mathematical framework for describing noncommutative metric spaces. This framework appears to shed surprising light on the physical geometry underlying the standard model of particle physics and its relation with general relativity, and opens a intriguing perspectives towards quantum gravity.

### Others

A number of directions of research are currently developed by smaller research communities. Among these, Roger Penrose has introduced an approach to quantum gravity denoted "Twistor theory", in which the primary elements for the description of spacetime are not the spacetime points but the light rays. Rafael Sorkin has introduced an approach denoted "Coset (or Causal Set) theory", where quantum spacetime is described by probabilistic superpositions of sets of points whose only structure is their causal connection. Finally, an approach that must be mentioned is simply the possibility of taking general relativity and merge it with quantum mechanics following the conventional methods of quantum field theory, but circumventing the traditional difficulties by an intelligent retuning of the quantization method itself. Several authors have questioned the conventional assumption that this is manifestly impossible.

The diversity of the approaches is not a sign of excessive confusion, but a sign of a healthy research landscape. In the history of physics, well posed problems on which the community of scientists has focused its attention have rarely resisted more than a few decades. The common hope, today, is that the development of the theoretical research and, hopefully, some direct input from experiments and observations could soon bring to a convergence or a selection among these tentative theories, and therefore more clarity on the quantum nature of space and time.

## References

**Introductions to string theory:**

- Green, M., Schwarz, J., and Witten, E. (1987). Superstring Theory. Cambridge University Press, Cambridge, UK.
- Polchinski, J. (1990). String Theory. Cambridge University Press, Cambridge UK.

**Introductions to loop quantum gravity (the first of these two books contains a general discussion of the conceptual problems raised by quantum gravity and an appendix on the history of the theory):**

- Rovelli, C. (2004). Quantum Gravity. Cambridge University Press, Cambridge UK.
- Thiemann, T. (2007). Modern Canonical Quantum General Relativity. Cambridge University Press, Cambridge UK.
- Baez, J.C., Muniain, J.P. (1994). Gauge Fields, Knots, and Gravity. World Scientific Publishing Company, USA.

**Introductions to noncommutative geometry:**

- Connes, A. (1994). Noncommutative Geometry. Academic Press, Sand Diego CA.

**Conceptual issues and overview of many current approaches to quantum gravity:**

- Ashtekar, A. and Stachel J. (1991) Conceptual Problems of Quantum Gravity. Birkhauser, Boston-Basel-Berlin.
- Oriti, D. (2008). Approaches to Quantum Gravity: Towards a New Understanding of Space, Time and Matter. Cambridge UP, Cambridge, UK.

**Collections of philosophers texts of quantum gravity:**

- Callender, C., Huggett, N. editors (2001). Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity. Cambridge UP, Cambridge, UK.
- Rickles, D., French S. and Saatsi, J. editors (2007). The Structural Foundations of Quantum Gravity. Oxford University Press, USA.

**Popularisation works:**

- Smolin, L. (2000). Three Roads to Quantum Gravity. Oxford University Press, Oxford, UK.
- Penrose, R. (2005), The road to reality. Oxford University Press, Oxford UK.
- Green, B. (1999), The Elegant Universe. Vintage, New York, NY.
- Rovelli, C. (2004). What is Time? What is Space? Di Renzo editore, Rome, Italy.

**Internal references**

- Teviet Creighton and Richard H. Price (2008) Black holes. Scholarpedia, 3(1):4277.

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

- Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.

- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.