Loop quantum gravity
Loop quantum gravity (LQG), also known as quantum geometry and canonical quantum general relativity, is a proposed quantum theory of spacetime which blends together the seemingly incompatible theories of quantum mechanics and general relativity. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphism-invariant gauge theories.As a theory of quantum gravity, LQG is the main competitor of string theory, albeit a distant one: stringy people outnumber loopy people by a factor of roughly 10 and stringy papers outnumber loopy papers by a factor of roughly 50.
LQG in itself is less ambitious than string theory, purporting only to be a quantum theory of gravity; string theory, on the other hand, automatically accommodates matter particles, gauge vector bosons and the graviton, which suggested early in its development that strings might be able to model all known fundamental physics. Should LQG succeed as a quantum theory of gravity, however, the known matter fields would have to be incorporated into the theory using the broader formalism. Lee Smolin, one of the fathers of LQG, has explored the possibility that string theory and LQG are two different approximations to the same ultimate theory.
The main successes of loop quantum gravity are: a nonperturbative quantization of 3-space geometry, with quantized area and volume operators; a calculation of the entropy of physical black holes; and a proof by example that it is not necessary to have a theory of everything in order to have a candidate for a quantum theory of gravity. Many of the core results in LQG are established at the level of rigour of mathematical physics. Its main shortcomings are: not yet having a picture of dynamics but only of kinematics; not yet able to perform particle physics calculations; not yet able to recover the classical limit. These difficulties may all be related.
The incompatibility between 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 dynamical. 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 dynamical, 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 controversial, as it is contrary to the way quantum field theory on Minkowski space is developed by Steven Weinberg's book Quantum Field Theory.
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. It is interesting that 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. Since then, though, progress was rapid on both fronts, leading ultimately to string theory and loop quantum gravity.
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.
History of LQG
Main article: history of loop quantum gravity
General relativity is the theory of gravitation published by Albert Einstein in 1915. According to it, the force of gravity is a manifestation of the local geometry of spacetime. Mathematically, the theory is modelled after Riemann's metric geometry, but the Lorentz group of spacetime symmetries (an essential ingredient of Einstein's own theory of special relativity) replaces the group of rotational symmetries of space. LQG inherits this geometric interpretation of gravity, and posits that a quantum theory of gravity is fundamentally a quantum theory of spacetime.
In the 1920s the French mathematician Elie Cartan formulated Einstein's theory in the language of bundles and connections, a generalization of Riemann's geometry to which Cartan made important contributions. The so-called Einstein-Cartan theory of gravity not only reformulated but also generalized general relativity, and allowed spacetimes with torsion as well as curvature. In Cartan's geometry of bundles the concept of parallel transport is more fundamental than that of distance, the centerpiece of Riemannian geometry. A similar conceptual shift occurs between the invariant interval of Einstein's general relativity and the parallel transport of Einstein-Cartan theory.
In the 1960s physicist Roger Penrose explored the idea of space arising from a quantum combinatorial structure. His investigations resulted in the development of spin networks. Because this was a quantum theory of the rotational group and not the Lorentz group, Penrose went on to develop twistors.
In 1986 physicist Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. Using this reformulation, he was able to quantize gravity using well-known techniques from quantum gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point.
The quantization of gravity in the Ashtekar formulation was based on Wilson loops, a technique developed in the 1970's to study the strong-interaction regime of quantum chromodynamics. It is interesting in this connection that Wilson loops were known to be ill-behaved in the case of standard quantum field theory on (flat) Minkowski space, and so did not provide a nonperturbative quantization of QCD. However, because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for nonperturbative quantization of gravity.
Ashtekar's work resulted, for the first time, in a setting where the Wheeler-DeWitt equation could be written in terms of a well-defined Hamiltonian operator on a well-defined Hilbert space, and led to construction of the first known exact solution, the so-called Chern-Simons or Kodama state. The physical interpretation of this state remains obscure.
Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial. Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics, as opposed to string theory, which is established at the level of rigour of physics.
After the spin network basis was described, progress was made on the analysis of the spectra of various operators resulting in a predicted spectrum for area and volume (see below). Work on the semiclassical limit, the continuum limit, and dynamics was intense after this but progress slower.
On the semiclassical limit front, the goal is to obtain and study analogues of the harmonic oscillator coherent states (candidates are known as weave states).
LQG was initially formulated as a a quantization of the Hamiltonian ADM formalism, according to which the Einstein equations are a collection of constraints (Gauss, Diffeomorphism and Hamiltonian). The kinematics are encoded in the Gauss and Diffeomorphism constraints, whose solution is the space spanned by the spin network basis. The problem is to define the Hamiltonian constraint as a self-adjoint operator on the kinematical state space. The most promising work in this direction is Thomas Thiemann's Phoenix program.
Spin foams are new framework intended to tackle the problem of dynamics and the continuum limit simultaneously. Heuristically, it would be expected that evolution between spin network states might be described by discrete combinatorial operations on the spin networks, which would then trace a two-dimensional skeleton of spacetime. This approach is related to state-sum models of statistical mechanics and topological quantum field theory such as the Turaeev-Viro model of 3D quantum gravity, and also to the Regge calculus approach to calculate the Feynman path integral of general relativity by discretizing spacetime.
Some radical approaches to spin foams include the work on causal sets by Fotini Markopoulou and Rafael Sorkin, among others.
The ingredients of loop quantum gravity
Loop quantization
At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions (Baez and Krasnov), an arbitrary gauge group (or even quantum group), and supersymmetry (Smolin), and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.
Lorentz invariance
LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just like it is broken in general relativity (unless one is dealing with Minkowski spacetime, which is one particular solution of the Einstein field equations). On the other hand, there has been much talk about possible local and global violations of Lorentz invariance beyond those expected in straightforward general relativity. Of interest in this connection would be to see whether the LQG analogue of Minkowski spacetime breaks or preserves global lorentz invariance, and Carlo Rovelli and coworkers have revently been investigating the Minkowski state of LQG using spin-foam techniques.
These questions (and the ones raised bythe following discusion) will all remain open as long as the classical limits of various LQG models (see below for the sources of variation) cannot be calculated.
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact SO(3) or SU(2) gauge group as opposed to the non-compact SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus-far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The infamous Immirzi parameter is necessary to resolve an ambiguity in the process of complexification. These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through.
It should be pointed out that the reasons why one can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level is that the respective Lie algebras are the same. In fact, all four groups have the same complexified Lie algebra, which makes matters even more confusing (these subtleties are usually ignored in elementary particle physics). The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified Lie groups are indistinguishable in the absence of matter fields.
To make matters more complicated, it can be show that a positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group. At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated (in the case of the rotation group, instead of having representations labelled by all half-integral spins, one is left with all representations with total spin j less than some constant). It is entirely possible to formulate LQG in terms of q-deformed Lie algebras instead of ordinary Lie algebras, and in the case of the Lorentz group the result would, again, be indistinguishable for sufficiently small velocity paramenters.
In the spin-foam formalism the Barrett-Crane model, which was for a while the most promising state-sum model of 4D Lorentzian quantum gravity, was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks.
These and other considerations, including difficulties interpreting what it would mean to apply a Lorentz transformation to a spin network state, led Lee Smolin and others to suggest that spin network states must break Lorentz invariance. Lee Smolin and Joao Magueijo then went on to study doubly-special relativity, in which not only there is a constant velocity c but also a constant distance l. They showed that there are nonlinear representations of the Lorentz lie algebra with these prperties (the usual Lorentz group being obtained from a linear representation). Doubly-special relativity predicts deviations from the special relativity dispersion relation at large energies (corresponding to small wavelengths of the order of the constant length l in the doubly-special theory). Giovanni Amelino-Camelia then proposed that the mystery of ultra-high-energy cosmic rays might be solved by assuming such violations of the special-relativity dispertion relation for photons.
General relativity and diffeomorphism invariance
General covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. This symmetry is one of the defining features of general relativity. LQG preserves this symmetry by requiring that the physical states must be invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffemorphisms; however the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics, and a generally accepted calculational framework to account for this constraint is yet to be found.
Background independence
Despite the broken Lorentz invariance, LQG is formally background independent. The equations of LQG are not embedded in or presuppose space and time (except for its topology that cannot be changed), but rather they give rise to and create space and time at the Planckian and trans-Planckian distances. This reflects a philosophical view that gravity is the very geometric fabric of space and time, and that a quantum theory of gravity must be a quantum theory of space and time while all other particles and forces must be separated: LQG predicts that unification of forces can never occur. At any rate, any theory of quantum gravity must give an account of space and time. Unfortunately, it has not been yet shown that LQG's description of spacetime at the Planckian and trans-Planckian scales can lead to spacetime as described by general relativity.
Mathematics
As with any theory of quantum gravity, research into LQG employs a variety of techniques and fields of mathematics. A partial list would include
- Heyting algebra
- mathematical category theory
- noncommutative geometry
- topos theory
- C* Algebra
- Regge calculus
Open problems
The classical limit
Any successful theory of quantum gravity must provide physical predictions that closely match known observation, and reproduce the results of quantum field theory and gravity. To date Einstein's theory of general relativity is the most successful theory of gravity. It has been shown that quantizing the field equations of general relativity will not necessarily recover those equations in the classical limit. It remains unclear whether LQG yields results that match general relativity in the domain of low-energy, macroscopic and astronomical realm. To date, LQG has been shown to yield results that match general relativity in 1+1 and 2+1 dimensions where the metric tensor carries no physical degrees of freedom. To date, it has not been shown that LQG reproduces classical gravity in 3+1 dimensions. Thus, it remains unclear whether LQG successfully merges quantum mechanics with general relativity.
Time
Additionally, in LQG, time is not continuous but discrete and quantized, just as space is: there is a minimum moment of time, Planck time, which is on the order of 10^{−43} seconds, and shorter intervals of time have no physical meaning. This carries the physical implication that relativity's prediction of time dilation due to accelerating speed or gravitational field, must be quantized, and must consist of multiples of Planck time units. (This helps resolve the time zero singularity problem: see subsection "The big bang")
Particle physics
While classical particle physics posit particles traveling through space and time that is continuous and therefore infinitely divisible, LQG predicts that space-time is quantized or granular. The two different models of space and time affects the way ultra high energy cosmic rays interacts with the background, with quantized spacetime predicting that the threshold for allowable energies for such high energy particles be raised. Such particles have been observed, however, alternative explanations have not been ruled out.
LQG does not constrain the spectrum of non-gravitational forces and elementary particles. Unlike the situation in string theory, all of them must be added to LQG by hand. It has proved difficult to incorporate elementary scalar fields, Higgs mechanism, and CP-violation into the framework of LQG.
Quantum field theory
Quantum field theory is perturbative and background dependent. One problem LQG may be able to address in QFT is the ultraviolet catastrophe.
The term ultraviolet catastrophe has also been applied to similar situations in quantum electrodynamics in which summing across all energies results in an infinite value because the higher energy terms do not decrease quickly enough to create finite values.
In LQG, the background quantum field theory depends on is quantized, and hence, there is apparently no physical "room" for the ultraviolet infinities to occur. However, this argument may be compromised if LQG does not admit a limit of smooth geometry at long distance scales. It should bear in mind that LQG is constructed as an alternative to perturbative quantum field theory on a fixed background. In its present form, it does not allow a perturbative calculation of graviton scattering or other processes and it is not clear whether it ever will.
Graviton
In quantum field theories, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. In order to do this gravitons have to be always-attractive (gravity never pushes), work over any distance (gravity is universal) and come in unlimited numbers (to provide high strengths near stars). In quantum theory, this defines an even-spin (spin 2 in this case) boson with a rest mass of zero.
It remains open to debate whether loop quantum gravity requires, or does not require, the graviton, or whether the graviton can be accounted for in its theoretical framework. As of today, the appearance of smooth space and gravitons in LQG has not been demonstrated, and henceforth the questions about graviton scattering cannot be answered in LQG.
Loop quantum gravity's implications
Space atoms
In LQG, the fabric of spacetime is a foamy network of interacting loops mathematically described by spin networks. These loops are about 10^{-35} meters in size, called the Planck scale. The loops knot together forming edges, surfaces, and vertices, much as do soap bubbles joined together. In other words, spacetime itself is quantized. Any attempt to divide a loop would, if successful, cause it to divide into two loops each with the original size. In LQG, spin networks represent the quantum states of the geometry of relative spacetime. Looked at another way, Einstein's theory of general relativity is (as Einstein predicted) a classical approximation of a quantized geometry.
Kinematics
One of the key results of loop quantum gravity is quantization of areas: according to several related derivations based on loop quantum gravity, the operator of the area of a two-dimensional surface should have discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals
Similar quantization applies to the volume operators but the mathematics behind these derivations is less convincing.
Quantum cosmology
An important principle in quantum cosmology that LQG adheres to is that there are no observers outside the universe. All observers must be a part of the universe they are observing. However, because light cones limit the information that is available to any observer, the Platonic idea of absolute truths does not exist in a LQG universe. Instead, there exists a consistency of truths in that every observer will report consistent (not necessarily the same) results if truthful.
Another important principle is the issue of the "cosmological constant", which is the energy density inherent in a vacuum. Cosmologists working on the assumption of zero cosmological constant predicted that gravity would slow the rate at which the universe is expanding following the big bang. However, astronomical observations of the magnitude and cosmological redshift of Type I supernovae in remote galaxies implies that the rate at which the universe is expanding is actually increasing. General relativity has a constant, Lambda, to account for this, and the observations, recently supported by independent data on the cosmic microwave background, appear to require a positive cosmological constant. In string theory, there are many vacua with broken supersymmetry which have positive cosmological constant, but generically their value of Lambda is much larger than the observed value. In LQG, there have been proposals to include a positive cosmological constant, involving a state referred to as the Kodama state after Hideo Kodama, a state described by a Chern-Simons wave function. Some physicists, for example Edward Witten, have argued by analogy with other theories that this state is unphysical. This issue is considered unresolved by other physicists.
Standard quantum field theory and supersymmetric string theories make a prediction based on calculation of the vacuum energy density that differs from what has actually been observed by 120 orders of magnitude. To date, this remains an unsolved mystery that a successful quantum theory of gravity would hopefully avoid
Black hole thermodynamics
While experimental tests for LQG may be years in the future, one conceptual test any candidate for QG must pass is that it must derive the correct formula Hawking derived for the black hole entropy.
With the proper Immirzi parameter, LQG can calculate and reproduce the Hawking formula for all black holes. While string/M-theory does not need the Immirzi parameter, it can as yet only derive the Hawking formula for extremal black holes and near-extremal black holes -- black holes with a net electric charge, which differ from the nearly neutral black holes formed from the collapse of electrically neutral matter such as neutron stars. To date, the Immirzi parameter cannot be derived from more fundamental principles, and is an unavoidable artefact of quantization of general relativity's field equations.
LQG's interpretation of black hole entropy is that the spacetime fabric that make up the black hole horizon is quantized per Planck area, and the Bekenstein-Hawking entropy represents the degrees of freedom present in each Planck quanta. LQG does not offer an explanation why the interior of the black hole carries no volume-extensive entropy. Instead, it assumes that the interior does not contribute. The spacetime is truncated at the event horizon, and consistency requires to add Chern-Simons theory at the event horizon. A calculation within Chern-Simons theory leads to the desired result for the entropy, proportional to the horizon area.
Additionally, the spectrum of radiation of particles emanating from the event horizon of a black hole has been calculated from LQG's theoretical framework and precisely predicted. This prediction disagrees with Hawking's semiclassical calculation, but the use of a semiclassical calculation that is so far unconfirmed by experiment as a benchmark for an exact nonperturbative fully quantum calculation may be problematic. Modulo the Immirzi parameter, which is the only free parameter of LQG, it matches it on average, and additionally predicts a fine structure to it, which is experimentally testable and potentially an improvement.
The big bang
Several LQG physicists have shown that LQG can, at least formally, get rid of the infinities and singularities present when general relativity is applied to the big bang. While standard physics tools break down, LQG have provided internally self-consistent models of a big bounce in the time preceding the big bang.
Theory of everything: unification of the four forces
Grand unification theory refers to a theory in particle physics that unifies the strong interaction and electroweak interactions. A so-called theory of everything (TOE) is a putative theory that unifies the four fundamental forces of nature: gravity, the strong nuclear force, the weak nuclear force, and electromagnetism. Since the strong and electroweak interactions are described by quantum field theory, such a theory would require gravity also to be quantized, bringing with it the inconsistencies noted above.
One candidate for a consistent quantum gravity is string theory, which in addition to gravity contains gauge vector bosons and matter particles reminiscent of those experimentally observed. This has led to attempts (so far unsuccessful) to construct TOE's within its framework. In contrast, LQG is just a theory of one part of the Universe, namely quantum gravity.
Unification in field theory or string theory is difficult or impossible to test directly, due to the extremely large energy (greater than 10^{16} GeV) at which unification is manifest. However, indirect tests exist, such as proton decay and the convergence of the coupling constants when extrapolated to high energy through the renormalization group. The simplest unified models (without supersymmetry) have failed such tests, but many models are still viable. Incorporating the correct strength of gravity in string unification is particularly challenging. While unified theories have greater explanatory and predictive power, it may be that nature does not favour them.
Supersymmetry and extra dimensions
For string theory to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In contrast, LQG is a theory of quantum gravity which does not require supersymmetry. In particle theory, supersymmetry is recognized as a way to stabilize the hierarchy between the unification scale and the electroweak scale (or the Higgs particle mass), and can also provide a natural dark matter candidate; it may also introduce additional theoretical problems.
To date, no supersymmetric partner particles have been experimentally observed. If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino that is often believed to be the lightest superpartner, (the superpartner of the photon, Z boson, or Higgs boson), quite possibly as early as 2007 when Europe's Large Hadron Collider (LHC) will be in operation with sufficient energies to produce such particles, it may be possible to modify LQG's spin networks to accommodate these discoveries by requiring the spin networks to carry more quantum numbers.
String theory also requires extra spatial dimensions which have to be "hidden" somewhat as in Kaluza-Klein theory. By contrast, LQG in its current formulation predicts no additional spatial dimensions, nor anything else about particle physics: the minimal LQG is formulated in 3 spatial dimensions and one dimension of time. Again, as yet there is no experimental evidence for extra spatial dimensions, however it is possible (though seen as unlikely) that Kaluza-Klein modes may be seen at the LHC, or that there may be other future signals of extra dimensions.
Lee Smolin, one of the originators of LQG, has proposed that loop quantum gravity incorporating either supersymmetry or extra dimensions, or both, be called loop quantum gravity II, in light of experimental evidence.
Chaos theory and classical physics
Chaos theory sensitivity on the initial conditions means that two nonlinear systems with however small a difference in their initial state eventually will end up with a finite difference between their states (however, two deterministic systems with identical initial conditions will remain identical). Since most chaos theory is formulated in partial differential equations, it is thought that for two systems to have the same exact initial conditions, they must be reproduced with infinite accuracy.
Loop quantum gravity suggests that the Planck scale represents the physical cut-off allowed for such sensitivity -- a chaotic's system's sensitivity to initial conditions need only be reproduced at the Planck scale.
Similarly, many of the equations in classical physics, such as the equations of motion, must be seen as a classical approximation to quantized geometry, and hence, a particle traveling through space is "jumping" from one space-time quantum to the next, much as the electron orbits of a hydrogen atom are quantized.
Philosophy around LQG
LQG begins with different philosophical assumptions from those of particle physics and string theory. LQG has also influenced some philosophers, especially in the philosophy of science.
One of the long standing debates in a branch of philosophy known as metaphysics concerns the "ultimate" nature of space and time. LQG preserves Einstein's position on spacetime by preserving diffeomorphism invariance, while breaking Lorentz invariance. On the philosophy of space and time, two well-established positions have been developed: one, that space and time is absolute, and the other, space and time is relational. LQG takes the latter position. While philosophy and metaphysics offers no clear cut method of determining which of the two positions is more plausible, LQG offers predictions that could at least in principle, if not yet technologically feasible, be empirically tested or experimentally falsified.
An underlying philosophy of LQG is that fundamental physics should provide an account for space and time.
Certain predictions that originated from black hole thermodynamics, such as the holographic principle, and its parsimonious methodology, has resulted in fruitful debate in the philosophy of science and physics community.
Research and progress in Quantum Gravity has stimulated philosophy of science and mathematics, including LQG and string theory.
Differences between LQG and string/M-theory
String theory and LQG are the products of different communities. String theory emerged from the particle physics community and was originally formulated as a theory that depends on a background spacetime, flat or curved, which obeys Einstein's equations. This is now known to be just an approximation to a mysterious and not well-formulated underlying theory which may or may not be background independent.
In contrast, LQG was formulated with background independence in mind. However, it has been difficult to show that classical gravity can be recovered from the theory. Thus, LQG and string theory seem somewhat complementary. String theory easily recovers classical gravity, but so far it lacks a universal, perhaps background independent, description. LQG is a background independent theory of something, but the classical limit has yet not proven tractable. This has led some people to conjecture that LQG and string theory may both be aspects of some new theory, or that, perhaps there is some synthesis of the techniques of each that will lead to a complete theory of quantum gravity. For now, this is mostly a fond hope with little evidence.
Experimental tests of LQG in the near future
LQG may make predictions that can be experimentally testable in the near future.
The path taken by a photon through a discrete spacetime geometry would be different from the path taken by the same photon through continuous spacetime. Normally, such differences should be insignificant, but Giovanni Amelino-Camelia points out that photons which have traveled from distant galaxies may reveal the structure of spacetime. LQG predicts that more energetic photons should travel ever so slightly faster than less energetic photons. This effect would be too small to observe within our galaxy. However, light reaching us from gamma ray bursts in other galaxies should manifest a varying spectral shift over time. In other words, distant gamma ray bursts should appear to start off more bluish and end more reddish. Alternatively, highly energetic photons from gamma ray bursts should arrive a split second sooner than less energetic photons. LQG physicists eagerly await results from space-based gamma-ray spectrometry experiments (GLAST).
2007 will see the launch of GLAST, and (hopefully) the completion and operation of LHC. The results of these experiments will profoundly develop the course of QG. These experiments may establish spontaneously broken supersymmetry, Higgs boson and the Higgs field, extra spatial dimensions, and/or violations of Lorentz invariance.
If GLAST detects violations of Lorentz invariance in the form of energy-dependent photon velocity, in agreement with theoretical calculations, such observations would strongly support LQG. However, string theory would not necessarily be disfavoured, since although it predicts an underlying exact Lorentz symmetry, it is possible that this may be spontaneously broken through a nonzero expectation value of tensor fields.
Other topics where observation may affect the future theoretical development of quantum gravity are dark matter and dark energy.
LQG and the sociology and politics of science
To date, there is no experimental evidence to support any theory of quantum gravity, whether string/M-theory or loop quantum gravity. As previously noted, however, the string community outnumbers the loop community by at least a factor of 10:1, and the literature on string theory exceeds the literature on LQG roughly by a factor of 50:1. Furthermore, string theory has achieved far more success than loop quantum gravity in attracting graduate and doctoral students, professors, research positions, and funding, and more recognition in popular culture. Contributing to this situation is the fact that string theory has been under development for about 30 years and started out as a natural extension of ordinary perturbative quantum field theory, while loop quantum gravity has been around for 15 to 20 years and represents a significant methodological and conceptual departure from perturbative quantum field theory.
Although this asymmetry is deeply rooted in different approaches of both communities to scientific issues, the question how this imbalance has come about, and the nature, controversies, and interactions between the two communities, has nevertheless provided fascinating material in the sociology, economics, and politics of science, academia, and higher learning.
String theorist Brian Greene, author of the popular books "The Elegant Universe" (a Pulitzer finalist) and "The Fabric of the Cosmos" only dedicated one page to LQG, and has stated that string theory is the only viable theory to Quantum gravity. Lee Smolin, one of the originators of LQG, has stated that theorists in both camps are often convinced their approach is the only viable approach.
Recently, Portuguese cosmologist Joao Magueijo wrote in his book "Faster Than the Speed of Light: The Story of a Scientific Speculation" of the intense rivalry, even hostility between these opposing camps of quantum gravity theorists. The hostility stems in part from funding: academic positions open to PhDs for string theorists are unavailable to loop quantum gravity theorists. Currently, universities have employed more string theorists than LQG theorists. It also stems from the fact that string/M-theory and LQG in their present forms, makes different physical predictions, and appear to be incompatible, make enormous academic demands on its researchers, and that it is psychologically difficult for anyone to accept he or she may have dedicated their graduate studies and academic careers on the wrong theory.
On the physics newsgroup sci.physics.research, vigorous debates used to take place between string theorists, notably Lubos Motl and loop quantum gravity theorist and (now former) s.p.r. moderator John Baez, with general relativists such as Steve Carlip taking sides "against" strings. As a result Lubos Motl spearheaded the creation of a new newsgroup, sci.physics.strings. These heated debates did not preclude scientific collaboration of sorts. After Baez announced in his column "This Week's Finds in Mathematical Physics" that Olaf Dreyer had found a connection between the Immirzi parameter and the asymptotic behavior of black hole quasinormal modes (in numerical general relativity), Lubos Motl wasted no time proving that the exact asymptotic behavior was as predicted numerically.
Hopefully, experimental results as early as 2007 to 2010 may help guide researchers and funding.
Research in LQG and related areas
Active research directions
- Spin foam models
- 2+1 and 3+1 theories
- Barrett-Crane model
- relation to the canonical approach
- the Barbero-Immirzi parameter
- canonical and spin foam geometries
- the continuum limit
- renormalization group flows
- the Hamiltonian constraint
- 2+1 and 3+1 theories
- spin-foam and canonical approach
- quantum cosmology
- Semi-classical corrections to Einstein equations
- factor ordering
- finding solutions and physical inner product
- Thiemann's phoenix project.
- Semi-classical issues
- kinematical and dynamical semi-classical states
- quantum field theory on quantum geometry
- quantum cosmology
- Minkowski coherent state and Minkowski spin foam
- Loop quantum phenomenology
- Lorentz invariance
- Doubly-special relativity
- quantum cosmology
- Kodama state and de Sitter background
- Conceptual issues
- observables through matter coupling
- string theory in polymer representation
- matter couplings on semi-classical states
- the problem of time
- spin foam histories
- quantum groups in LQG
People
- Giovanni Amelino-Camelia
- Abhay Ashtekar
- John Baez
- Julian Barbour
- John Barrett
- Martin Bojowald
- Luca Bombelli
- Daniel Christensen
- Alejandro Corichi
- Louis Crane
- Laurent Freidel
- Rodolfo Gambini
- Florian Girelli
- Viqar Husain
- Giorgio Immirzi
- Christopher Isham
- Kirill Krasnov
- Jerzy Lewandowski
- Etera Livine
- Renate Loll
- Seth Major
- Fotini Markopoulou-Kalamara
- Donald Marolf
- Merced Montesinos
- Hugo Morales-Tecotl
- Karim Noui
- Robert Oeckl
- Daniele Oriti
- Alejandro Perez
- Jorge Pullin
- Michael Reisenberger
- Carlo Rovelli
- Hanno Sahlmann
- Lee Smolin
- Daniel Sudarsky
- Thomas Thiemann
- Oliver Winkler
- José Antonio Zapata;
Places
- New Brunswick
- Mississipi
- Louisiana
- Uruguay
- UC Riverside
- University of Western Ontario
- Kansas
- Penn State
- Hamilton
- Mexico
- Perimeter Institute for Theoretical Physics
Graduate and doctoral studies in LQG
The following universities have physics departments that offer graduate-level and doctoral studies for undergraduate physics and mathematics majors interested in pursuing research and advance degrees in LQG.
- Penn state www.psu.edu
Bibliography
- Popular books:
- Julian Barbour, The End of Time.
- Lee Smolin, Three Roads to Quantum Gravity
- Magazine Articles
- Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004
- Introductory/expository works:
- John Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994), ISBN 9810220340
- Carlo Rovelli, A Dialog on Quantum Gravity, preprint available as hep-th/0310077
- Advanced books, reports, conference proceedings:
- Robert M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Chicago University Press (1994), ISBN 0226-87027-8
- Robert M. Wald, General Relativity, Chicago University Press, ISBN 0-226-87033-2
- Steven Weinberg, Gravitation and Cosmology: principles and applications of the general theory of relativity, Wiley (1972), ISBN 0-471-92567-5
- Misner, Thorne and Wheeler, Gravitation, Freeman, (1973), ISBN 0-7167-0344-0
- A. Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
- Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity
- John Baez (ed.), Knots and Quantum Gravity
External links
- Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/
- Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/