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# Loop quantum gravity

Loop quantum gravity (LQG), also known as loop gravity, quantum geometry and canonical quantum general relativity, is a proposed quantum theory of spacetime which attempts to blend 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.

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Leading advocates of loop quantum gravity consider it the main competitor of string theory as a theory of quantum gravity. The current position (as of 2004) is not symmetrical, in that string theorists are generally not concerned with loop quantum gravity, or they are skeptical that there is any real competition. String theory has its many critics, but has been the dominant force in quantum gravity since the mid-1980s.

## Loop quantum gravity in general, and its ambitions

LQG in itself was initially less ambitious than string theory, purporting only to be a quantum theory of gravity. String theory, on the other hand, appears to predict not only gravity but also various kinds of matter and energy that lie inside spacetime. Many string theorists believe that it is not possible to quantize gravity in 3+1 dimensions without creating these artifacts. But this is not proven, and it is also not proven that the matter artifacts of string theory are exactly the same as observed matter. Should LQG succeed as a quantum theory of gravity, the known matter fields would have to be incorporated into the theory a posteriori. 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 claimed successes of loop quantum gravity are: (1) that it is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators; (2) that it includes a calculation of the entropy of black holes; and (3) that it is a viable gravity-only alternative to string theory. However, these claims are not universally accepted. While many of the core results are rigorous mathematical physics, their physical interpretations are speculative. LQG may or may not be viable as a refinement of either gravity or geometry; entropy is calculated for a kind of hole which may or may not be a black hole.

## The incompatibility between quantum mechanics and general relativity

Main article: quantum gravity

Quantum field theory studied on curved (non-Minkowskian) backgrounds has shown that some of the core assumptions of quantum field theory cannot be carried over. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see Unruh effect).

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 emergent. The other view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.

Loop quantum gravity is an effort to formulate a background-independent quantum theory. Topological quantum field theory is a background-independent quantum theory, but it lacks causally-propagating local degrees of freedom needed for 3 + 1 dimensional gravity.

## History of LQG

Main article: history of loop quantum gravity

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. He was able to quantize gravity using 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. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states.

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.

## 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.

In a nutshell, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions, because classical general relativity can be formulated as a BF theory with constraints, and it is hoped that a consistent quantization of gravity may arise from perturbation theory of BF spin-foam models.

### Lorentz invariance

For detailed discussion see the Lorentz covariance page

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 as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.

### Diffeomorphism invariance and background independence

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 the so-called problem of time in general relativity, and a generally accepted calculational framework to account for this constraint is yet to be found.

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory 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 are expected to give rise to space and time at large distances compared to the Planck length. It has not been yet shown that LQG's description of spacetime at the Planckian scale has the right continuum limit described by general relativity with possible quantum corrections.

## 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: "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 background dependent. One problem LQG may be able to address in QFT is the ultraviolet catastrophe.

The term ultraviolet catastrophe is also applied to similar situations in quantum electrodynamics. There, summing over all energies results in an infinite value because the higher energy terms do not decrease quickly.

In LQG, the background is quantized, and there is apparently no physical "room" for the ultraviolet infinities of quantum field theory to occur. This argument, however, may be compromised if LQG does not admit a limiting smooth geometry at long distance scales. 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.

### 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 hence questions about graviton scattering cannot be answered.

### People

See list of loop quantum gravity researchers

### 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

### 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

Kinematics in loop quantum gravity is the physics of space and time at the Planck scale. It is expressed in terms of area and volume operators, and spin foam formalism.

Area and volume operators

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 A of a two-dimensional surface Σ should have discrete spectrum. Every spin network is an eigenstate of each such operator, and the area eigenvalue equals

$A_{\Sigma} = 8\pi G_{\mathrm{Newton}} \gamma \sum_i \sqrt{j_i(j_i+1)}$

where the sum goes over all intersections i of Σ with the spin network. In this formula, GNewton is the gravitational constant, γ is the Immirzi parameter and $j_i=0,0.5,1,1.5,\dots$ is the spin associated with the link i of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.

Similar quantization applies to the volume operators but the mathematics behind these derivations is less convincing.

Spin foams

#### 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 makes up the black hole horizon is quantized per Planck area, and the Bekenstein-Hawking entropy represents the degrees of freedom present in each Planck quantum. 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 1016 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

See supersymmetry for detailed discussion

LQG in its current formulation predicts no additional spatial dimensions, nor anything else about particle physics. 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

Sensitivity on initial conditions, in the light of chaos theory 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. Loop quantum gravity suggests that the Planck scale represents the physical cut-off allowed for such sensitivity.

### Differences between LQG and string/M-theory

String theory and LQG are the products of different communities within theoretical physics. It is not generally agreed whether they are in any sense compatible, and their differences have sometimes been represented as different ways of doing physics. This is a sharp debate, or at times presented as such: in other words matters are currently subject to dialectic rather than experimental test.

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

Observation may affect the future theoretical development in quantum gravity in the areas of dark matter and dark energy. The year 2007 will see the launch of GLAST (space-based gamma-ray spectrometry experiments), and perhaps the completion and operation of LHC.

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. Giovanni Amelino-Camelia points out that photons which have traveled from distant galaxies may reveal the structure of spacetime.

If GLAST detects violations of Lorentz invariance in the form of energy-dependent photon velocity, in agreement with theoretical calculations, such observations would support LQG. However, string theory would not necessarily be disfavoured.

## Objections to the theory

Objections to the theory of loop quantum gravity


As a physical theory, loop quantum gravity has been subject to some heavy criticisms. Some objections to the ideas of loop quantum gravity are given here.

### Too many assumptions

OBJECTION Loop quantum gravity makes too many assumptions about the behavior of geometry at very short distances. It assumes that the metric tensor is a good variable at all distance scales, and it is the only relevant variable. It even assumes that Einstein's equations are more or less exact in the Planckian regime.

The spacetime dimensionality (four) is another assumption that is not questioned, much like the field content. Each of these assumptions is challenged in a general enough theory of quantum gravity, for example all the models that emerge from string theory.

These assumptions have neither theoretical nor experimental justification. Particular examples will be listed in a separate entry.

The most basic, underlying assumption is that the existence of a meaningful classical theory, of general relativity, implies that there must exist a "quantization" of this theory. This is commonly challenged. Many reasons are known why some classical theories do not have a quantum counterpart. Gauge anomalies are a prominent example. General relativity is usually taken to be another example, because its quantum version is not renormalizable.

It is known, therefore, that a classical theory is not always a good starting point for a quantum theory. Theorists of loop quantum gravity work with the assumption that "quantization" can be done, and continue to study it even if their picture seems inconsistent.

### Commentary from the renormalization group aspect

OBJECTION According to the logic of the renormalization group, the Einstein-Hilbert action is just an effective description at long distances; and it is guaranteed that it receives corrections at shorter distances. String theory even allows us to calculate these corrections in many cases.

There can be additional spatial dimensions; they have emerged in string theory and they are also naturally used in many other modern models of particle physics such as the Randall-Sundrum models. An infinite amount of new fields and variables associated with various objects (strings and branes) can appear, and indeed does appear according to string theory. Geometry underlying physics may become noncommutative, fuzzy, non-local, and so on. Loop quantum gravity ignores all these 20th and 21st century possibilities, and it insists on a 19th century image of the world which has become naive after the 20th century breakthroughs.

### As a predictive theory

OBJECTION Loop quantum gravity is not a predictive theory. It does not offer any possibility to predict new particles, forces and phenomena at shorter distances: all these objects must be added to the theory by hand. Loop quantum gravity therefore also makes it impossible to explain any relations between the known physical objects and laws.

Loop quantum gravity is not a unifying theory. This is not just an aesthetic imperfection: it is impossible to find a regime in real physics of this Universe in which non-gravitational forces can be completely neglected, except for classical physics of neutral stars and galaxies that also ignores quantum mechanics. For example, the electromagnetic and strong force are rather strong even at the Planck scale, and the character of the black hole evaporation would change dramatically had the Nature omitted the other forces and particles. Also, the loop quantum gravity advocates often claim that the framework of loop quantum gravity regularizes all possible UV divergences of gravity as well as other fields coupled to it. That would be a real catastrophe because any quantum field theory - including all non-renormalizable theories with any fields and any interactions - could be coupled to loop quantum gravity and the results of the calculations could be equal to anything in the world. The predictive power would be exactly equal to zero, much like in the case of a generic non-renormalizable theory. There is absolutely no uniqueness found in the realistic models based on loop quantum gravity. The only universal predictions - such as the Lorentz symmetry breaking discussed below - seem to be more or less ruled out on experimental grounds.

### Self-consistency

OBJECTION Unlike string theory, loop quantum gravity has not offered any non-trivial self-consistency checks of its statements and it has had no impact on the world of mathematics. It seems that the people are constructing it, instead of discovering it. There are no nice surprises in loop quantum gravity - the amount of consistency in the results never exceeds the amount of assumptions and input. For example, no answer has ever been calculated in two different ways so that the results would match. Whenever a really interesting question is asked - even if it is apparently a universal question, for example: "Can topology of space change?" - one can propose two versions of loop quantum gravity which lead to different answers.

There are many reasons to think that loop quantum gravity is internally inconsistent, or at least that it is inconsistent with the desired long-distance limit (which should be smooth space). Too many physical wisdoms seem to be violated. Unfortunately the loop quantum gravity advocates usually choose to ignore the problems. For example, the spin foam (path-integral) version of loop quantum gravity is believed to break unitarity. The usual reaction of the loop quantum gravity practitioners is the statement that unitarity follows from time-translation symmetry, and because this symmetry is broken (by a generic background) in GR, we do not have to require unitarity anymore. But this is a serious misunderstanding of the meaning and origin of unitarity. Unitarity is the requirement that the total probability of all alternatives (the squared length of a vector in the Hilbert space) must be conserved (well, it must always be 100%), and this requirement - or an equally-strong generalization of it - must hold under any circumstances, in any physically meaningful theory, including the case of the curved, time-dependent spacetime. Incidentally, the time-translation symmetry is related, via Noether's theorem, to a time-independent, conserved Hamiltonian, which is a completely different thing than unitarity.

A similar type of "anything goes" approach seems to be applied to other no-go theorems in physics.

### Gap to high-energy physics

OBJECTION Loop quantum gravity is isolated from particle physics. While extra fields must be added by hand, even this ad hoc procedure seems to be impossible in some cases. Scalar fields can't really work well within loop quantum gravity, and therefore this theory potentially contradicts the observed electroweak symmetry breaking, the violation of the CP symmetry, and other well-known and tested properties of particle physics.

Loop quantum gravity also may deny the importance of many methods and tools of particle physics - e.g. the perturbative techniques; the S-matrix, and so on. Loop quantum gravity therefore potentially disagrees with 99% of physics as we know it. Unfortunately, the isolation from particle physics follows from the basic opinions of loop quantum gravity practitioners and it seems very hard to imagine that a deeper theory can be created if the successful older theories, insights, and methods (and exciting newer ones) in the same or closely related fields are ignored.

### Smooth space as limiting case

OBJECTION Loop quantum gravity does not guarantee that smooth space as we know it will emerge as the correct approximation of the theory at long distances; there are in fact many reasons to be almost certain that the smooth space cannot emerge, and these problems of loop quantum gravity are analogous to other attempts to discretize gravity (e.g. putting gravity on lattice).

While string theory confirms general relativity or its extensions at long distances - where GR is tested - and modifies it at the shorter ones, loop quantum gravity does just the opposite. It claims that GR is formally exact at the Planck scale, but implies nothing about the correct behavior at long distances. It is reasonable to assume that the usual ultraviolet problems in quantum gravity are simply transmuted into infrared problems, except that the UV problems seem to be present in loop quantum gravity, too.

### Clash with special relativity

OBJECTION Loop quantum gravity violates the rules of special relativity that must be valid for all local physical observations. Spin networks represent a new reincarnation of the 19th century idea of the luminiferous aether - environment whose entropy density is probably Planckian and that picks a privileged reference frame. In other words, the very concept of a minimal distance (or area) is not compatible with the Lorentz contractions. The Lorentz invariance was the only real reason why Einstein had to find a new theory of gravity - Newton's gravitational laws were not compatible with his special relativity.

Despite claims about the background independence, loop quantum gravity does not respect even the special 1905 rules of Einstein; it is a non-relativistic theory. It conceptually belongs to the pre-1905 era and even if we imagine that loop quantum gravity has a realistic long-distance limit, loop quantum gravity has even fewer symmetries and nice properties than Newton's gravitational laws (which have an extra Galilean symmetry, and can also be written in a "background independent" way - and moreover, they allow us to calculate most of the observed gravitational effects well, unlike loop quantum gravity). It is a well-known fact that general relativity is called "general" because it has the same form for all observers including those undergoing a general accelerated motion - it is symmetric under all coordinate transformations - while "special" relativity is only symmetric under a subset of special (Lorentz and Poincaré) transformations that interchange inertial observers. The symmetry under any coordinate transformation is only broken spontaneously in general relativity, by the vacuum expectation value of the metric tensor, not explicitly (by the physical laws), and the local physics of all backgrounds is invariant under the Lorentz transformations.

Loop quantum gravity proponents often and explicitly state that they think that general relativity does not have to respect the Lorentz symmetry in any way - which displays a misunderstanding of the symmetry structure of special and general relativity (the symmetries in general relativity extend those in special relativity), as well as of the overwhelming experimental support for the postulates of special relativity. Loop quantum gravity also depends on the background in a lot of other ways - for example, the Hamiltonian version of loop quantum gravity requires us to choose a pre-determined spacetime topology which cannot change.

One can imagine that the Lorentz invariance is restored by fine-tuning of an infinite number of parameters, but nothing is known about the question whether it is possible, how such a fine-tuning should be done, and what it would mean. Also, it has been speculated that special relativity in loop quantum gravity may be superseded by the so-called doubly special relativity, but doubly special relativity is even more problematic than loop quantum gravity itself. For example, its new Lorentz transformations are non-local (two observers will not agree whether the lion is caught inside the cage) and their action on an object depends on whether the object is described as elementary or composite.

### Global justification of variables

OBJECTION The discrete area spectrum is not a consequence, but a questionable assumption of loop quantum gravity. The redefinition of the variables - the formulae to express the metric in terms of the Ashtekar variables (a gauge field) - is legitimate locally on the configuration space, but it is not justified globally because it imposes new periodicities and quantization laws that do not follow from the metric itself. The area quantization does not represent physics of quantum gravity but rather specific properties of this not-quite-legitimate field redefinition. One can construct infinitely many similar field redefinitions (siblings of loop quantum gravity) that would lead to other quantization rules for other quantities. It is probably not consistent to require any of these new quantization rules - for instance, one can see that these choices inevitably break the Lorentz invariance which is clearly a bad thing.

### Testability of the discrete area spectrum

OBJECTION The discrete area spectrum is not testable, not even in principle. Loop quantum gravity does not provide us with any "sticks" that could measure distances and areas with a sub-Planckian precision, and therefore a prediction about the exact sub-Planckian pattern of the spectrum is not verifiable. One would have to convert this spectrum into a statement about the scattering amplitudes.

### The S-matrix

OBJECTION Loop quantum gravity provides us with no tools to calculate the S-matrix, scattering cross sections, or any other truly physical observable. It is not surprising; if loop quantum gravity cannot predict the existence of space itself, it is even more difficult to decide whether it predicts the existence of gravitons and their interactions. The S-matrix is believed to be essentially the only gauge-invariant observable in quantum gravity, and any meaningful theory of quantum gravity should allow us to calculate it, at least in principle.

### Ultraviolet divergences

OBJECTION Loop quantum gravity does not really solve any UV problems. Quantized eigenvalues of geometry are not enough, and one can see UV singular and ambiguous terms in the volume operators and most other operators, especially the Hamiltonian constraint. Because the Hamiltonian defines all of dynamics, which contains most of the information about a physical theory, it is a serious object. The whole dynamics of loop quantum gravity is therefore at least as singular as it is in the usual perturbative treatment based on semiclassical physics.

We simply do have enough evidence that a pure theory of gravity, without any new degrees of freedom or new physics at the Planck scale, cannot be consistent at the quantum level, and loop quantum gravity advocates need to believe that the mathematical calculations leading to the infinite and inconsistent results (for example, the two-loop non-renormalizable terms in the effective action) must be incorrect, but they cannot say what is technically incorrect about them and how exactly is loop quantum gravity supposed to fix them. Moreover, the loop quantum gravity proponents seem to believe that the naive notion of "atoms of space" is the only way to fix the UV problems. String theory, which allows us to make real quantitative computations, proves that it is not the case and there are more natural ways to "smear out" the UV problems. In fact, a legitimate viewpoint implies that the discrete, sharp character of the metric tensor and other fields at very short distances makes the UV behavior worse, not better.

Moreover, as explained above, the "universal solution of the UV problems by discreteness of space" implies at least as serious loss of predictive power as in a generic non-renormalizable theory. Even if loop quantum gravity solved all the UV problems, it would mean that infinitely many coupling constants are undetermined - a situation analogous to a non-renormalizable theory.

### Black hole entropy

OBJECTION Despite various claims, loop quantum gravity is not able to calculate the black hole entropy, unlike string theory. The fact that the entropy is proportional to the area does not follow from loop quantum gravity. It is rather an assumption of the calculation. The calculation assumes that the black hole interior can be neglected and the entropy comes from a new kind of dynamics attached to the surface area - there is no justification of this assumption. Not surprisingly, one is led to an area/entropy proportionality law. The only non-trivial check could be the coefficient, but it comes out incorrectly (see the Immirzi discrepancy).

The Immirzi discrepancy was believed to be proportional to the logarithm of two or three, and a speculative explanation in terms of quasinormal modes was proposed. However it only worked for one type of the black hole - a clear example of a numerical coincidence - and moreover it was realized in July 2004 that the original calculation of the Immirzi parameter was incorrect, and the correct value (described by Meissner) is not proportional to the logarithm of an integer. The value of the Immirzi parameter - even according to the optimists - remains unexplained. Another description of the situation goes as follows: Because the Immirzi parameter represents the renormalization of Newton's constant and there is no renormalization in a finite theory - and loop quantum gravity claims to be one - the Immirzi parameter should be equal to one which leads to a wrong value of the black hole entropy.

### Nonseparable Hilbert space

OBJECTION While all useful quantum theories in physics are based on a separable Hilbert space; i.e. a Hilbert space with a countable basis, loop quantum gravity naturally leads to a non-separable Hilbert space, even after the states related by diffeomorphisms are identified. This space can be interpreted as a very large, uncountable set of superselection sectors that do not talk to each other and prevent physical observables from being changed continuously. All known procedures to derive a different, separable Hilbert space are physically unjustified.

### Foundational lacks

OBJECTION Loop quantum gravity has no tools and no solid foundations to answer other important questions of quantum gravity - the details of Hawking radiation; the information loss paradox; the existence of naked singularities in the full theory; the origin of holography and the AdS/CFT correspondence; mechanisms of appearance and disappearance of spacetime dimensions; the topology changing transitions (which are most likely forbidden in loop quantum gravity); the behavior of scattering at the Planck energy; physics of spacetime singularities; quantum corrections to geometry and Einstein's equations; the effect of the fluctuating metric tensor on locality, causality, CPT-symmetry, and the arrow of time; interpretation of quantum mechanics in non-geometric contexts including questions from quantum cosmology; the replacement for the S-matrix in de Sitter space and other causally subtle backgrounds; the interplay of gravity and other forces; the issues about T-duality and mirror symmetry.

Loop quantum gravity is criticised as a philosophical framework that wants us to believe that these questions should not be asked. As if general relativity is virtually a complete theory of everything (even though it apparently can't be) and all ideas in physics after 1915 can be ignored.

### Prejudices claimed

OBJECTION The criticisms of loop quantum gravity regarding other fields of physics are misguided. They often dislike perturbative expansions. While it is a great advantage to look for a framework that allows us to calculate more than the perturbative expansions, it should never be less powerful. In other words, any meaningful theory should be able to allow us to perform (at least) approximative, perturbative calculations (e.g. around a well-defined classical solution, such as flat space). Loop quantum gravity cannot do this, definitely a huge disadvantage, not an advantage as some have claimed. A good quantum theory of gravity should also allow us to calculate the S-matrix.

### Background independence

OBJECTION Loop quantum gravity's calls for "background independence" are misled. A first constraint for a correct physical theory is that it allows the (nearly) smooth space[time] — or the background — which we know to be necessary for all known physical phenomena in this Universe. If a theory does not admit such a smooth space, it can be called "background independent" or "background free", but it may be a useless theory and a physically incorrect theory.

It is a very different question whether a theory treats all possible shapes of spacetime on completely equal footing or whether all these solutions follow from a more fundamental starting point. However, it is not a priori clear on physical grounds whether it must be so (It can be just an aesthetic feature of a particular formulation of a theory, not the theory itself.), and moreover, for a theory that does not predict many well-behaved backgrounds the question is meaningless altogether. Physics of string theory certainly does respect the basic rules of general relativity exactly - general covariance is seen as the decoupling of unphysical (pure gauge) modes of the graviton. This exact decoupling can be proved in string theory quite easily. It can also be seen in perturbative string theory that a condensation of gravitons is equivalent to a change of the background; therefore physics is independent of the background we start with, even if it is hard to see for the loop quantum gravity advocates.

### Claims on non-principled approach

OBJECTION Loop quantum gravity is not science because every time a new calculation shows that some quantitative conjectures were incorrect, the loop quantum gravity advocates invent a non-quantitative, ad hoc explanation why it does not matter. Some borrow concepts from unrelated fields, including noiseless information theory and philosophy, and some explanations why previous incorrect results should be kept are not easily credible.