Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. This theory is one of a family of theories called canonical quantum gravity. The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism-invariant gauge theory. In plain English, LQG tries to establish a quantum theory of gravity in which the very space in which all other physics occurs becomes quantized. In theoretical physics, quantum geometry is the set of new mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at very short distance scales (comparable to Planck length). ... For other uses of this term, see Spacetime (disambiguation). ... Fig. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... This article does not cite any references or sources. ...

Loop quantum gravity (LQG) is a proposed theory of spacetime which is constructed with the idea of spacetime quantization via the mathematically rigorous theory of loop quantization. It preserves many of the important features of general relativity, while at the same time employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics. In physics, Planck units are physical units of measurement originally proposed by Max Planck. ...

LQG is not the only theory of quantum gravity. The critics of this theory say that LQG is a theory of gravity and nothing more, though some LQG theorists are trying to show that the theory can describe matter as well. There are other theories of quantum gravity, and a list of them can be found on the quantum gravity page. This article does not cite any references or sources. ...

Loop quantum gravity in general, and its ambitions

Many string theorists believe that it is impossible to quantize gravity in 3+1 dimensions without creating matter and energy artifacts. This is not proven, and it is also unproven that the matter artifacts, predicted by string theory, are not 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 originators of LQG, has explored the possibility that string theory and LQG are two different approximations to the same ultimate theory. For other uses of this term, see Spacetime (disambiguation). ... Lee Smolin at Harvard. ...

The main claimed successes of loop quantum gravity are:

1. It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators.
2. It includes a calculation of the entropy of black holes.
3. 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 remain speculative. LQG may possibly be viable as a refinement of either gravity or geometry. For example, entropy calculated in (2) is for a kind of hole which may, or may not, be a black hole. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... For a less technical and generally accessible introduction to the topic, see Introduction to entropy. ... This article is about the astronomical body. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ...

Some alternative approaches to quantum gravity, such as spin foam models, are closely related to loop quantum gravity. In physics, a spin foam is a four-dimensional graph made out of two-dimensional faces that represents one of the configurations that must be summed to obtain Feynmans path integral (functional integration) describing the alternative formulation of quantum gravity known as loop gravity or loop quantum gravity. ...

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 must be modified. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see Unruh effect). While the Unruh effect can be described in the case of a fixed background geometry on which propagate non-gravitational degrees of freedom, attempting to solve the full problem -- where one allows gravitational degrees of freedom such as the graviton to propagate -- with the usual methods of quantum field theory is mathematically problematic. In particular, one finds that the theory is non-renormalizable, a technical term that implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive. This article does not cite any references or sources. ... Quantum field theory (QFT) is the quantum theory of fields. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none, that is, the accelerating observer will find themselves in a warm background. ... In physics, the graviton is mainly still considered to be a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ... In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ...

History of LQG

Main article: history of loop quantum gravity

In 1986, 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. General relativity is the theory of gravitation published by Albert Einstein in 1915. ... Year 1986 (MCMLXXXVI) was a common year starting on Wednesday (link displays 1986 Gregorian calendar). ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... In theoretical physics, Ashtekar (new) variables (named after Abhay Ashtekar who invented them) represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU(2) gauge field and its complementary variable. ... This article is in need of attention from an expert on the subject. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. ... In quantum field theory, the vacuum state, usually denoted , is the element of the Hilbert space with the lowest possible energy, and therefore containing no physical particles. ...

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. Year 1990 (MCMXC) was a common year starting on Monday (link displays the 1990 Gregorian calendar). ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ... A spin network is a (directed) graph whose edges are associated with irreducible representations of a compact Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. ... A knot invariant is a useful tool in knot theory. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics. A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ...

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,^[1] an arbitrary gauge group (or even quantum group), and supersymmetry,^[2] 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 mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...

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. C*-algebras are an important area of research in functional analysis. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

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. Since classical general relativity can be formulated as a BF theory with constraints, scientists hope that a consistent quantization of gravity may arise from the perturbation theory of BF spin-foam models.

This discrete structure may require modifications of quantum mechanics, and a line of research called polymer quantum mechanics has been pursued.

Lorentz invariance

Main article: Lorentz covariance

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. In physics, Lorentz covariance is a key property of spacetime that follows from the special theory of relativity, where it applies globally. ... Look up quantization in Wiktionary, the free dictionary. ... A contour plot of the effective potential (the Hills Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... This article is in need of attention from an expert on the subject. ... In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...

Diffeomorphism invariance and background independence

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General covariance is the invariance of physical laws under arbitrary coordinate transformations. This condition is most noteworthy in the context of general relativity where it has some profound implications, as Einstein discovered. The argument is easy and involves only the very basics of GR, as we will see below. More details and discussions can be found in Rovelli's book or the papers by Rovelli and Gaul^[3] and by Smolin.^[4] Image File history File links Broom_icon. ... This article or section is in need of attention from an expert on the subject. ...

It begins with a mathematical observation. Here is written the SHO differential equation twice

Eq(1)

Eq(2)

except in Eq(1) the independent variable is x and in Eq(2) the independent variable is y. Once we find out that a solution to Eq(1) is f(x) = cosx, we immediately know that g(y) = cosy solves Eq(2). This observation combined with general covariance has profound implications for GR.

Assume pure gravity first. Say we have two coordinate systems, x-coordinates and y-coordinates. General covariance demands the equations of motion have the same form in both coordinate systems, that is, we have exactly the same differential equation to solve in both coordinate systems except in one the independent variable is x and in the other the independent variable is y. Once we find a metric function g_ab(x) that solves the EQM in the x-coordinates we immediately know (by exactly the same reasoning as above!) that the same function written as a function of y solves the EOM in the y-coordinates. As both metric functions have the same functional form but belong to different coordinate systems, they impose different spacetime geometries. Thus we have generated a second distinct solution! Now comes the problem. Say the two coordinate systems coincide at first, but at some point after t = 0 we allow them to differ. We then have two solutions, they both have the same initial conditions yet they impose different spacetime geometries. The conclusion is that GR does not determine the proper-time between spacetime points! The argument I have given (or rather a refinement of it) is what's known as Einstein's hole argument. It is straightforward to include matter - we have a larger set of differential equations but they still have the same form in all coordinates systems, the same argument applies and again we obtain two solutions with the same initial conditions which impose different spacetime geometries. It is very important to note that we could not have generated these extra distinct solutions if spacetime were fixed and non-dynamical, and so the resolution to the hole argument, background independence, only comes about when we allow spacetime to be dynamical. Before we can go on to understand this resolution we need to better understand these extra solutions. We can interpret these solutions as follows. For simplicity we first assume there is no matter. Define a metric function whose value at P is given by the value of g_ab at P₀, i.e. This article or section is in need of attention from an expert on the subject. ...

Eq(3) .

(see figure 1(a)). Now consider a coordinate system which assigns to P the same coordinate values that P₀ has in the x-coordinates (see figure 1(b)). We then have

Eq(4)

where u₀,u₁,u₂,u₃ are the coordinate values of P₀ in the x-coordinate system.

Figure 1

When we allow the coordinate values to range over all permissible values, Eq(4)is precisely the condition that the two metric functions have the same functional form! We see that the new solution is generated by dragging the original metric function over the spacetime manifold while keeping the coordinate lines "attached", see Fig 1. It is important to realise that we are not performing a coordinate transformation here, this is what's known as an active diffeomorphism (coordinate transformations are called passive diffeomorphisms). It should be easy to see that when we have matter present, simultaneously performing an active diffeomorphism on the gravitational and matter fields generates the new distinct solution. Image File history File links No higher resolution available. ...

The resolution to the hole argument (mainly taken from Rovelli's book) is as follows. As GR does not determine the distance between spacetime points, how the gravitational and matter fields are located over spacetime, and so the values they take at spacetime points, can have no physical meaning. What GR does determine, however, are the mutual relations that exist between the gravitational field and the matter fields (i.e. the value the gravitational field takes where the matter field takes such and such value). From these mutual relations we can form a notion of matter being located with respect to the gravitational field and vice-versa, (see Rovelli's for exposition). What Einstein discovered was that physical entities are located with respect to one another only and not with respect to the spacetime manifold. This is what background independence is! And the context for Einstein's remark "beyond my wildest expectations".

Since the Hole Argument is a direct consequence of the general covariance of GR, this led Einstein to state:

"That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, ..."^[5]

LQG preserves this symmetry under active diffeomorphisms by requiring that the physical states remain invariant under the generators of active diffeomorphisms. The interpretation of this condition is well understood for purely spatial active diffemorphisms. However, the understanding of active 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. A generally accepted calculational framework to account for this constraint is yet to be found. Categories: Pages needing attention | Stub | Physics | Classical mechanics | Theoretical physics ...

The term "active diffeomorphism" has been used, instead of just "diffeomorphism", to emphasize that this is not a case of simple coordinate transformations. It is active diffeomorphisms which are the gauge transformations of GR and they should not be confused with the freedom of choosing coordinates on the space-time M. Invariance under coordinate transformations is not a special feature of GR as all physical theories are invariant under coordinate transformations. (Indeed, the mathematical definition of a diffeomorphism is a transformation which relates manifolds with equivalent topological and differentiable structure, but not necessarily equivalent metrics. For example, a diffeomorphism can turn a doughnut into a tea cup.)

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 invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length. At present, it remains unproven that LQG's description of spacetime at the Planckian scale has the right continuum limit, described by general relativity with possible quantum corrections. Background independence is a condition in theoretical physics, especially in quantum gravity, that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. ... The Planck length, denoted by , is the unit of length approximately 1. ...

Although a number of vocal string theoreticians have derided background independence and expressed that it plays little, if any, role in their vision of a quantum theory of gravity, Edward Witten has spoken of the need for a background independent formulation of string theory a number of times, for example in 1993, Edward Witten (born August 26, 1951) is an American theoretical physicist and professor at the Institute for Advanced Study. ...

"Finding the right framework for an intrinsic, background independent formulation of string theory is one of the main problems in the subject, and so far has remained out of reach." ... "This problem is fundamental because it is here that one really has to address the question of what kind of geometrical object the string represents."^[6]

Arguments on the need of a background independent formulation of string theory can be found in Lee Smolin's paper.^[4] Lee Smolin at Harvard. ...

LQG and big bang singularity

Main article: loop quantum cosmology

In 2006, Abhay Ashtekar released a paper stating that according to loop quantum gravity, the singularity of the Big Bang is avoided. What the researchers found was a prior collapsing universe. Since gravity becomes repulsive near Planck density according to their simulations, this resulted in a "big bounce" and the birth of our current universe.^[7] These topics are an active research in loop quantum cosmology.^[8][9][10] There are very few or no other articles that link to this one. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... For other uses, see Big Bang (disambiguation). ... The Planck density is the natural unit of density, denoted by ρP. ρP = Planck mass / (Planck length)3 = ≈ 5. ... This article does not cite any references or sources. ... There are very few or no other articles that link to this one. ...

LQG and particle physics

Main article: preon

There have been recent proposals that loop quantum gravity may be able to reproduce features resembling the Standard Model. So far only the first generation of fermions (leptons and quarks) with correct parity properties have been modelled using preons constituted of braids of spacetime as the building blocks.^[11] However there is no derivation of the Lagrangian that would describe the interactions of such particles. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.^[12] Other recent results suggest that LQG's framework may allow for the derivation of certain spin-1 bosons such as the photon, and gluon, and possibly the spin-2 graviton. This line of research follows the reductionist paradigm of finding a building block of elementary particles. In particle physics, preons are postulated point-like particles, conceived to be subcomponents of quarks and leptons. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... A lepton is also a unit of currency. ... For other uses of this term, see: Quark (disambiguation) 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, the quarks are subatomic particles thought to be elemental and indivisible. ... In particle physics, preons are postulated point-like particles, conceived to be subcomponents of quarks and leptons. ... Molecule of alanine used in NMR implementation of error correction. ... In quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space, arising from Werner Heisenbergs uncertainty principle. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... In physics, the graviton is mainly still considered to be a hypothetical elementary particle that mediates the force of gravity in the framework of quantum field theory. ...

Xiao-Gang Wen and Michael Levin are two solid-state physicists who have attempted to model elementary particles such as electrons and photons as resulting from a discrete lattice structure of spacetime in analogy to phonons in solid state physics. In the paper "Photons and electrons as emergent phenomena" they attempt to model elementary particles as emergent properties of a String-net condensation in analogy to phonons in solid state physics, and LQG's spin networks have the properties necessary to reproduce the Standard Model as the result of the collective behavior of a group of spin network.^[13][14] This approach differs from the preon approach in that Wen and Levin see particles as an emergent property of quantum spacetime, rather than built up of smaller substructure as is the case with both preon and string theory. In condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical explanation for topological order by Michael Levin and Xiao-Gang Wen. ... A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system. ... In particle physics, preons are postulated point-like particles, conceived to be subcomponents of quarks and leptons. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point...

Independent of the above discussion, there exist various proposals on how to incorporate fermions (matter) within LQG's framework. A single framework that can account for the standard model and gravity is known in physics as a theory of everything. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... This article or section is in need of attention from an expert on the subject. ...

LQG and the Graviton

There have been recent results in LQG using the spinfoam formalism by Carlo Rovelli, Eugenio Bianchi, Leonardo Modesto, and Simone Speziale^[15][16]that LQG does give rise to gravitons, and allows gravitons to interact as expected, reproducing Newton's law of gravity.

The Kodama state

Main article: Chern-Simons theory

In 1988, Hideo Kodama wrote down the equations of the Kodama state, but as it described a positive (de-Sitter) spacetime, which was believed to be inconsistent with observation, it was largely ignored. In physics, Chern-Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. ... Year 1988 (MCMLXXXVIII) was a leap year starting on Friday (link displays 1988 Gregorian calendar). ...

Lee Smolin's paper, "Quantum gravity with a positive cosmological constant"^[17] suggests that the Kodama state is a proposal which purports to show that LQG has a good semiclassical limit which reproduces the dynamics of general relativity with a positive (de-Sitter) cosmological constant, 4 dimensions, and gravitons, and an exact solution to ordinary constraints on background independent quantum gravity, providing evidence that loop quantum gravity is indeed a quantum gravity with the correct semiclassical description. Edward Witten published a paper in response to Lee Smolin's arguing that the Kodama state is unphysical, due to an analogy to a state in Chern-Simons theory wavefunction resulting in negative energies,^[18] and cites Smolin's paper. Recently, Andrew Randono has published two papers that cite Witten's paper,^[19][20] and address these objections, by generalizing the Kodama state, with the conclusion that the Immirzi parameter, when generalized with a real value, fixed by matching with black hole entropy, describes parity violation in quantum gravity, and is CPT invariant, and is normalizable, and chiral, consistent with known observations of both gravity and quantum field theory. The physical inner product may resemble the Macdowell Mansouri formulation of gravity. Eyo Eyo Ita published papers that build on Randono's generalized Kodama state, and argue that a generalized Kodama state can be built that can couple to matter and the Hamiltonian constraint can reproduce the dynamics of general relativity, resulting in a finite, full quantum gravity http://arxiv.org/abs/gr-qc/0703057 Lee Smolin at Harvard. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Edward Witten (born August 26, 1951) is an American theoretical physicist and professor at the Institute for Advanced Study. ... In physics, Chern-Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. ... The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. ... Black hole entropy is entropy carried by a black hole. ... CPT-symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously. ... In quantum mechanics, wave functions which describe real particles must be normalisable: the probability of the particle to occupy any place must equal 1. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ...

Problems

While there has been a recent proposal relating to observation of naked singularities,^[21] and doubly special relativity, as a part of a program called loop quantum cosmology, as of now there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity. This problem plagues all current theories of quantum gravity (except those that have been proven wrong). Doubly-Special Relativity is a new theory of special relativity first postulated in a paper by Giovanni Amelino-Camelia. ... There are very few or no other articles that link to this one. ...

Making predictions from the theory of LQG has been extremely difficult computationally, also a recurring problem with modern theories in physics.

Another problem is that a crucial free parameter in the theory known as the Immirzi parameter can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value. A prediction directly from theory would be preferable. The Immirzi parameter (also known as the Barbero-Immirzi parameter) is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. ... Jacob David Bekenstein (born May 1, 1947) is a physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ...

LQG has gained limited support in the physics community; at present, many more physicists still work in string theory than in LQG. Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point...

See also

• Heyting algebra
• mathematical category theory
• noncommutative geometry
• topos theory
• C*-algebra
• Regge calculus
• double special relativity
• invariance mechanics

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... For discussion of topoi in literary theory, see literary topos. ... C*-algebras are an important area of research in functional analysis. ... In theoretical physics, Regge calculus is a simplified form of general relativity, introduced by the Italian theoretician Tullio Regge in the early 1960s. ... Doubly-special relativity, also deformed special relativity, is a new theory of special relativity first postulated in a paper by Giovanni Amelino-Camelia, though is at least implicit in a paper of Paul Merriam. ... Invariance mechanics, in its simplest form, is the rewriting of the laws of quantum field theory in terms of invariant quantities only. ...

References

John Carlos Baez (b. ... “Einstein” redirects here. ... In general, the principle of relativity is the requirement that the laws of physics be the same for all observers. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 231st day of the year (232nd in leap years) in the Gregorian calendar. ...

Bibliography

• Topical Reviews
  • Carlo Rovelli, Loop Quantum Gravity, Living Reviews in Relativity 1, (1998), 1, online article, 2001 15 August version.
  • Thomas Thiemann, Lectures on loop quantum gravity, e-print available as gr-qc/0210094
  • Abhay Ashtekar and Jerzy Lewandowski, Background independent quantum gravity: a status report, e-print available as gr-qc/0404018
  • Carlo Rovelli and Marcus Gaul, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, e-print available as gr-qc/9910079.
  • Lee Smolin, The case for background independence, e-print available as hep-th/0507235.
• Popular books:
  • Julian Barbour, The End of Time
  • Lee Smolin, Three Roads to Quantum Gravity
  • Carlo Rovelli, Che cos'è il tempo? Che cos'è lo spazio?, Di Renzo Editore, Roma, 2004. French translation: Qu'est ce que le temps? Qu'est ce que l'espace?, Bernard Gilson ed, Brussel, 2006. English translation: What is Time? What is space?, Di Renzo Editore, Roma, 2006.
• Magazine articles:
  • Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004
• Easier introductory, expository or critical works:
  • Abhay Ashtekar, Gravity and the quantum, e-print available as gr-qc/0410054
  • John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)
  • Carlo Rovelli, A Dialog on Quantum Gravity, e-print available as hep-th/0310077
• More advanced introductory/expository works:
  • Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online
  • Thomas Thiemann, Introduction to modern canonical quantum general relativity, e-print available as gr-qc/0110034
  • Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).
  • Abhay Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
  • Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press (1996)
  • Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view, e-print available as hep-th/0501114
  • "Loop and Spin Foam Quantum Gravity: A Brief Guide for beginners arXiv:hep-th/0601129 H. Nicolai and K. Peeters.
  • Edward Witten, Quantum Background Independence In String Theory, e-print available as hep-th/9306122.
• Conference proceedings:
  • John C. Baez (ed.), Knots and Quantum Gravity
• Fundamental research papers:
  • Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 2244-2247, 1986
  • Abhay Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D36, 1587-1602, 1987
  • Roger Penrose, Angular momentum: an approach to combinatorial space-time in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971
  • Carlo Rovelli and Lee Smolin, Knot theory and quantum gravity, Phys. Rev. Lett., 61 (1988) 1155
  • Carlo Rovelli and Lee Smolin, Loop space representation of quantum general relativity, Nuclear Physics B331 (1990) 80-152
  • Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005

Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... is the 227th day of the year (228th in leap years) in the Gregorian calendar. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ... Julian Barbour (born 1937) is a British physicist. ... The End of Time: The Next Revolution in Physics is a book by Julian Barbour published in 1999 that denies time exists as anything but an illusion. ... Lee Smolin at Harvard. ... Three Roads to Quantum Gravity is a 2001 book by theoretical physicist Lee Smolin. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ... Scientific American is a popular-science magazine, published (first weekly and later monthly) since August 28, 1845, making it the oldest continuously published magazine in the United States. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... John Carlos Baez (b. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... Rodolfo Gambini is a physicist of the Montevideo University. ... Jorge Pullin is the Horace Hearne Chair in theoretical Physics at the Louisiana State University. ... Edward Witten (born August 26, 1951) is an American theoretical physicist and professor at the Institute for Advanced Study. ... John Carlos Baez (b. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... Abhay Ashtekar (born July 5, 1949) completed his undergraduate education in India. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ... Carlo Rovelli is an Italian-born physicist who now works at the University of the Mediterraneum and the Centre de Physique Theorique in Marseille, France. ... Lee Smolin at Harvard. ...

External links

• Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/ (no longer active, try the Wayback machine)
• Resources for LQG and spin foams: http://jdc.math.uwo.ca/spin-foams/
• Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
• Zeno meets modern science. Article from Acta Physica Polonica B by Z.K. Silagadze.
• Referenced in Bob the Angry Flower: http://www.angryflower.com/dating.gif
• Abhay Ashtekar's home page. It has some excellent popular articles suitable for beginners about Space, Time, GR, and LQG. http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html
• April 2006 Scientific American Special Issue, A Matter of Time, has Lee Smolin LQG Article Atoms of Space and Time http://www.sciam.com/special/toc.cfm?issueid=40&sc=rt_nav_list
• Loop Quantum Gravity. Lee Smolin. Online at http://www.edge.org/3rd_culture/bios/smolin.html
• Loop Quantum Gravity on arxiv.org
• A list of LQG references catered to fresh graduates
• Wired News: Moving Beyond String Theory
• September 2006 The Economist article Looping the loop
• Loop Quantum Gravity Lectures Online by Lee Smolin

Bob the Angry Flower is a comic strip that tells the story of an easily angered anthropomorphic flower named Bob and his interactions with the world, often in search of either global domination or love. ... Wired is a full-color monthly magazine and on-line periodical published in San Francisco, California since March 1993. ... The Economist is a weekly news and international affairs publication owned by The Economist Newspaper Ltd and edited in London, UK. It has been in continuous publication since September 1843. ...

Papers

• Graviton propagator in loop quantum gravity-- We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter.
• Quantum Gravity and the Standard Model-- Shows that a class of background independent models of quantum spacetime have local excitations that can be mapped to the first generation fermions of the standard model of particle physics.