Loop quantum gravity
Encyclopedia
Loop quantum gravity (LQG), also known as loop gravity and quantum geometry
, is a proposed quantum theory of spacetime
which attempts to reconcile the theories of quantum mechanics
and general relativity
. Loop quantum gravity suggests that space
can be viewed as an extremely fine fabric or network "woven" of finite quantised loops of excited gravitational fields called spin network
s. When viewed over time, these spin networks are called spin foam
, which should not be confused with quantum foam
. A major quantum gravity
contender with string theory
, loop quantum gravity incorporates general relativity without requiring string theory's higher dimensions.
LQG preserves many of the important features of general relativity, while simultaneously employing quantization of both space and time at the Planck scale
in the tradition of quantum mechanics. The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism
-invariant gauge theory
. Roughly, LQG tries to establish a quantum theory of gravity
in which the very space itself, where all other physical phenomena occur, becomes quantized.
LQG is one of a family of theories
called canonical quantum gravity. The LQG theory also includes matter and forces, but does not address the problem of the unification of all physical forces the way some other quantum gravity theories such as string theory do.
, asymptotic safety, and loop quantum gravity (LQG).
Essentially, LQG introduces new variables which replace the (in general relativity (GR)) well-known metric that describes spacetime and curvature. These new variables are rather close to fields that are known from gauge theory like QED
and QCD
. In a certain sense gravity appears similar to QCD, but with an additional property of gravity that allows the application of a second mathematical technique which replaces the fundamental fields with "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.
Next, it becomes possible to eliminate the embedding of circles and surfaces into spacetime through so-called diffeomorphism invariance (which does not exist in other field theories). This is accomplished by replacing these entities with a so-called spin network—a graph with nodes and links among them, where each link and node carry numerical values which represent abstract entities from which certain properties of spacetime can be reconstructed. Conceptually, spacetime can be though of as cells, each with a certain volume carried by a node; each cell has certain surfaces and the link between different nodes (sitting inside these cells) carry the areas of the surfaces.
It would not be correct to envision these cells as sitting in spacetime since there is no spacetime anymore, only nodes and links (and certain numerical values associated with the nodes and links). Spacetime is no longer fundamental, but rather an entity emerging from the more fundamental graphs and their associated nodes and links. The graphs are called spin networks because the numerical values with which they are associated have properties well-known from quantum mechanical spins. This is a mathematical property only, and does not indicate that there are actual spinning objects.
It is possible to compare this emerging spacetime to the surface of a lake. We know that the water surface consists of atoms, and that no water between these atoms. Consequently, the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense, spin networks are fundamental entities from which spacetime, surfaces and their properties (such as volume, area and curvature)can be constructed. Dynamics of spacetime (such as curvature and gravitational waves in General Relativity) are replaced by dynamics of spin networks: within a given graph new nodes with new links can appear (or disappear) according to specific mathematical rules.
A spin network is not a mechanical object which comprises spacetime. Instead, quantized spacetime is a superposition of an infinite number of spin networks. This is well-known in quantum mechanics; there is no reason why an atom should be in a certain state; we can achieve that via preparation or measurement, but in principle a single atom can be in an arbitrary complex quantum state which is a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".
Therefore classical spacetime is recovered by two averaging process.
However there may be different regimes—such as black hole and Big Bang singularities—in which this classical picture and averaging no longer yield correct results. Potentially, in these regimes only spin networks exist without classical properties such as smooth spacetime, area or volume. Analogous to the individual atoms comprising a lake surface, there is no water surface anymore. There seems to be no smooth spacetime near Big Bang or black hole singularities. To understand these new non-classical regimes of spacetime a fundamentally new picture is required, such as LQG and certain other approaches, which represent new mathematical models from which the well-known classical General Relativity spacetime can be reconstructed, but which also remain well-defined and consistent in more extreme regimes.
reformulated Einstein's field equations of general relativity. 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, Carlo Rovelli
and Lee Smolin
understood that it was possible to use Wilson loop
s 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, Rovelli
and Smolin
obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose
's spin network
s, and showed that the geometry is quantized, that is, the (non-gauge-invariant) quantum operators representing area and volume have a discrete spectrum. 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 invariant
s such as the Jones polynomial.
calculations, one finds that gravitation is non-renormalizable in contrast to the electroweak and strong interactions of the Standard Model of particle physics. This implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive.
In general relativity, the Einstein field equations
assign a geometry (via a metric
) to space-time. Before this, there is no physical notion of distance or time measurements. In this sense, general relativity is said to be background independent. An immediate conceptual issue that arises is that the usual framework of quantum mechanics, including quantum field theory
, relies on a reference (background) space-time. Therefore, one approach to finding a quantum theory of gravity is to understand how to do quantum mechanics without relying on such a background; this is the approach of the canonical quantization/loop quantum gravity/spin foam approaches.
Starting with the initial-value-formulation of general relativity (cf. the section on General relativity#Evolution equations), the result is an analogue of the Schrödinger equation
called the Wheeler-deWitt equation
, which some argue is ill-defined. A major break-through came with the introduction of what are now known as Ashtekar variables
, which represent geometric gravity using mathematical analogues of electric
and magnetic field
s. The resulting candidate for a theory of quantum gravity is Loop quantum gravity, in which space is represented by a network structure called a spin network
, evolving over time in discrete steps.
Though not proven, it may be impossible to quantize gravity in 3+1 dimensions
without creating matter and energy artifacts. Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory a posteriori. Many of the approaches now being actively pursued (by Renate Loll
, Jan Ambjørn
, Lee Smolin
, Sundance Bilson-Thompson
, Laurent Freidel
, Mark B. Wise
and others) combine matter with geometry.
The main successes of loop quantum gravity are:
These claims are not universally accepted among physicists, who are presently divided between different approaches to the problem of quantum gravity. LQG may possibly be viable as a refinement of either gravity or geometry. Many of the core results are rigorous mathematical physics
; their physical interpretations remain speculative. Three speculative physical interpretations of LQG's core mathematical results are loop quantization, Lorentz invariance, General covariance and background independence, discussed below. Another physical test for LQG is to reproduce the physics of general relativity coupled with quantum field theory, discussed under problems.
-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, an arbitrary gauge group (or even quantum group
), and supersymmetry, 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. 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.
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
.
, also known as "diffeomorphism invariance", is the invariance of physical laws under arbitrary coordinate transformations. An example of this are the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain 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. 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 invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length.
has the right continuum limit, described by general relativity with possible quantum corrections. It is thus unclear if the theory is in agreement with any experiment ever made. Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint
, but there is no candidate Hamiltonian
. Other technical problems includes finding off-shell closure of the constraint algebra and physical inner product vector space
, coupling to matter fields of Quantum field theory
, fate of the Renormalization
of the graviton
in Perturbation theory
that lead to Ultraviolet divergence
beyond 2-loops (see One-loop Feynman diagram
) in Feynman diagram
. The fate of Lorentz invariance in loop quantum gravity
remains an open problem.
While there has been a recent proposal relating to observation of naked singularities
, 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 (a problem that plagues all current theories of quantum gravity). Because of the above mentioned lack of a semiclassical limit, LQG cannot even reproduce the predictions made by the Standard Model.
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.
Current LQG research directions attempt to address these known problems, and includes spinfoam models and entropic gravity.
Quantum geometry
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...
, is a proposed quantum theory of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
which attempts to reconcile the theories of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. Loop quantum gravity suggests that space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
can be viewed as an extremely fine fabric or network "woven" of finite quantised loops of excited gravitational fields called spin network
Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of...
s. When viewed over time, these spin networks are called spin foam
Spin foam
In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral description of quantum gravity...
, which should not be confused with quantum foam
Quantum foam
Quantum foam, also referred to as spacetime foam, is a concept in quantum mechanics, devised by John Wheeler in 1955. The foam is supposed to be the foundations of the fabric of the universe. Additionally, it can be used as a qualitative description of subatomic spacetime turbulence at extremely...
. A major quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
contender with string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, loop quantum gravity incorporates general relativity without requiring string theory's higher dimensions.
LQG preserves many of the important features of general relativity, while simultaneously employing quantization of both space and time at the Planck scale
Planck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
in the tradition of quantum mechanics. The technique of loop quantization was developed for the nonperturbative quantization of diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
-invariant gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. Roughly, LQG tries to establish a quantum theory of gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
in which the very space itself, where all other physical phenomena occur, becomes quantized.
LQG is one of a family of theories
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
called canonical quantum gravity. The LQG theory also includes matter and forces, but does not address the problem of the unification of all physical forces the way some other quantum gravity theories such as string theory do.
Broad Overview of LQG
The native mechanisms in quantum gravity, successfully applied to other fields, fail to quantize gravity, indicating that something fundamental has to be changed for quantum gravity. Several different approaches attempt to solve this problem, such as string theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, asymptotic safety, and loop quantum gravity (LQG).
Essentially, LQG introduces new variables which replace the (in general relativity (GR)) well-known metric that describes spacetime and curvature. These new variables are rather close to fields that are known from gauge theory like QED
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
and QCD
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
. In a certain sense gravity appears similar to QCD, but with an additional property of gravity that allows the application of a second mathematical technique which replaces the fundamental fields with "fluxes through surfaces" or "fluxes along circles". These surfaces and circles are embedded into spacetime.
Next, it becomes possible to eliminate the embedding of circles and surfaces into spacetime through so-called diffeomorphism invariance (which does not exist in other field theories). This is accomplished by replacing these entities with a so-called spin network—a graph with nodes and links among them, where each link and node carry numerical values which represent abstract entities from which certain properties of spacetime can be reconstructed. Conceptually, spacetime can be though of as cells, each with a certain volume carried by a node; each cell has certain surfaces and the link between different nodes (sitting inside these cells) carry the areas of the surfaces.
It would not be correct to envision these cells as sitting in spacetime since there is no spacetime anymore, only nodes and links (and certain numerical values associated with the nodes and links). Spacetime is no longer fundamental, but rather an entity emerging from the more fundamental graphs and their associated nodes and links. The graphs are called spin networks because the numerical values with which they are associated have properties well-known from quantum mechanical spins. This is a mathematical property only, and does not indicate that there are actual spinning objects.
It is possible to compare this emerging spacetime to the surface of a lake. We know that the water surface consists of atoms, and that no water between these atoms. Consequently, the surface is only an emerging phenomenon, the true fundamental objects are the atoms. In the same sense, spin networks are fundamental entities from which spacetime, surfaces and their properties (such as volume, area and curvature)can be constructed. Dynamics of spacetime (such as curvature and gravitational waves in General Relativity) are replaced by dynamics of spin networks: within a given graph new nodes with new links can appear (or disappear) according to specific mathematical rules.
A spin network is not a mechanical object which comprises spacetime. Instead, quantized spacetime is a superposition of an infinite number of spin networks. This is well-known in quantum mechanics; there is no reason why an atom should be in a certain state; we can achieve that via preparation or measurement, but in principle a single atom can be in an arbitrary complex quantum state which is a superposition of "an atom sitting here, an atom moving in a certain direction over there, an atom moving in this or that direction, ...".
Therefore classical spacetime is recovered by two averaging process.
- First there seems to be a regime where the superposition of spin networks is peaked around a single classical spacetime, i.e. where one single spin network dominates the superposition of infinitely many spin networks. And second,
- From this single spin network one can reconstruct spacetime in the same sense as one can reconstruct the water surface from the individual atoms.
However there may be different regimes—such as black hole and Big Bang singularities—in which this classical picture and averaging no longer yield correct results. Potentially, in these regimes only spin networks exist without classical properties such as smooth spacetime, area or volume. Analogous to the individual atoms comprising a lake surface, there is no water surface anymore. There seems to be no smooth spacetime near Big Bang or black hole singularities. To understand these new non-classical regimes of spacetime a fundamentally new picture is required, such as LQG and certain other approaches, which represent new mathematical models from which the well-known classical General Relativity spacetime can be reconstructed, but which also remain well-defined and consistent in more extreme regimes.
History of LQG
In 1986, Abhay AshtekarAbhay Ashtekar
Abhay Vasant Ashtekar is an Indian theoretical physicist. He is the Eberly Professor of Physics and the Director of the Institute for Gravitational Physics and Geometry at Pennsylvania State University. As the creator of Ashtekar variables, he is one of the founders of loop quantum gravity and its...
reformulated Einstein's field equations of general relativity. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
(technically, a connection
Connection (mathematics)
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. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
) and a coordinate frame (called a vierbein) at each point. Because the Ashtekar formulation was background-independent, Carlo Rovelli
Carlo Rovelli
Carlo Rovelli is an Italian physicist who has worked in Italy, the USA, and France. His work is mainly in the field of quantum gravity. He is among the founders of the Loop Quantum Gravity theory.-Life and career:...
and Lee Smolin
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...
understood that it was possible to use Wilson loop
Wilson loop
In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop...
s as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
invariance of the vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
plays an essential role in the regularization of the Wilson loop states.
Around 1990, Rovelli
Carlo Rovelli
Carlo Rovelli is an Italian physicist who has worked in Italy, the USA, and France. His work is mainly in the field of quantum gravity. He is among the founders of the Loop Quantum Gravity theory.-Life and career:...
and Smolin
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...
obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS 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...
's spin network
Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of...
s, and showed that the geometry is quantized, that is, the (non-gauge-invariant) quantum operators representing area and volume have a discrete spectrum. 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 invariant
Knot invariant
In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
s such as the Jones polynomial.
Key concepts of loop quantum gravity
In the framework of quantum field theory, and using the standard techniques of perturbativePerturbation theory (quantum mechanics)
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. The idea is to start with a simple system for which a mathematical solution is known, and add an...
calculations, one finds that gravitation is non-renormalizable in contrast to the electroweak and strong interactions of the Standard Model of particle physics. This implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive.
In general relativity, the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
assign a geometry (via a metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
) to space-time. Before this, there is no physical notion of distance or time measurements. In this sense, general relativity is said to be background independent. An immediate conceptual issue that arises is that the usual framework of quantum mechanics, including quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, relies on a reference (background) space-time. Therefore, one approach to finding a quantum theory of gravity is to understand how to do quantum mechanics without relying on such a background; this is the approach of the canonical quantization/loop quantum gravity/spin foam approaches.
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
called the Wheeler-deWitt equation
Wheeler-deWitt equation
In theoretical physics, the Wheeler–DeWitt equation is a functional differential equation. It is ill defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics...
, which some argue is ill-defined. A major break-through came with the introduction of what are now known as Ashtekar variables
Ashtekar variables
In theoretical physics, Ashtekar variables represent an unusual way to rewrite the metric on the three-dimensional spatial slices in terms of a SU gauge field and its complementary variable...
, which represent geometric gravity using mathematical analogues of electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s. The resulting candidate for a theory of quantum gravity is Loop quantum gravity, in which space is represented by a network structure called a spin network
Spin network
In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of...
, evolving over time in discrete steps.
Though not proven, it may be impossible to quantize gravity in 3+1 dimensions
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
without creating matter and energy artifacts. Should LQG succeed as a quantum theory of gravity, the known matter fields will have to be incorporated into the theory a posteriori. Many of the approaches now being actively pursued (by Renate Loll
Renate Loll
Renate Loll is a physicist who works at the , The Netherlands. She received her Ph.D. from , London, in 1989. In 2001 she joined the permanent staff of the ITP, after spending several years at the Max Planck Institute for Gravitational Physics in Golm, Germany...
, Jan Ambjørn
Jan Ambjørn
Jan Ambjørn is a Danish physicist regarded as the primary founder of Causal Dynamical Triangulation Theory .Ambjørn began in the early 1990s searching for a physics model that bonded quantum mechanics and relativistic gravity in a way that didn't require supersymmetry...
, Lee Smolin
Lee Smolin
Lee Smolin is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M...
, Sundance Bilson-Thompson
Sundance Bilson-Thompson
Sundance Osland Bilson-Thompson is an Australian theoretical particle physicist. He has developed the idea that certain preon models may be represented topologically, rather than by treating preons as pointlike particles. His ideas have attracted interest in the field of loop quantum gravity, as...
, Laurent Freidel
Laurent Freidel
Laurent Freidel is a physicist, working at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. He is working in loop quantum gravity and spin foam models of quantum gravity...
, Mark B. Wise
Mark B. Wise
Mark Brian Wise is a Canadian-American theoretical physicist. He has conducted research in elementary particle physics and cosmology...
and others) combine matter with geometry.
The main successes of loop quantum gravity are:
- It is a nonperturbativePerturbation theory (quantum mechanics)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. The idea is to start with a simple system for which a mathematical solution is known, and add an...
quantizationQuantization (physics)In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of 3-space geometry, with quantized area and volume operatorsOperator (physics)In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
. - It includes a calculation of the entropyEntropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of black holes. - It replaces the Big BangBig BangThe Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
spacetime singularity with a Big BounceBig BounceThe Big Bounce is a theoretical scientific model of the formation of the known universe. It is implied by the cyclic model or oscillatory universe interpretation of the Big Bang where the first cosmological event was the result of the collapse of a previous universe.- Expansion and contraction...
.
These claims are not universally accepted among physicists, who are presently divided between different approaches to the problem of quantum gravity. LQG may possibly be viable as a refinement of either gravity or geometry. Many of the core results are rigorous mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
; their physical interpretations remain speculative. Three speculative physical interpretations of LQG's core mathematical results are loop quantization, Lorentz invariance, General covariance and background independence, discussed below. Another physical test for LQG is to reproduce the physics of general relativity coupled with quantum field theory, discussed under problems.
Loop quantization
At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
-invariant gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, 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, an arbitrary gauge group (or even quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
), and supersymmetry, and results in a quantization of the kinematics
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
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. 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.
Lorentz invariance
LQG is a quantizationQuantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of a classical Lagrangian field theory
Lagrangian point
The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
with torsion
Torsion
The word torsion may refer to the following:*In geometry:** Torsion of a curve** Torsion tensor in differential geometry** The closely related concepts of Reidemeister torsion and analytic torsion ** Whitehead torsion*In algebra:** Torsion ** Tor functor* In medicine:** Ovarian...
. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. A positive cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
can be realized in LQG by replacing the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
with the corresponding quantum group
Quantum group
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
.
General covariance and background independence
General covarianceGeneral covariance
In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
, also known as "diffeomorphism invariance", is the invariance of physical laws under arbitrary coordinate transformations. An example of this are the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain 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
Hamiltonian constraint
In loop quantum gravity, dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint. The identity of the Hamiltonian constraint is a major open question in quantum gravity, as is extracting of physical observables from any such specific constraint.The Thiemann operator...
) 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.
Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent
Background independence
Background independence, also called universality, is the concept or assumption, fundamental to all physical sciences, that the nature of reality is consistent throughout all of space and time...
. 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.
Problems
Presently, no semiclassical limit recovering general relativity has been shown to exist. This means it remains unproven that LQG's description of spacetime at the Planck scalePlanck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
has the right continuum limit, described by general relativity with possible quantum corrections. It is thus unclear if the theory is in agreement with any experiment ever made. Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint
Hamiltonian constraint
In loop quantum gravity, dynamics such as time-evolutions of fields are controlled by the Hamiltonian constraint. The identity of the Hamiltonian constraint is a major open question in quantum gravity, as is extracting of physical observables from any such specific constraint.The Thiemann operator...
, but there is no candidate Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
. Other technical problems includes finding off-shell closure of the constraint algebra and physical inner product vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, coupling to matter fields of Quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, fate of the Renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
of the graviton
Graviton
In physics, the graviton is a hypothetical elementary particle that mediates the force of gravitation in the framework of quantum field theory. If it exists, the graviton must be massless and must have a spin of 2...
in Perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
that lead to Ultraviolet divergence
Ultraviolet divergence
In physics, an ultraviolet divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very high energy , or, equivalently, because of physical phenomena at very short distances. An infinite answer to a question that should have a...
beyond 2-loops (see One-loop Feynman diagram
One-loop Feynman diagram
In physics, a one-loop Feynman diagram is a connected Feynman diagram with only one cycle . Such a diagram can be obtained from a connected tree diagram by taking two external lines of the same type and joining them together into an edge....
) in Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
. The fate of Lorentz invariance in loop quantum gravity
Lorentz invariance in loop quantum gravity
Loop quantum gravity is a quantization of a classical Lagrangian field theory. It 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...
remains an open problem.
While there has been a recent proposal relating to observation of naked singularities
Naked singularity
In general relativity, a naked singularity is a gravitational singularity, without an event horizon. In a black hole, there is a region around the singularity, the event horizon, where the gravitational force of the singularity is strong enough so that light cannot escape. Hence, the singularity...
, and doubly special relativity, as a part of a program called loop quantum cosmology
Loop quantum cosmology
Loop quantum cosmology is a finite, symmetry-reduced model of loop quantum gravity that predicts a quantum bridge between contracting and expanding cosmological branches...
, 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 (a problem that plagues all current theories of quantum gravity). Because of the above mentioned lack of a semiclassical limit, LQG cannot even reproduce the predictions made by the Standard Model.
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
Immirzi parameter
The Immirzi parameter is a numerical coefficient appearing in loop quantum gravity, a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units...
can only be computed by demanding agreement with Bekenstein
Jacob Bekenstein
Jacob David Bekenstein is an Israeli theoretical physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.-Biography:...
and Hawking's
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
calculation of the black hole entropy
Black hole thermodynamics
In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons...
. 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.
Current LQG research directions attempt to address these known problems, and includes spinfoam models and entropic gravity.
External links
- "Loop Quantum Gravity" by Carlo Rovelli Physics World, November 2003
- Quantum Foam and Loop Quantum Gravity
- Abhay Ashtekar: Semi-Popular Articles . Some excellent popular articles suitable for beginners about space, time, GR, and LQG.
- Loop Quantum Gravity: Lee Smolin.
- Loop Quantum Gravity on arxiv.org
- A list of LQG references catered to fresh graduates
- Loop Quantum Gravity Lectures Online by Lee Smolin
- Spin networks, spin foams and loop quantum gravity
- Wired magazine, News: Moving Beyond String Theory
- April 2006 Scientific American Special Issue, A Matter of Time, has Lee Smolin LQG Article Atoms of Space and Time
- September 2006, The Economist, article Looping the loop
- 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.
- Did pre-big bang universe leave its mark on the sky? - According to a model based on "loop quantum gravity" theory, a parent universe that existed before ours may have left an imprint (New Scientist, 10 April 2008)