loop quantum gravity

 

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.

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.

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 several LQG theorists are presently working 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.

Loop quantum gravity in general, and its ambitions

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 Loll, Smolin, Bilson-Thompson, Freidel, Wise and others^[1] combine matter with geometry. Several of these current efforts would be proven wrong if evidence were found of extra spatial dimensions.

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 replaces the Big Bang spacetime singularity with a Big Bounce.

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.

Some alternative approaches to quantum gravity, such as spin foam models, are closely related to loop quantum gravity.

The apparent incompatibility between quantum mechanics and general relativity

Main article: quantum gravity

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.

Furthermore, in the framework of quantum field theory, and using the standard techniques of perturbative calculations, one finds that gravitation is non-renormalizable in contrast to the electroweak and strong interactions of the Standard Model of particle physics. This technical term implies that there are infinitely many free parameters in the theory and thus that it cannot be predictive. However, current research has provided modern techniques in calculating physical processes that go beyond the old, standard definitions of ``non-renormalizability", so the story of gravitation in the framework of quantum field theory is not over.

Another interface between general relativity and quantum mechanics occurs in quantum field theory studied on curved (non-Minkowskian) backgrounds. The vacuum, when it exists, is shown in general relativity to depend on the path of the observer through space-time (see Unruh effect). The Unruh effect can be described semi-classically in the case of a fixed background geometry on which propagate non-gravitational quantum mechanical particles (quanta). It's then natural to wonder about the inclusion of quantum gravitational effects, including interactions with the graviton (in analogy with the electron's interactions with the electromagnetic field of the nucleus within an atom).

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.

Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics.

The ingredients of loop quantum gravity

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions,^[2] an arbitrary gauge group (or even quantum group), and supersymmetry,^[3] 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.

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.

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^[4] and by Smolin.^[5]

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, so 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.

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.

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.

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 that is 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, ..."^[6]

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.

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.

LQG and big bang singularity

Main article: loop quantum cosmology

Abhay Ashtekar and Martin Bojowald have released papers 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.

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 by Sundance Bilson-Thompson using preons constituted of braids of spacetime as the building blocks.^[8] 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.^[9] 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.

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 networks.^{[10] [11]} 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 substructures as is the case with Bilson-Thompson's preon theory.

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.

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^[12][13]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: Immirzi parameter

In 1988, Hideo Kodama wrote down the equations of the Kodama state, but as it described a positive (de Sitter universe) spacetime, which was believed to be inconsistent with observation, it was largely ignored.

Lee Smolin's paper, "Quantum gravity with a positive cosmological constant"^[14] suggests that the Kodama state is a ground state which has a good semiclassical limit which reproduces the dynamics of general relativity with a positive (de Sitter) cosmological constant, 4 dimensions, and gravitons, and is 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,^[15] and citing Smolin's paper. Recently, Andrew Randono has published two papers that cite Witten's paper,^[16][17] 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 has 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. ^{[18] [19] [20]}

^{[21] [22]}

Problems

While there has been a recent proposal relating to observation of naked singularities,^[23] 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).

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.

See also

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

References

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:
  • 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.
  • Julian Barbour, The End of Time
• 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
  • H. Nicolai and K. Peeters, "Loop and Spin Foam Quantum Gravity: A Brief Guide for beginners arXiv:hep-th/0601129

• 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

External links

• Quantum Foam and Loop Quantum Gravity: http://universe-review.ca/R01-07-quantumfoam.htm
• Resources for LQG and spin foams: http://jdc.math.uwo.ca/spin-foams/
• 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
• Physics Forum, Beyond the Standard Model. http://www.physicsforums.com/forumdisplay.php?f=66
• 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
• Loop Quantum Gravity Lectures Online by Lee Smolin
• Wired News: Moving Beyond String Theory
• September 2006 The Economist article Looping the loop
• Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
• [http://uk.arxiv.org/abs/physics/0505042
• Zeno meets modern science.] Article from Acta Physica Polonica B by Z.K. Silagadze.
• Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/ (no longer active, try the Wayback machine)

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