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. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphism-invariant gauge theory. In plain English this is a quantum theory of gravity in which the very space that all other physics occurs in is quantized.

Loop quantum gravity (LQG) is a proposed theory of spacetime which is built from the ground up with the idea of spacetime quantization via the mathematically rigorous theory of loop quantization. It preserves many of the important features of general relativity, such as local Lorentz invariance, 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 some LQG theorists have tried 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

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 exactly the same as observed matter. Should LQG succeed as a quantum theory of gravity, the known matter fields would have to be incorporated into the theory a posteriori. Lee Smolin, one of the fathers of LQG, has explored the possibility that string theory and LQG are two different approximations to the same ultimate theory.

The main claimed successes of loop quantum gravity are:

1) It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators
2) 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 or may not 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 incompatibility between quantum mechanics and general relativity

Main article: quantum gravity

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

Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity. The first is that the geometric interpretation of general relativity is not fundamental, but emergent. The other view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.

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

History of LQG

Main article: history of loop quantum gravity

In 1986, 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 (John Baez and Kirill Krasnov), an arbitrary gauge group (or even quantum group), and supersymmetry (Smolin), and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.

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

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. 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

For detailed discussion see the Lorentz covariance page.

LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.

Diffeomorphism invariance and background independence

General covariance, 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 gr-qc/9910079 by Rovelli and Gaul and hep-th/0507235 by Smolin.

It begins with an utterly straightforward mathematical observation. Here is written the SHO differential equation twice

Eq(1) ${d^2 f(x) \over dx^2} + f(x) = 0$

Eq(2) ${d^2 g(y) \over dy^2} + g(y) = 0$

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 $f(x) = \cos x$ , we immediately know that $g(y) = \cos
y$ 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 $\tilde{g}_{ab}$ whose value at $P$ is given by the value of $g_{ab}$ at $P_0$ , i.e.

Eq(3) $\tilde{g}_{ab}(P) = g_{ab}(P_0)$ .

(see figure 1(a)). Now consider a coordinate system which assigns to $P$ the same coordinate values that $P_0$ has in the x-coordinates (see figure 1(b)). We then have

Eq(4) $\tilde{g}_{ab} (y_0=u_0,y_1=u_1, y_2=u_2, y_3=u_3) = g_{ab}
(x_0=u_0,x_1=u_1, x_2=u_2 , x_3=u_3),$

where $u_0,u_1,u_2,u_3$ are the coordinate values of $P_0$ 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 diffeomorphism|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 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, ..." (Einstein, 1916, p.117).

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 "diffeomrophism", 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 invaraint under coordinate transformations. (Indeed, the mathematical definition of a diffeomorphism is a transfromation which relates topologically equivalent spaces, not geometrically equivalent spaces. For example, a diffepmorphism 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.

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,

"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." hep-th/9306122

Arguments on the need of a background independent formulation of string theory can be found in Lee Smolin's paper hep-th/0507235.

LQG and big bang singularity

In 2006, Abhay Ashtekar released a paper claiming 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. However, it has to be noted that similar solutions of the Big Bang singularity have been previously proposed in String Theory and M-Theory (for examples see: ^*" target="_blank" >and [http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:hep-th/0203031).

Problems

While there has been a recent proposal relating to observation of naked singularities, as of now, not a single experimental observation yet verifies or refutes any aspect of LQG. This problem plagues all current theories of quantum gravity. The second 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.

Finally, LQG has gained limited support in the physics community, perhaps because of its limited scope. So far, it seeks to describe a quantum theory including gravity and more or less arbitrary other forces and forms of matter. String theory and M-theory are more ambitious, since also they seek a more or less unique theory which predicts the detailed behavior of elementary particles and the forces besides gravity. While they have not succeeded in doing so yet, nowadays more people work in string theory than in Loop Gravity. The debate is hot, as testified by the rapid oscillations of the content of this page. Only time and experimentation can decide the matter.

Criticisms of LQG

Problems associated with LQG can be found in Objections to Loop Quantum Gravity

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

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

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

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:
- Albert Einstein, H. A. Lorentz, H. Weyl, and H. Minkowski, The Principle of relativity (1916).
- 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

External links

Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/
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

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