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Loop quantum gravity

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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.

This is not the most popular theory of quantum gravity; many physicists have philosophical problems with it. For one thing, the critics of this theory say that LQG is one theory of gravity and nothing more. There are many 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

LQG in itself was initially less ambitious than string theory, purporting only to be a quantum theory of gravity. String theory, on the other hand, appears to predict not only gravity but also various kinds of matter and energy that lie inside spacetime. Many string theorists believe that it is impossible to quantize gravity in 3+1 dimensions without creating these 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:

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.

It should be noted that several 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, also known as diffeomorphism invariance, is the invariance of physical laws under arbitrary coordinate transformations. A good 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. 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.

Problems

As of now, there is not a single experiment which verifies or refutes any aspect of LQG. This problem plagues most current theories of quantum gravity. LQG is affected especially, because it applies on a small scale to the weakest forces in nature. There is no work around for this problem, as it is the biggest problem any scientific theory can have; theory without experiment is just faith. 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, the general feeling is that these competing theories are more potent. Loop theorists disagree, because they believe that we need a proper theory of quantum gravity as a prerequisite for any theory of everything. Only time and experimentation can decide the matter.

Other problems associated with LQG can be found in Talk:Loop Quantum Gravity Archive 3

See also

Bibliography

arXiv:gr-qc/03113V2

External links

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