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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. 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 ===
{{main|Lorentz covariance}}


Unlike General Relativity, LQG is acknoledged by both critics and supporters (see, ''e.g.'', ) not to satisfy local Lorentz invariance -- although controversy exists on this point (see, ''e.g.'', ). Tests of LQG-induced local Lorentz violation through modification of the photon ] have been suggested through searches for energy-dependent photon arrival from distant ] .


===LQG and big bang singularity=== ===LQG and big bang singularity===

Revision as of 22:08, 27 September 2006

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, 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. It includes a calculation of the entropy of black holes.
  3. It is a viable gravity-only alternative to string theory.

However, these claims are not universally accepted. While many of the core results are rigorous mathematical physics, their physical interpretations remain speculative. LQG may possibly be viable as a refinement of either gravity or geometry. For example, entropy calculated in (2) is for a kind of hole which may, or may not, be a black hole.

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.


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.

LQG and standard model

There have been recent proposals that Loop quantum gravity may be able to reproduce the standard model. So far only the first generation of fermions (leptons and quarks) with correct charge and parity properties have been modelled using preons constituted of braids of spacetime as the building blocks. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.

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. These efforts have so far been unsuccessful, although at present more physicists work in string theory than in LQG.

Criticisms of LQG

  • 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.

See also

References

  1. "Researchers Look Beyond the Birth of the Universe". Eberly College of Science. 12 May 2006.
  2. Khoury, Justin. "From Big Crunch to Big Bang". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  3. Seiberg, Nathan. "From Big Crunch To Big Bang - Is It Possible?". {{cite journal}}: Cite journal requires |journal= (help)
  4. Cornalba, L. "A New Cosmological Scenario in String Theory". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  5. Bilson-Thompson, Sundance O. "Quantum gravity and the standard model". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  6. Castelvecchi, Davide (2006). "You are made of space-time". New Scientist (2564). {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
  7. "404 error". Institute of Physics. Retrieved 2006-08-19. {{cite web}}: Cite uses generic title (help)

Bibliography

External links

Papers

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