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Loop quantum gravity

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Loop quantum gravity (LQG), also known as loop gravity, quantum geometry and canonical quantum general relativity, is a proposed quantum theory of spacetime which attempts to blend together the seemingly incompatible theories of quantum mechanics and general relativity. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphism-invariant gauge theories.

Theories about quantum gravity

LQG is controversial. Naturally, its practitioners consider it the main competitor of string theory as a theory of quantum gravity. LQG is, however, not even considered a contender by string theorists. See problems with loop quantum gravity for a more thorough discussion of some criticisms. String theory is itself also subject to various kinds of critical sniping, but has been a dominant presence since the mid-1980s in the quantum gravity area. In numerical terms, stringy people outnumber loopy people by a factor of roughly 10 and stringy papers outnumber loopy papers by a factor of roughly 50. The 'minority status' of LQG is not of course of any weight scientifically.

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, automatically accommodates matter particles, gauge vector bosons and the graviton, which suggested early in its development that strings might be able to model all known fundamental physics. Should LQG succeed as a quantum theory of gravity, however, the known matter fields would have to be incorporated into the theory using the broader formalism. 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 successes of loop quantum gravity are: a nonperturbative quantization of 3-space geometry, with quantized area and volume operators; a calculation of the entropy of physical black holes; and a proof by example that it is not necessary to have a theory of everything in order to have a candidate for a quantum theory of gravity. Many of the core results in LQG are established at the level of rigour of mathematical physics, and rely heavily on algebraic quantum field theory. Its main shortcomings are: not yet having a picture of dynamics but only of kinematics; not yet able to perform particle physics calculations; not yet able to recover the classical limit. These difficulties may all be related.

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 the fruit of the effort to formulate a background-independent quantum theory. Topological quantum field theory provided an example of background-independent quantum theory, but lacked the causally-propagating local degrees of freedom needed for a description of 3 + 1 dimensional gravity.

History of LQG

Main article: history of loop quantum gravity

In 1986 physicist 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 (Baez and 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, because classical general relativity can be formulated as a BF theory with constraints, and it is hoped that a consistend quantization of gravity may arise from 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 (for example, the equations of general relativity) under arbitrary coordinate transformations. This symmetry is one of the defining features of general relativity. LQG preserves this symmetry by requiring that the physical states must be 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, and 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 topology that cannot be changed), but rather they are expected to give rise to space and time at large distances compared to the Planck length. It has not been yet shown that LQG's description of spacetime at the Planckian scale has the right continuum limit described by general relativity with possible quantum corrections.

Open problems

The classical limit

Any successful theory of quantum gravity must provide physical predictions that closely match known observation, and reproduce the results of quantum field theory and gravity. To date Einstein's theory of general relativity is the most successful theory of gravity. It has been shown that quantizing the field equations of general relativity will not necessarily recover those equations in the classical limit. It remains unclear whether LQG yields results that match general relativity in the domain of low-energy, macroscopic and astronomical realm. To date, LQG has been shown to yield results that match general relativity in 1+1 and 2+1 dimensions where the metric tensor carries no physical degrees of freedom. To date, it has not been shown that LQG reproduces classical gravity in 3+1 dimensions. Thus, it remains unclear whether LQG successfully merges quantum mechanics with general relativity.

Time

Additionally, in LQG, time is not continuous but discrete and quantized, just as space is: there is a minimum moment of time, Planck time, which is on the order of 10−43 seconds, and shorter intervals of time have no physical meaning. This carries the physical implication that relativity's prediction of time dilation due to accelerating speed or gravitational field, must be quantized, and must consist of multiples of Planck time units. (This helps resolve the time zero singularity problem: see subsection "The big bang")

Particle physics

While classical particle physics posit particles traveling through space and time that is continuous and therefore infinitely divisible, LQG predicts that space-time is quantized or granular. The two different models of space and time affects the way ultra high energy cosmic rays interacts with the background, with quantized spacetime predicting that the threshold for allowable energies for such high energy particles be raised. Such particles have been observed, however, alternative explanations have not been ruled out.

LQG does not constrain the spectrum of non-gravitational forces and elementary particles. Unlike the situation in string theory, all of them must be added to LQG by hand. It has proved difficult to incorporate elementary scalar fields, Higgs mechanism, and CP-violation into the framework of LQG.

Quantum field theory

Quantum field theory is background dependent. One problem LQG may be able to address in QFT is the ultraviolet catastrophe.

The term ultraviolet catastrophe has also been applied to similar situations in quantum electrodynamics in which summing across all energies results in an infinite value because the higher energy terms do not decrease quickly enough to create finite values.

In LQG, the background quantum field theory depends on is quantized, and hence, there is apparently no physical "room" for the ultraviolet infinities to occur. However, this argument may be compromised if LQG does not admit a limit of smooth geometry at long distance scales. It should bear in mind that LQG is constructed as an alternative to perturbative quantum field theory on a fixed background. In its present form, it does not allow a perturbative calculation of graviton scattering or other processes and it is not clear whether it ever will.

Graviton

In quantum field theories, the graviton is a hypothetical elementary particle that transmits the force of gravity in most quantum gravity systems. In order to do this gravitons have to be always-attractive (gravity never pushes), work over any distance (gravity is universal) and come in unlimited numbers (to provide high strengths near stars). In quantum theory, this defines an even-spin (spin 2 in this case) boson with a rest mass of zero.

It remains open to debate whether loop quantum gravity requires, or does not require, the graviton, or whether the graviton can be accounted for in its theoretical framework. As of today, the appearance of smooth space and gravitons in LQG has not been demonstrated, and henceforth the questions about graviton scattering cannot be answered in LQG.



People

See list of loop quantum gravity researchers

See also

Bibliography

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