Misplaced Pages

Loop quantum gravity

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

This is an old revision of this page, as edited by Byrgenwulf (talk | contribs) at 19:06, 26 September 2006 (→See also: removed link to Heim theory). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Revision as of 19:06, 26 September 2006 by Byrgenwulf (talk | contribs) (→See also: removed link to Heim theory)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

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. 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.

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

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 and by Smolin.

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

Eq(1) d 2 f ( x ) d x 2 + f ( x ) = 0 {\displaystyle {\frac {d^{2}f(x)}{dx^{2}}}+f(x)=0}
Eq(2) d 2 g ( y ) d y 2 + g ( y ) = 0 {\displaystyle {\frac {d^{2}g(y)}{dy^{2}}}+g(y)=0}

except in Eq(1) the independent variable is x and in Eq(2) the independent variable is y {\displaystyle y} . Once we find out that a solution to Eq(1) is f ( x ) = cos ⁡ x {\displaystyle f(x)=\cos x} , we immediately know that g ( y ) = cos ⁡ y {\displaystyle 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 {\displaystyle x} -coordinates and y {\displaystyle 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 {\displaystyle x} and in the other the independent variable is y {\displaystyle y} . Once we find a metric function g a b ( x ) {\displaystyle g_{ab}(x)} that solves the EQM in the x {\displaystyle x} -coordinates we immediately know (by exactly the same reasoning as above!) that the same function written as a function of y {\displaystyle y} solves the EOM in the y {\displaystyle 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 {\displaystyle 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 g ~ a b {\displaystyle {\tilde {g}}_{ab}} whose value at P {\displaystyle P} is given by the value of g a b {\displaystyle g_{ab}} at P 0 {\displaystyle P_{0}} , i.e.

Eq(3) g ~ a b ( P ) = g a b ( P 0 ) {\displaystyle {\tilde {g}}_{ab}(P)=g_{ab}(P_{0})} .

(see figure 1(a)). Now consider a coordinate system which assigns to P {\displaystyle P} the same coordinate values that P 0 {\displaystyle P_{0}} has in the x-coordinates (see figure 1(b)). We then have

Eq(4) g ~ a b ( y 0 = u 0 , y 1 = u 1 , y 2 = u 2 , y 3 = u 3 ) = g a b ( x 0 = u 0 , x 1 = u 1 , x 2 = u 2 , x 3 = u 3 ) , {\displaystyle {\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 {\displaystyle u_{0},u_{1},u_{2},u_{3}} are the coordinate values of P 0 {\displaystyle P_{0}} in the x-coordinate system.

File:Activediffwiki.svg
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 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, ..."

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 invaraint under coordinate transformations. (Indeed, the mathematical definition of a diffeomorphism is a transformation which relates topologically equivalent spaces, not geometrically equivalent spaces. 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.

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."

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

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. Gaul, Marcus (2000). "Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance". Lect.Notes Phys. 541: 277–324. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  2. ^ Smolin, Lee. "The case for background independence". hep-th/0507235. {{cite journal}}: Cite journal requires |journal= (help)
  3. Einstein, Albert (1916). The Principle of Relativity. p. 117. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  4. Witten, Edward. "Quantum Background Independence In String Theory". Report-no: IASSNS-HEP-93/29. {{cite journal}}: Cite journal requires |journal= (help)
  5. "Researchers Look Beyond the Birth of the Universe". Eberly College of Science. 12 May 2006.
  6. Khoury, Justin. "From Big Crunch to Big Bang". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  7. Seiberg, Nathan. "From Big Crunch To Big Bang - Is It Possible?". {{cite journal}}: Cite journal requires |journal= (help)
  8. Cornalba, L. "A New Cosmological Scenario in String Theory". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  9. Bilson-Thompson, Sundance O. "Quantum gravity and the standard model". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  10. 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)
  11. "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.
Categories: