What is quantum gravity and quantum space
ASTRO / 167: Loop Quantum Gravity (Stars and Space)
Stars and Space 7/11 - July 2011
Loop quantum gravity
The quanta of gravity
By Kristina Giesel
For three fundamental forces in the universe - the electromagnetic, the weak and the strong - we already know that classical physics must be replaced by quantum physics on microscopic scales in order to adequately describe the observed processes in nature. But what about the fourth fundamental force, gravity? Does classical gravity have to be replaced by quantum gravity here too?
• Description of all natural phenomena is achieved with great precision using four types of interaction - forces. Problems only arise on tiny scales.
• Physicists have been working on a consistent description of the fundamental forces for a long time. To get closer to this goal, loop quantum gravity is looking for a quantization of spacetime.
• As a result of the loop quantum gravity, space-time no longer appears as a continuum, but as a foam-like structure.
There are two fundamental theories in modern theoretical physics. The older is the general theory of relativity formulated by Albert Einstein in 1915. It describes the interaction between mass and four-dimensional space-time. The other is quantum field theory, which was formulated over the course of the 20th century. It treats the particles and the force fields involved in their interaction in a uniform way, takes into account the special theory of relativity and describes the interactions of elementary particles in the subatomic range up to very small distances of about 10-19 Meters. This is considerably smaller than an atom (typically a tenth of a nanometer, 10-10 Meters) and still considerably smaller than an atomic nucleus (the size of a proton is 10-15 Meter). The physicists summarize the microscopic description of these three basic forces of quantum field theory in the so-called standard model of particle physics (see box "The four basic forces").
The four basic forces
There are four basic forces in nature: strong, weak, electromagnetic and gravitation.
• The strong nuclear power is the strongest of all interactions, but its range is very short. It only acts on hadrons, i.e. particles made up of quarks and their antiparticles, the antiquarks. The most famous hadrons are the proton and the neutron, which make up the atomic nucleus.
• The electromagnetic force plays the greatest role in our everyday life, as it acts between all electrically charged particles, for example electrons. Their strength decreases with the square of the distance, so their range is basically infinite. But wherever there are roughly the same number of positively and negatively charged particles, their electromagnetic forces cancel each other out. They are therefore shielded over a large area and act mainly on the molecular and atomic level and determine the material properties of the elements and their compounds.
• The weak force has an even smaller range than the strong one and plays an important role in particle decays. These are processes in which one particle is converted into another, as happens in radioactive beta-minus decay. There a neutron decays into a proton, an electron and an anti-electron neutrino. But it also affects leptons, which are indivisible particles, such as the electron or its antiparticle, the positron, and can convert different leptons into one another.
• The gravity is by far the weakest of the four forces. As in the case of electromagnetic force, its strength decreases with the square of the distance, so its range is infinite. It affects all masses. And since there are no masses with opposite "gravitational charges", the gravitational forces of all masses add up.
relative strength: 1
Range: 10-15 meter
relative strength: 0.01
relative strength: 10-15
Range: ≤ 10-15 meter
The four known forces or interactions have different strengths and ranges.
Quantum Field Theory (QFT) is the mathematical framework of the Standard Model. It can be used, for example, to calculate the probability of certain particle processes occurring, such as collisions. It also makes predictions about the existence of previously unknown particles and explains how new particles can be created from microscopic particles.
The microscopic particles of the Standard Model are described as quantum objects and are subject to the laws of quantum physics (see box). The standard model was tested experimentally in numerous experiments on particle accelerators, such as the Large Hadron Collider (LHC) at the European Nuclear Research Center CERN near Geneva in Switzerland (see picture). So far, the physicists have observed an excellent match between theory and experiment (see SuW 9/2008, p. 48 and 10/2010, p. 46).
The physicists describe a quantum as a certain smallest amount of a physical measurand that occurs in nature, for example the minimal amount of energy that can occur in a system under consideration. An excited atom sends a precisely defined amount of energy - a light quantum - in the form of a photon of a certain frequency with the energy E = h v off when one of its electrons is from a higher energy level E.2 to a lower one E.1, falls. The energy E. of the photon is then E = E2 - E.1.
Physicists also refer to the smallest particles that make up our matter as quanta or elementary particles, such as electrons, protons and neutrinos. This emphasizes the corpuscular, i.e. the particle, character. Matter particles show a wave character that is described with the so-called De Broglie wavelength (after Louis-Victor de Broglie, a French physicist, 1892-1987). Here one imagines classical particles as tightly localized wave packets.
The standard model is now being put to the test again because there are still a number of unanswered questions to be answered. The researchers are paying the greatest attention to the search for the Higgs boson. This particle is part of the standard model and explains why the elementary particles have mass (see SuW 10/2010, p. 46). However, the Higgs boson has not yet been observed directly. That is why particle physicists try to prove it with large-scale experiments. This is one of the key tasks at the Large Hadron Collider. The standard model of particle physics, however, only includes three of the four fundamental forces. Gravitation as the fourth force is not part of it.
Gravitation in Einstein's theory of relativity
So far, gravitation has been described extremely successfully with the theory developed by Albert Einstein almost a hundred years ago: the general theory of relativity. It describes processes on macroscopic scales, and its laws are subject to classical physics, but not to quantum physics. By macroscopic scales, physicists mean distances that are significantly larger than atoms; they are certainly also of a cosmological order of magnitude.
In contrast to the other three fundamental forces, gravity can be understood as a theory for the geometry of spacetime. That's what makes her so special. Because while the quantum field theory describes the behavior of the particles and forces on a given scene - in space and in time - the GTR defines the scene itself. Space and time are already treated in the special theory of relativity (SRT) as a four-dimensional space-time continuum , in which the three dimensions of space form a unit with one dimension of time. Albert Einstein published the SRT in 1905, ten years before his broader general theory of relativity (ART).
The geometry of this spacetime is flat in the SRT. In GTR, on the other hand, spacetime can also be curved. The curvature of a spacetime is caused by any form of energy, for example also by mass. Because, according to Einstein's famous formula, every mass corresponds m the energy E = m c², in which c stands for the speed of light. Therefore the photon is also zero, with rest mass zero, but also the energy E = h v, a source of gravity. Physicists refer to anything that can serve as a source of gravity as matter. How one can vividly imagine a curved space-time, the graphic in the box "Curved space-time" illustrates for two spatial dimensions.
Einstein's general theory of relativity says that matter bends space-time, i.e. changes its geometry. You can imagine it as follows: A heavy ball lies on an initially smooth and flat membrane, for example a rubber blanket. At its position (due to its weight) it bulges the membrane downwards, giving it a curvature. If it now moves along the membrane, this changes its curvature, depending on its position. A comparison of the movement of the ball along the membrane with a ball moving on a non-deformable smooth surface (e.g. a steel plate) shows that the movements are different. Obviously, the curvature of the base itself also has an influence on the movement of the ball. This non-linear interaction between geometry and matter is described by the Einstein equations.
Einstein's theory tells us how matter and the geometry of space-time interact with one another. On the one hand, matter bends space-time, on the other hand, matter moves differently in a curved space-time than in a flat one. This interaction between matter and gravity is highly non-linear, so far from simple proportionality. This can be seen in the shape of the central GTR equations, the so-called Einstein equations. They form a set of ten coupled non-linear partial differential equations that describe the dynamics of gravity and matter.
Just like the Standard Model, the theory of relativity was also excellently confirmed in numerous experiments. In our everyday life, the effects of GART do not normally play a role; but for example the one realized with the help of satellites in near earth orbits Global Positioning System (GPS), both the effects of the SRT and the ART must be taken into account so that the desired accuracy of position determinations in the meter range can be achieved.
Limits of predictive power
Both quantum field theory (QFT) and ART describe processes in nature very successfully on their respective length scales and form the pillars of today's theoretical physics. In certain areas of both theories, however, infinities occur, also called singularities in technical terminology. They exist when physical quantities take on infinitely large values and the theory therefore loses its predictive power. In QFT this is the case, for example, when two elementary particles come as close as desired, i.e. when ever smaller distances are considered. The infinities that appear at first, however, can be eliminated by a mathematically complex procedure. This process is called renormalization.
Such mathematical methods transform physical quantities, such as mass, into variable quantities that are dependent on the energy. This means that they take on different values depending on the energy. Another example of such an energy-dependent variable are the coupling constants, which, strictly speaking, are not constant at all. Coupling constants represent a measure of the strength with which a certain interaction occurs in nature. This so-called “running of the coupling constants” could also be observed experimentally (see SuW 10/2010, p. 46). The singularities in the standard model can be eliminated consistently, but if gravity is also included in the considerations, this is no longer so easy.
The most well-known GTR singularities occur inside black holes and also during the Big Bang. For black holes, the GTR predicts that the curvature of space takes on an infinitely large value at their center. At the time of the Big Bang, all matter in the universe must have been concentrated in one point, which also means an infinitely high curvature of space.
Obviously, the existence of singularities in GTR is inevitable; at least that's what mathematical theorems say - the so-called singularity theorems. They were proven by the English theorists Roger Penrose and Stephen Hawking.
Can singularities be interpreted as a sign of the limits of the predictability of a theory? A possible cause for their occurrence could be that both ART and QFT are applied on scales where they become invalid. A consequence of this would then be that an even more fundamental theory would have to exist. This would be able - at least that is the hope - to describe physics even under such extreme conditions as prevailed, for example, in the Big Bang. Furthermore, such a concept should contain the already existing theories, i.e. the GTR and QFT, as borderline cases, since these already provide a good description of nature on certain scales. For this reason, such a more fundamental theory is also called quantum gravity theory.
Expectations of the new theory
What do you expect from such a theory? Since a quantum gravity theory should contain GTR and QFT as borderline cases, their principles must also be part of the overarching concept. One of the basic principles of GTR says that the geometry of space-time itself is not static, but changes dynamically in the interplay with the energy density of matter. Theories that allow a dynamic spacetime are called background-independent by physicists. In contrast, background-dependent theories assume that there is an immutable and rigid spacetime that serves as a kind of background platform (see table below).
In background-dependent theories, the physical laws for matter with regard to a fixed space-time - the background - are formulated, and all repercussions of matter on space-time, which the GTR actually describes and includes, are not taken into account. Quantum field theory is such a concept because it describes the interactions between quantum particles on rigid, flat spacetime.
The two approaches could hardly be more opposed, and this is certainly the reason why it has not yet been possible to formulate a quantum gravity theory that meets all requirements.
Loop quantum gravity
A candidate for a quantum gravity theory is loop quantum gravity (LQG). Their approach is to consistently link the principles of ART and QFT and thereby enable a background-independent quantization of gravity. What does that mean? The aim is to obtain a quantum physical version of the well-known Einstein equations, which then describe the dynamic processes of quantum gravity theory. This approach leads to a "quantum geometry". In doing so, one abandons the idea that geometry is a purely classical, continuous construct and treats it as fundamentally as a quantum quantity.
Just as classical geometry and classical matter interact with one another in ART, loop quantum gravity is based on an interaction between quantum matter and quantum geometry. Both ingredients of the Einstein equations are therefore considered to be quantum objects. This is a very natural approach to an expanded theory, since nature has taught us that matter is fundamentally quantized (see the "Quantization" box). In our everyday life, the quantum properties of matter can usually be neglected because the everyday world runs on much larger length scales than on those microscopic length scales on which quantum effects dominate. Therefore we can regard matter as a classical object. Strictly speaking, this is only an approximation.
Since a quantum gravity theory describes physics on extremely small length scales, the quantum properties of matter become dominant. In addition, it does not necessarily follow that the geometry of space-time can still be viewed as a classical object.
What exactly does quantization actually mean in physics? In general, this denotes the transition from a classical theory to a quantum theory. The physically described quantities can then no longer assume any arbitrary value.
In addition to the mathematically very different framework for classical physics and quantum physics, there are also significant differences in the physical questions that can be asked in the context of a classical or a quantum mechanical theory.In classical physics, the position and momentum of a particle can be determined as precisely as required. In quantum physics, however, this is no longer possible (see SuW 10/2010, p. 48).
Quantum physics teaches us three things:
• In general, it is only possible to specify probabilities with which the momentum or the position of the particle assume a certain value.
• Furthermore, due to Heisenberg's uncertainty relation, position and momentum can no longer be determined at the same time as precisely as desired.
• Another typical property of quantum theories is that physical quantities, which in classical theory can assume any value, only occur in quantum theory with very specific, discrete values. So they appear in quantized form.
A prominent example of this is the energy of electrons in atoms and molecules. According to classical physics, accelerated electrical charges continuously radiate energy in the form of so-called bremsstrahlung. As a result, an electron orbiting around the atomic nucleus in the atomic shell should gradually lose its energy. It would therefore be increasingly attracted to the atomic nucleus and thereby inevitably fall into the nucleus at some point. Thus, in the context of classical physics, atoms and molecules would not be stable, which, however, contradicts our daily observations.
If you look at atoms in quantum physics, the electron is no longer allowed to radiate any amount of energy. Only very specific discrete amounts of energy are allowed, which are in harmony with the transitions of energy levels in the atomic shells. The exact energy level of an atom or molecule can be calculated using quantum theory. For an electron that falls into the nucleus, the position and momentum would also be precisely determined at the same time, which is excluded according to Heisenberg's uncertainty principle. So it turns out that quantum theory provides an explanation for why atoms and molecules are stable.
Physical quantities can only take on very specific, discrete values.
In general, the transition from a classical to a quantum theory is anything but clear, sometimes there are in principle an infinite number of different possibilities. In order to drastically limit the number of possibilities, the physicists combine a quantization with additional sensible physical assumptions, which then select a certain form of the quantization. In connection with loop quantum gravity, recent research has shown that it is precisely the additional assumption of background independence that clearly defines the quantization step.
The work of Hanno Sahlmann, currently at the Asia Pacific Center for Theoretical Physics in South Korea, as well as a theorem that the four physicists Jerzy Lewandowski, Andrzej Okolow (both at the University of Warsaw, Poland), Hanno Sahlmann and Thomas Thiemann von of the University of Erlangen-Nuremberg. It is called the LOST theorem after the first letters of their surname. Further research by the mathematician Christian Fleischhack at the University of Paderborn comes to the same result.
But what exactly is quantum geometry? In GTR, space-time is viewed as a continuum. This means that the parameters with which we associate space and time can take on any value. Such spacetimes are interesting, the geometry of which forms solutions to the classical Einstein equations. In the LQG, on the other hand, quantization replaces continuous spacetime with a discrete structure.
The beginnings of loop quantum gravity (LQG) go back to the Indian-born US physicist Abhay Ashtekar, who introduced the Ashtekar variables named after him into the formula apparatus at the end of the 1990s. With the help of these variables it was possible to reformulate the general theory of relativity in such a way that it is in the same mathematical language as that of the other classical theories that are quantized in the context of quantum field theory. This gave rise to the advantage of being able to apply existing quantization techniques to the ART.
When quantizing the strong interaction, a similar variable plays an important role, the so-called Wilson loop, named after the US Nobel Prize winner Kenneth Wilson, one of the pioneers of quantum field theory. The formulation of the theory in these loop variables simplifies the later elimination of unnecessary degrees of freedom. The use of these variables justifies the name loop quantum gravity. An essential difference between the LQG and the quantum field theory is that it enables background-independent quantization and is therefore not limited to a rigid, fixed, space-time considered classically.
In the LQG, the discrete structure of spacetime manifests itself as follows: The continuous space is replaced by a very fine network of one-dimensional edges. On large scales where quantum gravity effects are negligible, this fine network still appears as a continuum. However, if you zoom in to length scales close to the so-called Planck scale at 10-35 Meters (see box below and SuW 3/2007, p. 92), it can be seen that the space, which initially appears to be continuous, is actually made up of a network of very fine, polymer-like structures, the edges.
The limits of the validity of the standard model of particle physics and the general theory of relativity in the smallest dimensions can be determined by the combination of the Compton wavelength λm = h / (m c), that is the wave-mechanical expansion of a particle of mass m, with the gravitational radius HG = G m / c², the area from which a mass m does not release any information.
If one wants to determine the location of the particle more precisely than its gravitational radius and the Compton wavelength indicate, then, because of Heisenberg's uncertainty principle, so much energy must be expended that it exceeds the amount of energy to generate a new particle of the same mass. This results in the limit of validity of the known physics. Below a smallest distance lp = (h G / c²)1/2 ≈ 10-35 Meters, the so-called Planck length, all models fail. Other physical quantities are linked to the Planck length, for example the Planck mass mp = (h c / G)1/2 ≈ 2 · 10-8 Kilograms and the Planck time tp = lp/ c ≈ 5 · 10-44 Seconds, the length of time required to traverse the Planck length at the speed of light.
The graphic in the box below (in the shadow view without illustration) shows such a network as an example. The edges meet at nodes, the vertices. Each of these edges is assigned a so-called spin quantum number, which codes the properties of the quantum geometry and which are symbolized in the graphic by different colors of the edges. The spin quantum numbers are arbitrary positive multiples of the number ½. This is why one speaks of spin networks, the relevance of which for the LQG was first recognized by Carlo Rovelli from Italy and Lee Smolin from the USA. Abhay Ashtekar from the USA, the British theorist Chris Isham and Jerzy Lewandowski from Poland then created the mathematical framework for the LQG.
Loop quantum gravity (LQG) does not describe space as a continuous structure, but rather through a spin network of different edges that meet at nodes, so-called vertices. Each edge is assigned a spin quantum number, that is, positive multiples of the number ½. In the graphic, different colors symbolize different spin quantum numbers. The LQG describes certain properties of quantum geometry through the structure of the spin networks and their spin quantum numbers. Even if the space appears as a continuum on macroscopic length scales, one would, according to the LQG, on microscopic scales close to the Planck length of lp = 10-35 Meters find this polymer-like structure of the room. The spin networks of the LQG replace the classical spacetime of the general theory of relativity.
(Image of the original publication in Schattenblick not published)
The physicists now do not consider the edges themselves, but rather the "dual surfaces" linked to them. For example, for a network in which four edges always meet in a vertex (node), a tetrahedron is created, i.e. a closed body made up of four triangular surfaces (see box below).
Dual faces of the edges
Loop quantum gravity assigns a so-called dual surface to all edges of space-time that meet at a node. These dual surfaces are pierced perpendicularly by the edges of the spin network. The individual dual surfaces of the edges should then form a closed surface again. Which geometric object describes this surface depends on the number of edges that meet at the node. The graphic above shows a node where six edges (green) meet. The six dual square faces form the surface of a cube.
The graphic below (in the print edition), on the other hand, shows a node with only four edges. Here the four dual triangular faces form the surface of a tetrahedron. For such dual surfaces it holds that the sum of the dimensions of the one-dimensional edges (D.K = 1) and the two-dimensional surfaces (D.F. = 2) just gives three, which corresponds to the space dimension (3): D.K + D.F. = 3. For example, a point (with dimension zero, D.p = 0) dual to a three-dimensional object (3 - D.p = 3).
(Images of the original publication in Schattenblick not published)
The spin quantum numbers at the edges are related to the area of the triangles. The graphic below on the left shows different spin quantum numbers in different colors. In contrast to continuous classical geometry, quantum geometry only exists where the edges are excited by spin quantum numbers other than zero. If, on the other hand, the spin quantum number zero is assigned to an edge, then this edge does not even exist.
If an observer were able to look at the room with a Planck magnifying glass, he would see, according to the loop quantum gravitation, a structure of quantum geometry as shown in the computer graphics on p. 38 above. The graphic shows a snapshot of quantum space. The development of the quantum space over time, i.e. the dynamics, is described by the quantum Einstein equations. This is a quantum analogue of the classic Einstein equations, which the physicist Thomas Thiemann from the University of Erlangen-Nuremberg consistently formulated for the first time.
A temporal change in space modifies the spin quantum numbers or, equivalently, the area - in computer graphics, the colors change over time. In addition, new triangles can be created and existing ones can be destroyed. We can imagine quantum spacetime as a kind of spin foam in which triangles appear and disappear at different points as time progresses and which changes its color continuously.
A formulation of quantum dynamics in the form of spin-foam models, which, in contrast to the quantum Einstein equations, does not require the space-time continuum to be split into space and time, is another branch of loop quantum gravity that is currently being intensively researched.
Before we take a closer look at the properties of the quantum Einstein equations in a later section, we first examine geometric objects such as length, area and volume, which occur as quantum objects in the context of the LQG.
Quantum properties of length, area and volume
What does it mean when volume, area and length no longer show classical behavior, but are rather quantum objects? Let us consider an area of one square meter. We are now dividing the total area into smaller and smaller areas. In the context of classical physics, these partial areas can have any small area. In the LQG, however, this is no longer allowed. Rather, the areas of the partial areas always only take on certain discrete values, multiples of the square of the Planck length (lp2 = 10-70 Square meters) are. This means that the area can no longer be viewed as continuous, but is made up of the smallest units of area, the "area quanta".
This is a typical property of quantum systems. The quantum mechanical treatment of atoms has taught us, for example, that the electrons in the shell of the atoms can only have very specific energies. Not every possible energy value is allowed, which explains the discrete spectra of electromagnetic radiation observed in atoms, for example the emission lines of a neon tube.
In the case of a quantum gravity theory, which says something about the quantum properties of geometry, quantized surfaces are obviously something that one would have naively expected. The same also applies to the volume and the length, which both also only assume discrete values that correspond to the Planck volume (lp3 = 10-105 Cubic meters) or the Planck length (lp = 10-35 Meters) are linked. The enormous dimensions from the length scale of gravity and electromagnetic radiation in the universe to Planck's length in the subatomic range are demonstrated by the graphic on p. 38 below.
The special thing about the quantum area is that within the framework of the LQG there is a very small area that cannot be undercut. It is around lp2 = 10-70 Square meters, the square of the Planck length. In the case of the one square meter area, this consists of about 1070 smallest quantum areas. Whether the quantum volume or the quantum length also have a minimum value is currently still an open question. The reason for this is to be found in the fact that the mathematical-physical expressions for the quantum volume and the quantum length are far more complicated than those for the quantum area. The quantum volume is of particular interest, as it is essential for the formulation of the dynamics that the quantum Einstein equations describe. Their structure is as complicated as that of the classical Einstein equations in general relativity, which unfortunately makes it much more difficult to find solutions to the quantum Einstein equations.
In ten steps from the universe to the microcosm
1026 Meter / 1010 Lj
Our universe went through an extremely hot phase at the beginning. When it became transparent, photons escaped from the greatest accessible distance, more than 10 billion light years (ly).
1021 Meters / 100,000 ly
The galaxies are five orders of magnitude smaller than the universe and host almost all stars. Their typical size is around 100,000 light years, which is around 1021 Meter.
1016 Meters / 1 light year
The comet nuclei of the Oort cloud are located in the outermost regions of our solar system, around 50,000 times the distance between the earth and the sun.
With its equatorial diameter of 12,378 kilometers, the earth is around nine orders of magnitude smaller than the solar system. For us humans, however, it is still huge.
The size of humans differs by seven orders of magnitude from that of our planet. The Matterhorn in the background is just a pimple on its surface.
A human egg cell can barely be seen with the naked eye. Their diameter is a little more than 0.1 millimeters, that of the sperm head around three micrometers.
Depending on the excitation, an electron has different probabilities of being in the atom. Here is the wave function of the 3dz2-Orbitals of a single electron are shown.
The atomic nuclei of protons and neutrons sit in the center of the atoms. They are just as many orders of magnitude smaller as the galaxies compared to the universe.
The interaction particles of the weak force, the W bosons, show up at the highest energies that particle accelerators can currently generate. They couple with their weak charge to the vacuum and thus get their mass.
Loop quantum gravity no longer describes spacetime as a continuum, but in quantum terms. However, this only occurs with the Planck scale in the range of 10-35 Meters to days.
(Images of the original publication in Schattenblick not published)
Loop quantum cosmology
In the classical GTR, by assuming that the geometry of space-time has a certain symmetry, a lot can be learned about certain solutions to the Einstein equations. For cosmological spacetime, i.e. for the universe as a whole, it can be assumed, for example, that the space always looks the same to an observer, regardless of the point in space at which it is located. That is the "cosmological principle". The associated symmetry, which is referred to as homogeneous (independent of location) and isotropic (independent of direction), considerably restricts the possible shape of space-time geometry. In this context, one also speaks of symmetry reduction. This simplifies the Einstein equations and enables the equations to be solved explicitly. In the LQG the situation seems a little more complicated.
A symmetry reduction in the classical GR and a subsequent quantization of the reduced theory does not necessarily have to result in the same quantum theory as the reverse order in which first quantization is carried out and then an attempt is made to carry out the symmetry reduction in full quantum theory. The latter is much more complicated and has not yet succeeded in a satisfactory manner. Nevertheless, the symmetry-reduced cosmological sector of the LQG, which is also known as loop quantum cosmology (loop quantum cosmology, LQC) is an interesting area. The German physicist Martin Bojowald, who conducts research at Pennsylvania State University in the USA, developed essential parts of the LQC in order to explore typical properties of the LQG in simple models. Here, the simplified Einstein equations are quantized, which leads to much simpler quantum Einstein equations, which in turn can be solved explicitly.
A remarkable property of the LQC is that the Big Bang singularity that occurs in the GTR is no longer present. For this purpose, the physicists use the LQC to consider a universe with a very simple matter content, which today corresponds to the classic Einstein equations. If you follow its development backwards in time, this does not lead to the big bang, but through quantum effects of the geometry an additional repulsive force arises, which ricochets off the universe before it runs into a singularity and makes it expand again. This "quantum recoil" occurs at a critical matter density of around ρcrit ≈ 0.41 ρp ≈ 2 · 1096 Kilograms per cubic meter, where ρp ≈ 5,1 · 1096 Kilogram per cubic meter is the Planck density. It is a quantity derived from the Planck length and the Planck mass: ρp = mp/lp3 (see box »Planck scale«).
After the quantum recoil, the quantum Einstein equations describe a universe that expands, behaves almost classically with the exception of small quantum fluctuations and is thus in accordance with the Einstein equations. Thus, the quantization of the cosmological equations in the LQC models actually provides a theory without singularities. It therefore also enables statements to be made about the extreme conditions near the time of the Big Bang, which Einstein's ART remains hidden. Now the important and exciting question remains to be clarified: Do singularities also resolve in the complete LQG?
Black holes in the loop quantum gravity
Other classic solutions to the Einstein equations that have a singularity are black holes (see SuW 5/2010, p. 40). The name comes from the fact that their gravity is so strong that even light can no longer escape them from a certain proximity. The astrophysicists speak of the event horizon in this context. It is a spherical surface around a black hole. In the simplest form of black holes, the radius of this spherical shell depends only on their mass. It is called the Schwarzschild radius after the German physicist Karl Schwarzschild (1873-1916), who found this solution to the Einstein equations in 1915.
According to the general theory of relativity, particles that come from outside and pass the event horizon will inevitably hit the singularity inside the black hole after a finite time. Escape from the black hole is impossible after crossing the event horizon.
Similar to loop quantum cosmology, there are models of black holes that contain the cancellation of the singularity within the framework of a symmetry-reduced loop quantum gravity. Quantum geometric effects create a negative pressure that prevents the singularity from occurring.
In addition, it is an interesting question to investigate the entropy of a black hole as part of the LQG. The theorist Kirill Krasnov, currently at the University of Nottingham in England, developed the first approaches. Entropy is a thermodynamic state variable and represents a measure of the number of microscopic quantum states of a given macroscopic system. The higher the entropy, the more states the system can assume - the greater its disorder.
Conversely, a low entropy corresponds to a very ordered state. For macroscopic black holes there is a conjecture made by Jacob D. Bekenstein at the University of Jerusalem and Stephen W. Hawking at the University of Cambridge, according to which the entropy of a black hole and the surface of its event horizon are related. This relation can be derived in the context of loop quantum gravity for certain types of black holes. Here again, the quantum areas play an important role. Given a given value for the surface of the event horizon of a black hole, the loop quantum gravity can be used to calculate how many microscopic quantum states are connected to this surface. In this way, conclusions can be drawn about the entropy of the black hole. The entropy experiences a quantum physical interpretation within the quantum gravity.
In addition to the symmetry-reduced models and entropy, there are also certain properties that the full LQG must have in order to have physical relevance in addition to mathematical correctness.
Embedding in the loop quantum gravity
How are Einstein's general theory of relativity and ordinary quantum field theory embedded in loop quantum gravity? Each variant for a quantum gravity theory must contain both the classical GTR and the ordinary QFT as borderline cases in sub-areas of the theory, so-called sectors (see the graphic). The sector in the LQG in which the GTR can be found should be characterized by the fact that both the quantum fluctuations of geometry and that of matter are negligible. This corresponds to the left arrow in the graphic.
The usual quantum field theories, on the other hand, describe the dynamics of quantum matter on a fixed classical space-time. You should therefore be found in the LQG in a sector in which quantum geometry shows almost classic behavior and the effect of quantum matter on geometry can be neglected. The matter itself is still viewed as a quantum object. This corresponds to the right arrow in the graphic. Both limit cases are referred to as semiclassical (semi-classical) or low-energy limit values. Because in both cases the quantum properties of the geometry are negligible. As expected, this only occurs at energies that are far below the energy scale where quantum gravitational effects dominate.
Consequences for loop quantum gravity
What we would like to have in the LQG are quantum states, the dynamic behavior of which reproduces the classic Einstein equations with the exception of minimal quantum fluctuations. States in quantum theory can behave quite differently from what might be expected from the associated classical theory. The "state" of a quantum system describes the probabilities of all of its observable measurands.
In classical theory, both geometry and matter follow a certain trajectory over time, a so-called trajectory. Any arbitrary quantum state that develops will generally not follow the classical trajectory, but will have a different dynamic behavior. It is precisely the semiclassical states that have the property of following the classical trajectory apart from small quantum fluctuations.
These states are a mixture of many individual quantum states and play a role in several areas of physics, for example in quantum optics or in solid state physics. The special thing about them is that they enable a transition from quantum theory back to classical theory. So they allow to test whether a quantum theory is consistent with the associated classical theory. This is particularly important for quantum gravity theory, since in this case the occurrence of direct quantum effects can only be expected at extremely high energies.
Whether it is possible to construct such semiclassical states in a given quantum theory and thus to make the transition to the classical theory depends mainly on the form of the dynamic equations. The more complicated their structure, the more difficult it is usually to find suitable semiclassical states.
Even for ordinary quantum field theory this is a difficult task, for loop quantum gravity, on the other hand, quantum dynamics looks even more complicated, and the construction of semiclassical states represents a great mathematical-physical challenge.
The first progress was made around ten years ago with the construction of so-called coherent states for the LQG. This is a certain class of semiclassical states that have other specific properties and are also used in quantum mechanics. These states enable an appropriate semiclassical description of certain kinematic quantities, for example the geometric quantum quantities volume, area and length. It turns out that the quantum volume with regard to these states corresponds to the classical volume except for minimal quantum fluctuations. The same applies to quantum area and length.
For a long time it was not possible to check the quantum Einstein equations with these states for their semiclassical consistency. The physicists only achieved this in the last few years by further developing the existing semiclassical techniques. However, the analysis is still limited to very small time scales, since the coherent states under currently existing descriptions all dissolve after a certain time. So they lose their semiclassical properties and no longer follow the classical trajectory. For correspondingly small time intervals it could be shown that there are still states in the LQG that follow the trajectory expected according to the classical ART. This problem does not occur in loop quantum cosmology, since the corresponding symmetry-reduced quantum Einstein equations have a much simpler structure and therefore the coherent states remain dynamically stable even over longer time intervals.
In order to analyze what happens on larger time scales in the fully formulated loop quantum gravity, the already existing semiclassical states must first be generalized and better adapted to the quantum dynamics of the LQG. This step is also important to clarify the question of how ordinary quantum field theory occurs in the semiclassical limit of the LQG.
Here one needs semiclassical states that remain stable on the time scale that is important for the scattering processes of the quantum particles of the Standard Model.
The way to new knowledge
A better understanding of the dynamics of the LQG in the form of the quantum Einstein equations or in the context of so-called spin-foam models is currently being intensively researched. The theorists hope that this will provide further insights into the extent to which the LQG is consistent with the already existing theories that have proven themselves. Every attempt at a quantum gravity theory, not just the LQG, must face such consistency checks. If the new theory variant passes these tests, it can be used to research and analyze "uncharted territory" in physics, in which areas effects of quantum geometry change our previous understanding of physics.
So far, loop quantum gravity has already delivered promising results: within the framework of the symmetry-reduced models, it is possible to eliminate the singularities in the Big Bang and in the interior of black holes. It also allows a quantum theoretical explanation of the entropy of black holes. The theorists will continue to work intensively over the next few years to understand gravity in the context of quantum physics and thus to expand known physics to new territory.
Kristina Giesel researches and teaches at Louisiana State University in Baton Rouge, USA. Since her doctoral thesis, she has been working in the field of loop quantum gravity with a research focus on the semiclassical sector of theory. In the meantime she has accepted a position at the Friedrich-Alexander-Universität-Erlangen-Nürnberg.
Bojowald, M .: The original leap of space. In: Spektrum der Wissenschaft 5/2009, pp. 26-32
Smolin, L .: Three Roads to Quantum Gravity. Science Masters 2008
Thiemann, T., Pössel, M .: A cosmos without a beginning. In: Spektrum der Wissenschaft 6/2007, pp. 32-41
Sahlmann, H .: Loop Quantum Gravity, A Short Review. arXiv: 1001.4188 [gr-qc]
Giesel, K: On the Consistency of Loop Quantum Gravity with General Relativity. Dissertation, University of Potsdam, 2007
Captions of the images of the original publication not published in Schattenblick:
The physical description of the four basic forces of nature includes the most distant galaxies ten billion light years away as well as the tiniest structures that make up the quantum spacetime according to the loop quantum gravity on length scales close to the so-called Planck scale at 10-35 Meters. While gravity and electromagnetism are unlimited in their range, the effect of strong and weak nuclear forces is limited to atomic dimensions.
Fig. P. 32:
The Large Hadron Collider (LHC) is a gigantic particle accelerator. The acceleration section consists of two evacuated steel tubes, each five centimeters in diameter, which are inserted underground in an almost 27 kilometer long circular tunnel near Geneva in Switzerland. Superconducting magnets in tubes about one meter in size (here painted blue) keep the charged particles on track and cause two particle streams racing in opposite directions to collide at four measuring points.
Fig. P. 38:
The computer graphic shows a snapshot of quantum space, quantum spacetime. It is in the context of loop quantum gravity on length scales close to the Planck scale of lp = 10-35 Meters to be expected. The tetrahedra shown correspond to the dual surfaces of a spin network, in which four edges meet at a node. The colors of the individual triangles in the tetrahedra symbolize different values for the area of the triangles, which in turn indicates how many quantum areas there are at a given point in time. The quantum areas are related to the spin quantum numbers of the corresponding edges.
Fig. P. 40 above:
The two borderline cases of loop quantum gravity form the general theory of relativity and the ordinary quantum field theory. The ART describes how mass and space-time are related to each other. From the LQG's point of view, the geometry of spacetime is classically structured, i.e. not quantized. Quantum field theory, on the other hand, treats spacetime as a fixed, predetermined structure. Therefore, from the LQG's point of view, it is only an approximation.
© 2011 Kristina Giesel, Spectrum of Science Verlagsgesellschaft mbH, Heidelberg
Stars and Space 7/11 - July 2011, pages 30-41
Prof. Dr. Matthias Bartelmann (ZAH, University of Heidelberg),
Prof. Dr. Thomas Henning (MPI for Astronomy),
Dr. Jakob Staude
Editor Stars and Space:
Max Planck Institute for Astronomy
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published in Schattenblick on September 14, 2011
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