- Translation
Two candidates for a “theory of everything,” long considered incompatible, may turn out to be two sides of the same coin.
Eighty years have passed since physicists realized that the theories of quantum mechanics and gravity are incompatible, and the mystery of combining them remains unresolved. Over the past decades, researchers have studied this problem in two different ways - through string theory and through quantum gravity - which the scientists who practice them consider incompatible. But some scientists argue that to advance it is necessary to join forces.
Among the attempts to unify quantum theory and gravity, string theory has received the most attention. Its premise is simple: everything is made of little strings. The strings can be closed or open; they can vibrate, stretch, combine or disintegrate. And in this diversity lie explanations for all observable phenomena, including matter and space-time.
Loop quantum gravity (LQG), in contrast, places less emphasis on the matter present in spacetime and focuses more on the properties of spacetime itself. In the PKG theory, space-time is a network. The smooth background of Einstein's theory of gravity is replaced by nodes and links that are assigned quantum properties. Thus, the space consists of separate pieces. The PKG mainly studies these pieces.
This approach has long been considered incompatible with string theory. Indeed, their differences are obvious and profound. For starters, PCG studies chunks of spacetime, and string theory studies the behavior of objects in spacetime. These areas also share technical challenges. String theory requires that there be 10 dimensions in space; PKG does not work in higher dimensions. String theory assumes the existence of supersymmetry, in which all particles have as yet undiscovered partners. Supersymmetry is not characteristic of PCG.
These and other differences have divided the theoretical physics community into two camps. "The conferences are divided," says Dorge Pullin, a physicist at Louisiana State University and co-author of a textbook on PCG. – Loop players go to loop conferences, string players go to string conferences. Now they don’t even go to conferences on “physics”. I think it's quite unfortunate."
But some factors may move these camps closer. New theoretical discoveries have revealed possible similarities between PKG and string theory. A new generation of string theorists has moved beyond string theory and began searching for methods and tools that could be useful in creating a “theory of everything.” And the recent paradox of information loss in black holes has made everyone feel more humble.
Moreover, in the absence of experimental evidence for string theory or PKG, mathematical proof that they are two sides of the same coin would provide evidence that physicists are moving in the right direction in their quest for a “theory of everything.” The combination of PKG and string theory would make the new theory unique.
An unexpected connection
Attempts to solve some problems of PKG led to the first unexpected connection with string theory. Physicists studying PKG do not have a clear understanding of how to move from pieces of a spacetime network to a large-scale description of spacetime that matches Einstein's general relativity, our best theory of gravity. Moreover, their theory cannot accommodate the special case in which gravity can be neglected. This is the problem that besets any attempt to use space-time in pieces: in SRT, the linear dimensions of an object decrease depending on the movement of the observer relative to the object. Compression also affects the size of pieces of space-time, which are perceived differently by observers moving at different speeds. This discrepancy leads to problems with the central principle of Einstein's theory - that the laws of physics do not depend on the speed of the observer.“It's difficult to introduce discrete structures without running into SRT problems,” says Pullin. In a 2014 paper written with colleague Rudolfo Gambini, a physicist at the Universidad Republican de Uruguay in Montevideo, Pullin writes that aligning PKG with STR inevitably entails the appearance of interactions similar to those present in string theory.
That the two approaches had something in common had seemed likely to Pullin since a seminal discovery in the late 1990s by Juan Malzadena, a physicist at the Institute for Advanced Study in Princeton, New Jersey. Malzadena, in anti-De Sitter spacetime (AdS), reconciled the theory of gravity and conformal field theory (CFT) at the boundary of spacetime. Using the AdS/CFT approach, the theory of gravity can be described using a more understandable field theory.
The full version of dualism is still a hypothesis, but it has a well-understood limiting case that string theory does not deal with. Because strings do not play a role in this case, it can be used in any theory of quantum gravity. Pullin sees a common ground here.
PKG as imagined by an artist
Hermann Verlinde, a theoretical physicist at Princeton University who frequently works with string theory, says it is plausible that PKG methods could shed light on the gravitational side of dualism. In a recent paper, he described a simplified AdS/CFT model in two dimensions for space and one for time, or, as physicists say, in the “2+1” case. He discovered that the AdS space can be described using networks such as those used in PCG. Even though the entire design is still working in 2+1, it offers a new way of thinking about gravity. Verlinde hopes to generalize the model to more dimensions. “PKG was looked at too narrowly. My approach includes other areas as well. In an intellectual sense, this is a look into the future,” he said.
But even if it is possible to combine the methods of PKG and string theory to move forward with AdS space, the question remains: how useful will such a combination be? AdS space has a negative cosmological constant (this number describes the geometry of the Universe on large scales), while our Universe has a positive one. We do not live in a mathematical construct described by the AdS space.
Verlinde's approach is pragmatic. “For example, for a positive cosmological constant, we may need a new theory. The question then is how different it will be from this one. AdS is the best hint yet of the structure we are looking for, and we need to perform some trick to arrive at a positive constant.” He believes that scientists are not wasting their time with this theory: “Although AdS does not describe our world, it will give us lessons that will lead us in the right direction.”
Unification in the territory of a black hole
Verlinde and Pullin point to another possibility for the string theory and PKG communities to unite: the mysterious fate of information falling into a black hole. In 2012, four researchers from the University of California drew attention to a contradiction in the prevailing theory. They argued that if a black hole allowed information to escape from it, it would destroy the fine structure of empty space around the black hole's horizon, and create a high-energy barrier - a "firewall". But such a barrier is incompatible with the equivalence principle underlying general relativity, which states that an observer cannot tell whether he has crossed the horizon. This incompatibility has upset string theorists, who thought they understood the connection between black holes and information, and were forced to grab their notebooks again.But this problem is important not only for string theorists. “This whole firewall debate has been mostly in the string theorist community, which I don't understand,” Verlinde said. “Issues of quantum information, entanglement and the construction of mathematical Hilbert space are what the PCG experts have been working on.”
At this time, an event occurred unnoticed by most string specialists - the fall of the barrier erected by supersymmetry and additional dimensions. Thomas Tiemann's group at the University of Erlangen-Nuremberg (Germany) has extended PKG to higher dimensions and included supersymmetry - concepts that were previously the exclusive domain of string theory.
Recently, Norbert Bodendorfer, a former student of Tiemann working at the University of Warsaw, applied loop quantification methods from PCG to AdS space. He argues that PKG is useful for dealing with AdS/CFT duality in cases where string theorists cannot make gravitational calculations. Bodendorfer believes that the gap that existed between the PKG and the strings is disappearing. “Sometimes I got the impression that string theorists have very little understanding of PKG and don’t want to talk about it,” he said. “But younger specialists demonstrate open-mindedness. They are very interested in what is happening at the intersection of regions.”
“The biggest difference is how we define our questions,” says Verlinde. “The problem is more sociological than scientific, unfortunately.” He doesn't think the two approaches conflict: “I've always thought of string theory and PKG as being part of the same description. PKG is a method, not a theory. This is a method of thinking about quantum mechanics and geometry. This is a method that string theorists can, and already do, use. These things are not mutually exclusive."
But not everyone is convinced. Moshe Rozali, a string theorist at the University of British Columbia, remains skeptical about the PKG: “The reason I don't work on the PKG is because it has problems with SRT,” he says. “If your approach is disrespectful of the symmetries in special relativity from the very beginning, you will need a miracle at one of the intermediate steps.” However, according to Rosalie, some of the mathematical tools that come from PCG can be useful. “I don’t think it is possible to combine PKG and string theory. But people usually need methods, and in that sense they are similar. Mathematical methods may overlap."
Also, not all adherents of PCG expect a merger of the two theories. Carlo Rovelli, physicist at the University of Marseille and founder of the PCG theory believes in the dominance of his theory. “The string community is not as arrogant as it was ten years ago, especially after the disappointment of the lack of supersymmetric particles,” he says. – It is possible that the two theories could be part of one solution... but I think it’s unlikely. In my opinion, string theory has failed to deliver what it promised in the 1980s and is one of those ideas that look nice but don't describe the real world, of which the history of science has been replete. I don’t understand how people can still place their hopes in her.”
Pullin believes that it is premature to declare victory: “Adherents of the PCG say that their theory is the only correct one. I won't sign up for this. It seems to me that both theories are extremely incomplete."
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Definition 1
Loop quantum theory represents the knowledge of loop gravity of quantums. Its founders were such scientists as T. Jacobson, C. Rovelli, A. Ashtekar and L. Smolin.
The essence of loop quantum theory
According to this theory, time and space consist of discrete quantum cells connected in a certain way to each other. This allows them to create a discrete structure on small time scales, and on large time scales the time space becomes continuous.
Thus, space is made up of very small cells, smoothly connected to each other, forming the surrounding space for us. At the moments when these bundles form knots and tangles, elementary particles are formed.
Thanks to loop quantum gravity, scientists were able to find out the fact that the initial singularity disappears under the influence of quantum effects. Thus, the Big Bang ceases to be a veil of mystery behind which one cannot look. Science now allows us to look at the events that took place in the Universe before him.
The main objects in loop quantum theory are special cells of space, whose state and behavior are controlled by a certain field that exists in them. Its value becomes the so-called “internal time” for such cells. In other words, the transition from a weak field to a stronger one presupposes the existence of a “past” capable of influencing a certain “future”.
Consequently, the theory equates space to atoms: the numbers obtained when determining the volume form a discrete set, which allows the volume to change in separate portions. This, in turn, deprives space of continuity and allows the idea of its existence in the format of certain quantum units of volume and area.
Specifics of loop quantum theory
In the case of describing quantum mechanical phenomena, physicists calculate the probability of various processes that occur under certain circumstances. The same thing happens when using the theory of loop quantum gravity to describe changes in the geometry of space or the movement of fields and particles in a spin network.
Note 1
Scientist Thomas Tiemann was able to derive exact expressions for determining the quantum probability of spin network steps. The end result of such calculations was the emergence of a clear methodology for calculating the probability of any process whose origin is probable in this world within the framework of subordination to the laws of the above theory.
The theory of relativity assumes the inseparability of time and space from each other and their existence in the format of a single time space. Introducing the concept of time space into loop quantum theory, the spin networks that represent space become what is called "spin foam".
When one more measurement indicator is included - time - the lines of the spin network expand and transform into two-dimensional surfaces, while the nodes in the line dissolve. Transitions that provoke changes in the spin network are now presented in the form of special nodes, within which the foam lines are combined. A snapshot of an ongoing process is visually similar to the image of a cross-section of temporary space.
A similar slice of spin foam is a spin network, but one should not be mistaken about the movements of the slice plane in a continuous mode, similar to the smooth flow of time. Similar to the process of defining space as a discrete geometry of a spin network, time will be defined as a sequence of individual steps that are rearranged by the network.
Thus, certain conclusions can be drawn:
- About the discreteness of time, that is, it does not flow like a river, but is more reminiscent of a ticking clock, the interval between ticks of which is approximately equal to Planck’s time. In other words, time in the Universe is measured by myriads of clocks: in the region where a quantum step is carried out in the spin foam, the clock produces one “tick”.
- Loop quantum gravity contributes to characteristic predictions of new events and phenomena. In fact, it is considered to be completely compatible with the postulate of both a three-dimensional world and one time dimension.
- While compatible with a wide range of different versions of the matter contained in the world, it does not require symmetries, dimensions, or degrees of freedom except those explored by scientists.
At the same time, there are versions of loop quantum gravity that include supersymmetry and extend many of the results to higher dimensions. For this reason, when there are indications of the presence of supersymmetry or higher dimensions, problems do not arise for loop quantum theory. Instead, quantum loop gravity assumptions will apply to the structure of space over very small distances.
Thus, loop quantum gravity assumes the presence in reality of a smooth picture of the time space of classical general relativity only as a result of averaging of a discrete structure, within which regions and surfaces can have exclusively certain discrete quantized values of volumes and areas.
Note 2
Loop quantum gravity makes it possible to obtain specific assumptions for the discrete geometry of quanta (we are talking about short distances). Moreover, such assumptions begin to be formed on the basis of first principles, and therefore they exclude elements of adjustment.
In this sense, approaches in loop quantum gravity have certain differences in comparison with other approaches that postulate some form of discrete structure in the format of the starting position and without deriving it as a consequence of combining general relativity with quantum theory.
Differences between string theory and loop quantum gravity theory
Scientists note the fundamental differences between loop quantum theory and other theories. In particular, superstring theory. In the latter, the main objects are multidimensional membranes and strings moving in the time and space initially prepared for them. At the same time, this theory does not allow us to name the factors of the emergence of this multidimensional space.
The above theories are based on one-dimensional extended objects corresponding in their duality to the flow of lines of the gauge quantized field. Their differences are observed in three relationships:
- Strings are considered with the property of moving in a classical format, which is characterized by a fixed choice of metrics and other classical fields. The existence of loops is allowed to be considered at a more fundamental level, where other fields and the classical metric are absent.
- The gauge field in the case of loops is considered in the format of the gauge field of all Lorentz transformations or only some of them. With open strings, such a field will correspond to the Yang-Mills field.
- Loop quantum gravity allows quantization without corresponding assumptions. In fact, since global Lorentz invariance does not represent a symmetry of classical general relativity, it cannot be considered in cases of any exact quantization of this theory.
The founders of the “loop quantum theory of gravity” in the 80s of the 20th century are Lee Smolin, Abey Ashtekar, Ted Jacobson (English) and Carlo Rovelli. According to this theory, space and time consist of discrete parts. These small quantum cells of space are connected to each other in a certain way, so that on small scales of time and length they create a motley, discrete structure of space, and on large scales they smoothly transform into continuous smooth space-time.
Loop gravity and particle physics
One of the advantages of the loop quantum theory of gravity is the naturalness with which it explains the Standard Model of particle physics.
Thus, Bilson-Thompson and co-authors proposed that the theory of loop quantum gravity could reproduce the Standard Model by automatically unifying all four fundamental forces. At the same time, with the help of preons, presented in the form of brads (weavings of fibrous space-time), it was possible to construct a successful model of the first generation of fundamental fermions (quarks and leptons) with a more or less correct reproduction of their charges and parities.
The original Bilson-Thompson paper suggested that second- and third-generation fundamental fermions could be represented as more complex brads, and first-generation fermions as the simplest possible brads, although no specific representations of complex brads were given. It is believed that the electric and color charges, as well as the parity of particles belonging to generations of a higher rank, should be obtained in exactly the same way as for particles of the first generation. The use of quantum computing methods has made it possible to show that particles of this kind are stable and do not decay under the influence of quantum fluctuations.
Ribbon structures in the Bilson-Thompson model are represented as entities consisting of the same matter as spacetime itself. Although the Bilson-Thompson papers show how fermions and bosons can be obtained from these structures, the question of how the Higgs boson could be obtained using brading is not discussed in them.
L. Freidel ( L. Freidel), J. Kowalski-Glickman ( J. Kowalski-Glikman) and A. Starodubtsev in their 2006 article suggested that elementary particles can be represented using Wilson lines of a gravitational field, implying that the properties of particles (their mass, energy and spin) can correspond to the properties of Wilson loops - the basic objects of the theory of loop quantum gravity . This work can be seen as further theoretical support for the Bilson–Thompson preon model.
Using the model formalism spin foam, which is directly related to the theory of loop quantum gravity, and based only on the initial principles of the latter, it is also possible to reproduce some other particles of the Standard Model, such as photons, gluons and gravitons - regardless of the Bradson-Thompson scheme for fermions. However, as of 2006, it has not yet been possible to construct helon models using this formalism. The helon model does not contain brads that could be used to construct the Higgs boson, but in principle this model does not deny the possibility of the existence of this boson in the form of some kind of composite system. Bilson-Thompson notes that since particles with larger masses generally have a more complex internal structure (including the twisting of brads), this structure may be related to the mechanism of mass formation. For example, in the Bilson-Thompson model, the structure of a photon having zero mass corresponds to untwisted brads. True, it remains unclear whether the photon model obtained within the spin foam formalism corresponds to the Bilson-Thompson photon, which in his model consists of three untwisted ribbons (it is possible that several versions of the photon model can be constructed within the spin foam formalism).
Initially, the concept of “preon” was used to designate point subparticles included in the structure of half-spin fermions (leptons and quarks). As already mentioned, the use of point particles leads to the mass paradox. In the Bilson-Thompson model, ribbons are not “classical” point structures. Bilson-Thompson uses the term "preon" to maintain continuity in terminology, but with this term refers to a broader class of objects that are components of the structure of quarks, leptons and gauge bosons.
Important to understanding the Bilson-Thompson approach is that in its preon model, elementary particles such as the electron are described in terms of wave functions. The sum of quantum states of a spin foam having coherent phases is also described in terms of a wave function. It is therefore possible that using the spin foam formalism one can obtain wave functions corresponding to elementary particles (photons and electrons). Currently, combining the theory of elementary particles with the theory of loop quantum gravity is a very active area of research.
In October 2006, Bilson-Thompson modified his paper, noting that although his model was inspired by preon models, it is not preon in the strict sense of the word, so topological diagrams from his preon model are likely to be used in others fundamental theories, such as, for example, M-theory. Theoretical restrictions imposed on preon models are not applicable to his model, since in it the properties of elementary particles arise not from the properties of subparticles, but from the connections of these subparticles with each other (brads). One possibility is, for example, “embedding” preons into M-theory or into the theory of loop quantum gravity.
Sabine Hossenfelder proposed considering two alternative candidates for a “theory of everything” - string theory and loop quantum gravity - as sides of the same coin. To ensure that loop quantum gravity does not contradict the special theory of relativity, it is necessary to introduce interactions that are similar to those considered in string theory. .
Problems of theory
In a modified version of his paper, Bilson-Thompson admits that unresolved problems in his model remain the particle mass spectrum, spins, Cabibbo mixing, and the need to link his model to more fundamental theories.
A later version of the article describes the dynamics of brads using Pachner moves.
see also
Sources
Literature
Notes
- Smolin L. Atoms of space and time // In the world of science. - 2004. - No. 4. - P. 18-25. - URL: http://www.chronos.msu.ru/RREPORTS/smolin_atomy/smolin_atomy.htm Archived copy dated February 23, 2009 on the Wayback Machine
- , With. 219.
- S. Yu. Alexandrov Lorentz-covariant loop quantum gravity // TMF. - 2004. - t. 139, No. 3. - p. 363–380. - URL:
Loop quantum gravity theory
What happened before the Big Bang and where did time come from?
In the theory of quantum gravity, the smooth and continuous space we are accustomed to on ultra-small scales turns out to be a structure with a very complex geometry(image from www.aei.mpg.de)
The questions in the title are usually not discussed by physicists, since there is no generally accepted theory capable of answering them. However, recently, within the framework of loop quantum gravity, it was still possible to trace the evolution of a simplified model of the Universe back in time, right up to the moment of the Big Bang, and even look beyond it. Along the way, it became clear exactly how time arises in this model.
Observations of the Universe show that even on the largest scales it is not at all stationary, but evolves over time. If, on the basis of modern theories, we trace this evolution back in time, it turns out that the currently observable part of the Universe was previously hotter and more compact than now, and it began with the Big Bang - a certain process of the emergence of the Universe from a singularity: a special situation for which modern the laws of physics do not apply.
Physicists are not satisfied with this state of affairs: they want to understand and the process itself Big Bang. That is why numerous attempts are now being made to construct a theory that would be applicable to this situation. Since gravity was the most important force in the first moments after the Big Bang, it is believed that achieving this goal is possible only within the framework of an as yet undeveloped quantum theory of gravity.
At one time, physicists hoped that quantum gravity would be described using superstring theory, but the recent crisis in superstring theories has shaken this confidence. In this situation, other approaches to the description of quantum-gravitational phenomena began to attract more attention, and in particular, loop quantum gravity.
It is within the framework of loop quantum gravity that a very impressive result was recently obtained. It turns out that due to quantum effects the initial singularity disappears. The Big Bang ceases to be a special point, and it is possible not only to trace its course, but also to look into what happened before the Big Bang. A summary of these results was recently published in A. Ashtekar, T. Pawlowski, P. Singh, Physical Review Letters, 96, 141301 (12 April 2006), also available as gr-qc/0602086, and their detailed derivation is reported in the other day a preprint by the same authors gr-qc/0604013.
Loop quantum gravity is fundamentally different from conventional physical theories and even from superstring theory. The objects of superstring theory, for example, are various strings and multidimensional membranes, which, however, fly in pre-cooked for them space and time. The question of how exactly this multidimensional space-time arose cannot be resolved in such a theory.
In the loop theory of gravity, the main objects are small quantum cells of space, connected to each other in a certain way. The law of their connection and their state is controlled by a certain field that exists in them. The value of this field is for these cells a certain “ internal time": the transition from a weak field to a stronger field looks exactly as if there was some kind of “past” that would influence some kind of “future”. This law is designed in such a way that for a sufficiently large universe with a low concentration of energy and (that is, far from the singularity), the cells seem to “fuse” with each other, forming the “solid” space-time that is familiar to us.
The authors of the article claim that all this is already enough to solve the problem of what happens to the Universe as it approaches the singularity. The solutions to the equations they obtained showed that with extreme “compression” of the universe, space “scatters”, quantum geometry does not allow its volume to be reduced to zero, a stop inevitably occurs and expansion begins again. This sequence of states can be tracked both forward and backward in “time,” which means that in this theory, before the Big Bang, there is inevitably a “Big Bang” - the collapse of the “previous” universe. Moreover, the properties of this previous universe are not lost in the process of collapse, but are unambiguously transferred to our Universe.
The calculations described are, however, based on some simplifying assumptions about the properties of the universal field. Apparently, the general conclusions will hold even without such assumptions, but this still needs to be verified. It will be extremely interesting to follow the further development of these ideas.
Atoms of space and time
© Lee Smolin
"In the world of science", April 2004
Lee Smolin
If the amazing theory of loop quantum gravity is correct, then space and time, which we perceive as continuous, are actually made of discrete particles.
Since ancient times, some philosophers and scientists have suggested that matter might be made of tiny atoms, but until 200 years ago few believed their existence could be proven. Today we observe individual atoms and study the particles that make them up. The granular structure of matter is no longer news to us.
In recent decades, physicists and mathematicians have been asking the question: is space made up of discrete parts? Is it really continuous or is it more like a piece of fabric woven from individual fibers? If we could observe extremely small objects, would we see atoms of space, indivisible tiny particles of volume? But what about time: do changes in nature occur smoothly or does the world develop in tiny leaps, acting like a computer?
Over the past 16 years, scientists have gotten noticeably closer to answering these questions. According to a theory with the strange name “loop quantum gravity,” space and time really consist of discrete parts. Calculations performed within this concept paint a simple and beautiful picture that helps us explain the mysterious phenomena associated with black holes and the Big Bang. But the main advantage of the mentioned theory is that in the near future its predictions can be verified experimentally: we will detect the atoms of space if they really exist.
Quanta
Together with my colleagues, we developed the theory of loop quantum gravity (LQG), trying to develop the long-awaited quantum theory of gravity. To explain the extreme importance of the latter and its relation to the discreteness of space and time, I must talk a little about quantum theory and the theory of gravity.
The emergence of quantum mechanics in the first quarter of the 20th century. was associated with the proof that matter consists of atoms. Quantum equations require that certain quantities, such as the energies of an atom, can only take on certain discrete values. Quantum mechanics precisely describes the properties and behavior of atoms, elementary particles and the forces that bind them. The most successful quantum theory in the history of science underlies our understanding of chemistry, atomic and subatomic physics, electronics, and even biology.
During the same decades as quantum mechanics, Albert Einstein developed the general theory of relativity, which is a theory of gravity. According to it, the force of gravity arises as a result of the bending of space and time (which together form space-time) under the influence of matter. 
Imagine a heavy ball placed on a rubber sheet and a small ball rolling around near the big one. The balls can be considered as the Sun and the Earth, and the leaf as space. The heavy ball creates a depression in the rubber sheet, along the slope of which the smaller ball rolls towards the larger one, as if some force - gravity - is pulling it in this direction. In the same way, any matter or a bunch of energy distorts the geometry of space-time, attracting particles and light rays; This phenomenon is what we call gravity.
Separately, quantum mechanics and Einstein's general theory of relativity have been experimentally verified. However, a case in which both theories could be tested simultaneously has never been explored. The fact is that quantum effects are noticeable only on small scales, and in order for the effects of general relativity to become noticeable, large masses are required. It is possible to combine both conditions only under some extraordinary circumstances.
In addition to the lack of experimental data, there is a huge conceptual problem: Einstein’s general theory of relativity is completely classical, i.e. non-quantum. To ensure the logical integrity of physics, a quantum theory of gravity is needed, combining quantum mechanics with the general theory of relativity into a quantum theory of space-time.
Physicists have developed many mathematical procedures to transform classical theory into quantum theory. Many scientists tried in vain to apply them to the general theory of relativity.
Calculations carried out in the 1960s and 1970s indicated that quantum mechanics and general relativity could not be combined. It seemed that the situation could only be saved by the introduction of completely new postulates, additional particles, fields or objects of a different kind. The exoticism of a unified theory should manifest itself only in those exceptional cases when both quantum mechanical and gravitational effects become significant. In attempts to achieve a compromise, such directions as the theory of twistors, non-commutative geometry and supergravity were born.
String theory is very popular among physicists, according to which, in addition to the three well-known spatial dimensions, there are six or seven more that no one has yet been able to notice. String theory also predicts the existence of many new elementary particles and forces that have never been confirmed by observation. Some scientists believe that it is part of the so-called M-theory, but, unfortunately, no precise definition has yet been proposed. Therefore, many experts are convinced that available alternatives should be explored. Our loop quantum theory of gravity is the most developed of these.
Big loophole
In the mid-1980s. We, along with Abhay Ashtekar, Ted Jacobson, and Carlo Rovelli, decided to once again try to unify quantum mechanics and general relativity using standard methods. The fact is that there was an important loophole in the negative results obtained in the 1970s: the calculations assumed that the geometry of space was continuous and smooth, no matter how much detail we examined it. People viewed matter in exactly the same way before the discovery of atoms.
So, we decided to abandon the concept of smooth continuous space and not introduce any hypotheses other than the well-tested experimental provisions of the general theory of relativity and quantum mechanics. In particular, our calculations were based on two key principles of Einstein's theory.
The first of them - independence from the environment - proclaims that the geometry of space-time is not fixed, but is a changing, dynamic quantity. To determine the geometry, it is necessary to solve a series of equations that take into account the influence of matter and energy and. By the way, modern string theory is not independent of the environment: the equations describing strings are formulated in a certain classical (i.e., non-quantum) space-time.
The second principle, called “diffeomorphic invariance,” states that we are free to choose any coordinate system to display space-time and construct equations. A point in space-time is defined only by the events physically occurring in it, and not by its position in some special coordinate system (there are no special coordinates). Diffeomorphic invariance is an extremely important fundamental position of the general theory of relativity.
By carefully combining both principles with standard methods of quantum mechanics, we developed a mathematical language that allowed us to carry out the necessary calculations to determine whether space is discrete or continuous. To our delight, the calculations showed that space is quantized! This is how we laid the foundation for the theory of loop quantum gravity. By the way, the term "loop" was coined because some calculations involved small loops isolated in spacetime.
Many physicists and mathematicians have verified our calculations using various methods. Over the past years, the theory of loop quantum gravity has grown stronger thanks to the efforts of scientists from around the world. The work done allows us to trust the picture of space-time that I will describe below.
Our quantum theory is about the structure of spacetime on the smallest scales, and to understand it we need to look at its predictions for a small area or volume. When dealing with quantum physics, it is important to determine which physical quantities need to be measured. Imagine a certain region designated by boundary B (see figure below), which can be defined by a material object (for example, a cast iron shell) or directly by the geometry of space-time (for example, the event horizon in the case of a black hole). What happens when we measure the volume of the described area? What are the possible results allowed by both quantum theory and diffeomorphic invariance? If the geometry of space is continuous, then the region in question can have any size, and its volume can be expressed by any real positive number, in particular, arbitrarily close to zero. But if the geometry is granular, then the measurement result can only belong to a discrete set of numbers and cannot be less than some minimum possible volume. Let's remember what energy an electron orbiting an atomic nucleus can have? Within the framework of classical physics - any, but quantum mechanics allows only certain, strictly fixed discrete values of energy and. The difference is the same as between measuring the volume of a liquid forming a continuous flow (from the point of view of scientists of the 18th century), and determining the quantity of water whose atoms can be counted.
According to the theory of loop quantum gravity, space is like atoms: the numbers obtained by measuring the volume form a discrete set, i.e. the volume changes in individual portions. Another quantity that can be measured is the area of the boundary B, which also turns out to be discrete. In other words, space is not continuous and consists of certain quantum units of area and volume.
Possible values of volume and area are measured in units derived from the Planck length, which is related to the force of gravity, the magnitude of quanta and the speed of light. Planck length is very small: 10 -33 cm; it determines the scale at which the geometry of space can no longer be considered continuous. The smallest possible non-zero area is approximately equal to the square of the Planck length, or 10 -66 cm 2 . The smallest possible volume other than zero is a cube of Planck length or 10 -99 cm 3 . Thus, according to theory, each cubic centimeter of space contains approximately 10 99 atoms of volume. The volume quantum is so small that there are more such quanta in a cubic centimeter than cubic centimeters in the visible Universe (10 85).
Spin networks
What do volume and area quanta look like? Perhaps space consists of a huge number of tiny cubes or spheres? No, it's not that simple. We depict the quantum states of volume and area in the form of diagrams, which are not without their own beauty. Imagine an area of space shaped like a cube (see figure below Steps and foam
Particles and fields are not the only moving objects. According to the general theory of relativity, when matter and energy move, space is modified; waves can even pass through it, like ripples on a lake. In the theory of loop quantum gravity, such processes are represented by discrete transformations of the spin network, in which the connectivity of the graphs changes step by step (see the figure below).
When describing quantum mechanical phenomena, physicists calculate the probability of various processes. We do the same thing when we use the theory of loop quantum gravity to describe the change in the geometry of space or the movement of particles and fields in a spin network. Thomas Thiemann from the Institute for Theoretical Physics in Waterloo has derived precise expressions for calculating the quantum probability of spin network steps. As a result, a clear procedure has emerged for calculating the probability of any process that can occur in a world governed by the rules of our now fully formed theory. All that remains is to calculate and make predictions about what can be observed in certain experiments.
In the theory of relativity, space and time are inseparable and represent a single space-time. When the concept of spacetime is introduced into the theory of loop quantum gravity, the spin networks representing space turn into so-called spin foam. With the addition of another dimension - time - the lines of the spin network expand and become two-dimensional surfaces, and the nodes stretch into lines. Transitions where the spin network changes (steps described above) are now represented by nodes where foam lines converge. The view of spacetime as a spin foam has been proposed by several researchers, including Carlo Rovelli, Mike Reisenberger, John Barrett, Louis Crane, John Baez ) and Fotini Markopoulou.
A snapshot of what is happening is like a cross-section of space-time. A similar slice of spin foam represents a spin network. However, do not be mistaken that the slice plane moves continuously like a smooth flow of time. Just as space is defined by the discrete geometry of a spin network, time is defined by a sequence of individual steps that rearrange the network (see figure on page 55). Thus, time is also discrete. Time does not flow like a river, but ticks like a clock. The interval between “ticks” is approximately equal to Planck time, or 10 -43 s. More precisely, time in our Universe is measured by myriads of clocks: where a quantum step occurs in the spin foam, the clock makes one “tick”.
Predictions and tests
The theory of loop quantum gravity describes space and time on the Planck scale, which is too small for us. So how do we test it? First, it is very important to find out whether classical general relativity can be derived as an approximation to loop quantum gravity. In other words, if spin networks are like the threads from which a fabric is woven, then the question is whether it will be possible to correctly calculate the elastic properties of a piece of material by averaging over thousands of threads. Do we get a description of the “smooth fabric” of classical Einstein space if we average the spin network over many Planck lengths? Recently, scientists have successfully solved this complex problem for several special cases, so to speak, for certain material configurations. For example, low-frequency gravitational waves propagating in flat (uncurved) space can be considered as excitation of certain quantum states described in accordance with the theory of loop quantum gravity.
A good test for loop quantum gravity turned out to be one of the long-standing mysteries about the thermodynamics of black holes, and in particular about their entropy. Physicists have developed a thermodynamic model of a black hole, relying on a hybrid theory in which matter is treated quantum mechanically, but space-time is not. In particular, in the 1970s. Jacob D. Bekenstein deduced that the entropy of a black hole is proportional to its surface area (see the article “Information in the Holographic Universe,” “In the World of Science,” No. 11, 2003). Stephen Hawking soon came to the conclusion that black holes, especially small ones, should emit radiation.
To perform similar calculations within the framework of the theory of loop quantum gravity, we take the boundary of region B to be the event horizon of the black hole. By analyzing the entropy of the corresponding quantum states, we obtain exactly Bekenstein's prediction. With the same success, our theory not only reproduces Hawking's prediction about the radiation of a black hole, but also allows us to describe its fine structure. If a microscopic black hole is ever observed, theoretical predictions could be tested by studying its emission spectrum.
Generally speaking, any experimental verification of the theory of loop quantum gravity is fraught with enormous technical difficulties. The characteristic effects described by the theory become significant only on the Planck length scale, which is 16 orders of magnitude smaller than what can be studied in the near future at the most powerful accelerators (studying smaller scales requires higher energy).
However, scientists have recently proposed several accessible ways to test loop quantum gravity. The wavelength of light propagating in a medium undergoes distortion, which leads to refraction and dispersion of rays. Similar metamorphoses occur with light and particles moving through a discrete space described by a spin network.
Unfortunately, the magnitude of the effects mentioned is proportional to the ratio of the Planck length to the wavelength. For visible light it does not exceed 10 -28, and for cosmic rays with the highest energy it is about one billionth. In other words, the granularity of the structure of space has an extremely weak effect on almost any observable radiation. But the greater the distance the light travels, the more noticeable the consequences of the discreteness of the spin network. Modern equipment allows us to detect the radiation of gamma-ray bursts located billions of light years away (see the article “The Brightest Explosions in the Universe,” “In the World of Science,” No. 4, 2003).
Using the theory of loop quantum gravity, Rodolfo Gambini and Jorge Pullin found that photons of different energies should travel at slightly different speeds and reach the observer at different times (see figure below). Satellite observations of gamma-ray bursts will help us test this. The accuracy of modern instruments is 1,000 times lower than necessary, but already in 2006 the GLAST satellite observatory will be launched, the precision equipment of which will allow the long-awaited experiment to be carried out.
Is there a contradiction here with the theory of relativity, which postulates the constancy of the speed of light? Together with Giovanni Amelino-Camelia and Joao Magueijo, we developed modified versions of Einstein's theory that allow for the existence of high-energy photons traveling at different speeds. In turn, constancy of speed applies to low-energy photons, i.e. to long-wave light.
Another possible manifestation of the discreteness of space-time is associated with cosmic rays of very high energy and. More than 30 years ago, scientists established that cosmic ray protons with an energy of more than 3 * 10 19 eV should be scattered by the cosmic microwave background filling space, and therefore will never reach the Earth. However, the Japanese AGASA experiment recorded more than 10 events with cosmic rays of even higher energy and. It turned out that the discreteness of space increases the energy required for the dispersion reaction and allows high-energy protons to visit our planet. If the observations of Japanese scientists are confirmed, and no other explanation is found, then we can assume that the discreteness of space has been experimentally confirmed.
Space
The theory of loop quantum gravity forces us to take a new look at the origin of the Universe and helps us imagine what happened immediately after the Big Bang. In accordance with the general theory of relativity, there was the very first, zero moment of time in the history of the universe, which is not consistent with quantum physics. Calculations by Martin Bojowald based on the loop theory of quantum gravity indicate that the Big Bang was actually a Big Bounce, since the Universe was rapidly collapsing before it. Theorists are already working on new models of the early stages of the development of the Universe, which will soon be tested in cosmological observations. It is possible that you and I will be lucky enough to find out what happened before the Big Bang.
No less serious is the question of the cosmological constant: is the energy density u permeating “empty” space positive or negative? Observations of the cosmic microwave background and distant supernovae indicate that dark energy exists. Moreover, it is positive because the Universe is expanding at an accelerating rate. From the point of view of the theory of loop quantum gravity, there is no contradiction here: back in 1990, Hideo Kodama compiled equations that accurately describe the quantum state of the Universe with a positive cosmological constant.
A number of issues, including purely technical ones, have not yet been resolved. What adjustments should be made to the special theory of relativity at extremely high energies (if any)? Will the theory of loop quantum gravity help prove that various forces, including gravity, are aspects of a single fundamental force?
Perhaps loop quantum gravity is truly a quantum general theory of relativity, because it is based on no additional assumptions other than the basic principles of quantum mechanics and Einstein's theory. The conclusion about the discreteness of space-time described by the spin foam follows directly from the theory itself, and is not introduced as a postulate.
However, all I have discussed here is theory. Perhaps space is actually smooth and continuous on any scale, no matter how small. Then physicists will have to introduce additional radical postulates, as in the case of string theory. And since the experiment will ultimately decide everything, I have good news - the situation may become clearer in the near future.
Additional literature:
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THE MAIN CONCLUSION of the theory of loop quantum gravity relates to volumes and areas. Let us consider the region of space limited by the spherical shell B (see above). According to classical (non-quantum) physics, its volume can be expressed by any real positive number. However, according to the theory of loop quantum gravity, there is a nonzero absolute smallest volume (approximately equal to the cube of the Planck length, i.e. 10 99 cm 3), and the values of larger volumes are a discrete series of numbers. Likewise, there is a non-zero minimum area (roughly the square of the Planck length, or 10 66 cm 2 ) and a discrete range of larger allowable areas. The discrete spectra of admissible quantum areas (left) and quantum volumes (center) are broadly similar to the discrete quantum energy levels of the hydrogen atom (right).
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DIAGRAMS CALLED SPIN NETWORKS are used to represent quantum states of space at a minimum length scale. For example, a cube(a) is a volume surrounded by six square faces. The corresponding spin network (b) contains a point (node) representing the volume and six lines representing the edges. The number next to the node indicates the volume, and the number next to the line indicates the area of the corresponding face. In the case under consideration, the volume is equal to eight cubic Planck units, and each of the faces has an area of four square Planck units. (The rules of loop quantum gravity limit the permissible values of volumes and areas to certain values: only certain combinations of numbers can be located at lines and at nodes.)
If a pyramid (c) is placed on the top face of a cube, then the line representing that face in the spin network must connect the cube node to the pyramid node (d). The lines corresponding to the four free faces of the pyramid and the five free faces of the cube should extend from the corresponding nodes. (To simplify the diagram, numbers have been omitted.)
In general, in a spin network, one area quanta is depicted by one line (e), and an area composed of many quanta is denoted by many lines (f). Similarly, one volume quantum is represented by one node (g), while a larger volume contains many nodes (h). Thus, the volume inside a spherical shell is given by the sum of all nodes contained in it, and the surface area is equal to the sum of all lines passing through the boundary of the region.
Spin networks are more fundamental than constructions of polyhedra: any combination of polyhedra can be represented by a corresponding diagram, but some regular spin networks represent combinations of volumes and areas that cannot be constructed from polyhedra. Such spin networks arise when space is curved by strong gravitational fields or quantum fluctuations of geometry at Planck scales.
A CHANGE in the FORM of space when matter and energy moves in it and when gravitational waves pass through it is depicted by discrete rearrangements, steps of the spin network. In Fig. and a connected group of three volume quanta merges into one; The reverse process is also possible. In Fig. b two volumes share space and are connected to adjacent volumes in a different way. When depicted as polyhedra, two polyhedra are united along their common face and then split, as when crystals are split along another plane. Such steps in the spin network occur not only with large changes in the geometry of space, but also with continuous quantum fluctuations on the Planck scale.
Another way to represent steps is to add another dimension to the diagram - time. The result is spin foam (c). The lines of the spin network become planes, and the nodes turn into lines. A slice of spin foam at a certain point in time represents a spin network. Having made a series of such cuts, we will obtain frames of a film telling about the development of the spin network over time (d). But note that evolution, which at first glance seems smooth and continuous, actually occurs in spurts. All spin networks containing an orange line (the first three frames) display exactly the same geometry of space. The length of the lines does not matter - for the geometry, all that matters is how the lines are connected and what number each of them is marked with. This is what determines the relative position and size of the volume and area quanta. So, in Fig. d, during the first three frames the geometry remains constant - 3 volume quanta and 6 area quanta. Then space changes abruptly: 1 volume quantum and 3 area quanta remain, as shown in the last frame. Thus, the time determined by the spin foam does not change continuously, but in a sequence of sudden discrete steps.
And although for clarity such sequences are shown as film frames, it is more correct to consider the evolution of geometry as the discrete tapping of a clock. With one “tick” there is an orange quantum of area; the next time, it has disappeared: in fact, its disappearance is what defines the “tick.” The interval between successive "ticks" is approximately equal to Planck time (10 -43 s), but time does not exist between them; there can be no “between”, just as there is no water between two neighboring H 2 O molecules.
WHEN a gamma-ray burst occurs BILLIONS of light-years away, the instantaneous explosion produces a gigantic amount of gamma rays. In accordance with the theory of loop quantum gravity, a photon moving along a spin network occupies several lines at each moment of time, i.e. some space (in reality, there are a lot of lines per quantum of light, and not five, as shown in the figure). The discrete nature of space causes gamma rays to be higher energy and travel slightly faster. The difference is negligible, but during space travel the effect accumulates over billions of years. If gamma rays of different energies generated during the burst arrive on Earth at different times, this is evidence in favor of the theory of loop quantum gravity. The launch of the GLAST satellite is planned for 2006, on board which will be equipped with sufficiently sensitive equipment to detect dispersion gamma radiation.
Ecology of Cognition: “I just think there are too many good things that have happened in string theory for it to be completely wrong. People don't understand it very well, but I just don't believe in the giant cosmic design that created
“I just think there are too many good things that have happened in string theory for it to be completely wrong. People don't understand it very well, but I just don't believe in a gigantic cosmic design that created this incredible thing and that it has nothing to do with the real world,” Edward Whitten once said.
Without a doubt, from a mathematical point of view, there is no shortage of incredible, beautiful and elegant theories. But not all of them are suitable for our physical Universe. It seems that for every brilliant idea that accurately describes what we can observe and measure, there is at least one brilliant idea that tries to describe the same things but remains fundamentally wrong. Last week we asked a question that boils down to something like this.
Quantum gravity. We would like to know if there has been any progress in this area over the past five to ten years. It seems to us ordinary mortals that this field is a little stuck, and string theory has begun to fall into oblivion, since it is difficult to test and has 10^500 possible solutions. Is this true, or is there some progress happening somewhere behind the scenes that the press is simply not paying attention to?
First, it's worth drawing a big line between the idea of quantum gravity, the string theory solution (or proposed solution), and other alternatives.
Let's start with the Universe we know and love. On the one hand, there is the general theory of relativity, our theory of gravity. It argues that instead of being a simple action at a distance, as Newton bequeathed, with all masses in all places exerting forces on each other in inverse proportion to the square of the distance between them, it is based on a more subtle mechanism.
Mass, as Einstein established with the principle of equivalence and E=mc^2 in 1907, was a form of energy in the Universe. This energy, in turn, bends the very fabric of spacetime, changing the path of all objects and changing what the observer could observe as a Cartesian grid. Objects are not accelerated by an invisible force, but rather travel along a path determined by the influence of all the different forms of energy in the Universe.
This is gravity.
On the other hand, we have other laws of nature: quantum ones. There is electromagnetism, which is responsible for electrically charged particles, their movement and which is described by the force carrier photon, which acts as a mediator in these interactions and gives us phenomena that we associate with electrostatics and magnetism. There are also two nuclear forces: the weak nuclear force, which is responsible for phenomena like radioactive decay, and the strong nuclear force, which holds atomic nuclei together and allows protons and neutrons to exist.
Calculations for these forces usually occur in flat spacetime, which is where every student begins their study of quantum field theory. But this is not enough when we are present in curved space, as dictated by general relativity.
“So,” you say, “we’ll just carry out our field theory calculations against the backdrop of curved space!” This is known as semiclassical gravity, and this type of calculation allows us to calculate things like Hawking radiation. But even this is only available on the horizon of the black hole itself, and not where gravity will be in all its glory. There are many physical cases in which we would benefit from a quantum theory of gravity, and they all involve gravitational physics at the smallest scales, at tiny distances.
What, for example, happens in the central regions of black holes? You might think, “Oh, there's a singularity,” but a singularity is not so much a point with infinite density as it is an instance where the mathematical tool of general relativity spits out meaningless answers to questions about potentials and forces. What happens when an electron passes through a double slit? Does the gravitational field pass through both slits? Or after one? The general theory of relativity says nothing about this.
It is believed that there must be a quantum theory of gravity that will explain these and other problems inherent in a “smooth” theory of gravity like general relativity. In order to explain what happens at short distances in the presence of gravitational sources - or masses - we need a quantum, discrete, and therefore particle-based theory of gravity.
Thanks to the properties of general relativity itself, we already know something.
Known quantum forces are determined by the action of particles known as bosons, or particles with integer spin. Photons mediate the electromagnetic force, W and Z bosons mediate the weak nuclear force, and gluons mediate the strong nuclear force. All of these particles have a spin of 1, and for massive particles the spin can be -1, 0 or +1, while for massless particles (like gluons and photons) it can only be -1 or +1.
The Higgs boson is also a boson, but it does not act as an intermediary for forces and has a spin of 0. As far as we know gravity - GTR is a tensor theory of gravity - its mediator should be a massless particle with spin 2, which means its spin can take the value -2 or +2 only.
It turns out that we know something about the quantum theory of gravity even before we try to formulate it. We know this because whatever the quantum theory of gravity is, it must be in accordance with General Relativity when we are dealing with not the smallest distances to massive particles or objects, just as General Relativity must be reduced to Newtonian gravity in the weak field regime.
The big question, of course, is how to do this. How to quantize gravity so that it is correct (in the description of reality), correlated with general relativity and QFT, and leads to computable predictions of new phenomena that can be observed, measured or verified.
The leading contender, as you know, is string theory.
String theory is an interesting field that includes all the standard models of fields and particles, fermions and bosons. It includes a 10-dimensional tensor-scalar theory of gravity: with 9 spatial and 1 time dimensions and a scalar field parameter. If we remove six of these spatial dimensions (through a not fully understood process that people call compactification) and let the parameter (ω), which defines the scalar interaction, go to infinity, we can restore general relativity.
However, string theory has a number of phenomenological problems. One of them is that the theory implies a huge number of new particles, including all supersymmetric ones, which we have not yet discovered. It argues that there is no need for the "free parameters" that the Standard Model has (for particle masses), but replaces this problem with an even worse one. When we talk about 10^500 possible solutions, these solutions concern the expected values of the string fields, and there is no mechanism to reconstruct them; for string theory to work, you'd have to give up the dynamics and just say "it had to be anthropically chosen."
However, string theory is not the only player in this field.
Loop quantum gravity
PKG is an interesting way of looking at the problem: rather than trying to quantize particles, PKG argues that space itself is discrete. How gravity is usually represented: a stretched sheet with a bowling ball in the center. We also know that the sheet is usually quantized, that is, it is made up of molecules, which are made up of atoms, which are made up of nuclei (quarks and gluons) and electrons.
Space can be the same! Since it acts as a fabric, it consists of finite quantized elements. And, perhaps, woven from “loops”, which is where its name comes from. Connect these loops together and you get a network representing the quantum state of the gravitational field. According to this picture, not only matter is quantized, but also space itself. This scientific area is still actively being developed.
Asymptotically safe gravity
Asymptotic freedom was developed in the 1970s to explain the unusual nature of the strong force: it was a very weak force over extremely short distances, which became stronger as the charged particles moved further and further apart. Unlike electromagnetism, which had a small interaction constant, the strong interaction had a large one. Due to some interesting properties of quantum chromodynamics, if you associate with a neutral (colored) system, the strength of the interaction drops off quickly. This could be explained by the physical sizes of baryons (protons and neutrons, for example) and mesons (pions, for example).
Asymptotic freedom, on the other hand, solved the fundamental problem with this: you don't want small interactions, couplings (or couplings that tend to zero), but rather couplings that will simply be finite at the high-energy limit. All coupling constants vary with energy, and asymptotic freedom sets a high-energy fixed point for the constant (technically, for the renormalization group from which the coupling constant is extracted), and everything else can be calculated for low energies.
At least that's the idea. We have figured out how to do this for 1 + 1 dimensions (one spatial and one time), but not for 3 + 1. However, progress is being made, thanks in large part to Christoph Wetterich, who published two monumental works in the 90s. More recently, Wetterich used asymptotic freedom - just six years ago - to calculate a prediction of the mass of the Higgs boson before the LHC found it. The result?
Surprisingly, his predictions coincided perfectly with the LHC findings. This is such an excellent prediction that if asymptotic safety is correct and the masses of the top quark, W boson, and Higgs boson are finally established, physics will not need other fundamental particles to operate consistently up to Planck values.
Although asymptotically safe gravity has not received much attention, it remains a very attractive and promising theory, like string theory: it successfully quantizes gravity, reduces general relativity to the low energy limit, and remains UV-finite. It also beats string theory in one way: it doesn't have a whole mountain of new material that we can't prove yet.
Causal dynamic triangulation
This idea is quite new and was developed in 2000 by Renata Loll in collaboration with other scientists. It agrees with loop quantum gravity in that space is discrete, but is primarily concerned with how that space evolves. One of the interesting properties of this idea is that time must also be discrete. As a result, we get a four-dimensional space-time in the present time, but at very high energies and small distances (on the Planck scale) it appears as a two-dimensional structure. It is based on a mathematical structure called a simplex, which is an n-dimensional generalization of a triangle. A 2-simplex is a triangle, a 3-simplex is a tetrahedron, and so on. One of the "beautiful" features of this comes in the form of causality - a concept known to many - which is preserved in causal dynamic triangulation. It may be able to explain gravity, but it is not 100% clear whether the Standard Model of elementary particles can fit into this framework.
Emerging (induced) gravity
Perhaps the most controversial of the recent theories of quantum gravity is entropic gravity, proposed by Eric Verlinde in 2009, according to which gravity is not a fundamental force, but rather arises as a phenomenon related to entropy. In fact, the roots of emerging gravity go back to the discoverer of the conditions for the formation of matter-antimatter asymmetry, Andrei Sakharov, who proposed this idea back in 1967. The work is still in its infancy, but there has been some progress in the field over the last 5-10 years.
This is what we have today on quantum gravity. We are confident that without it we will not understand the workings of the Universe at a fundamental level, but we have no idea in which of the five (and other) directions presented the movement will be correct. published
